Foundations of Engineering Mechanics

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Soltakhanov, Sh. Kh., Yushkov, M.P.,

, 2009

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Zegzhda, S.A.

Mechanics of non-holonomic systems

Sh. Kh. Soltakhanov · M. P. Yushkov ·

S. A. Zegzhda

Mechanicsof non-holonomicsystemsA New Class of control systems

With 37 Figures

123

Series Editors:

V.I. Babitsky

University Loughborough

Department of Mechanical Engineering

Loughborough LE11 3TU, Leicestershire

United Kingdom

J. Wittenburg

Universitat Karlsruhe

Fakultat Maschinenbau

Institut fur Technische Mechanik

Kaiserstrasse 12

76128 Karlsruhe

Germany

Authors:

Prof. Dr. Shervani Kh. Soltakhanov

Academy of Sciences of the Chechen

Republic

Grozny

Chechen Republic

Russia 364906

Prof. Dr. Sergei A. Zegzhda

St. Petersburg State University

Dept. Mathematics & Mechanics

Universitetsky Pr. 28

St. Petersburg

Russia 198504

Prof. Dr. Mikhail P. Yushkov

St. Petersburg State University

Dept. Mathematics & Mechanics

Universitetsky Pr. 28

St. Petersburg

Russia 198504

ISBN: 978-3-540-85846-1 e-ISBN: 978-3-540-85847-8

Foundations of Engineering Mechanics ISSN: 1612-1384 e-ISSN: 1860-6237

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Reviewers: Dr phys.-math. sci., prof. A. V. Karapetyan (Moscow StateUniversity),Dr phys.-math. sci., prof. V. S. Novoselov (St.Petersburg StateUniversity)

Annotation

A general approach to the generation of equations of motion of holonomicand nonholonomic systems with the constraints of any order is proposed. Thesystem of equations of motion in generalized coordinates is regarded as onevector relation, represented in a space tangential to a manifold of all possiblepositions of system at given instant. The tangential space is partitioned bythe equations of constraints into two orthogonal subspaces. In one of themfor the constraints up to the second order, the law of motion is given by theequations of constraints and in another, for ideal constraints, it is describedby the vector equation not involving reactions of constraints. In the wholespace the law of motion involves the Lagrange multipliers. It is shown thatfor the holonomic and nonholonomic constraints up to the second order,these multipliers can be found as the function of time, positions of system,and its velocities. The application of the Lagrange multipliers to holonomicsystems permits us to construct two new methods for determining the normalfrequencies and normal forms of oscillations of elastic systems and also topropose a special form of equations of motion for system of rigid bodies. Thenonholonomic constraints, the order of which is greater than two, are regardedas program constraints such that their validity is provided by the existence ofgeneralized control forces, which are determined as the functions of time. Theclosed system of differential equations is obtained which makes it possible tofind both these control forces and generalized Lagrangian coordinates. Thetheory suggested is illustrated by examples of spacecraft motion. It is shownthat instead of using the Pontryagin maximum principle it is expedient toapply the generalized Gauss principle for solving the problems of vibrationsuppression (damping).

The book is primarily addressed to specialists in analytical mechanics.

v

Preface to the English edition

The first equations of motion of nonholonomic mechanics not includingthe Lagrange multipliers have been reported at a scientific seminar in 1895and published in 1897 by the world famous specialist in hydromechanics,academician of the Soviet Union Academy of Sciences Sergei AlekseevichChaplygin (1869–1942). One of his favourite pupils, who worked under thedirect supervision of S. A. Chaplygin since 1929 till 1941, was Professor Niko-lai Nikolaevich Polyakhov (1906–1987). In 1952–1987 N. N. Polyakhov wasthe head of the mechanics department of the Faculty of Mathematics andMechanics of Leningrad University and in charge of the chair of theoreticaland applied mechanics, then since 1977 he headed the chair of hydromechan-ics. As well as his teacher, Nikolai Nikolaevich successfully studied not onlyproblems of hydromechanics (he has created, in particular, the mathematicaltheory of a water propeller), but being at the head of the chair of theoreticaland applied mechanics, he also turned to studying nonholonomic mechanics.

N. N. Polyakhov published his first works in this direction in 1970–1974.Since 1975 the investigations under the supervision of N. N. Polyakhov andwith his personal participation had been regularly conducted. They weresummed up in Chapters "Motion with constraints"and "Variational princi-ples in mechanics"of the treatise for universities "Theoretical Mechanics"byN. N. Polyakhov, S. A. Zegzhda, and M. P. Yushkov, which was published in1985 by the Leningrad University Press and reprinted in 2000 by the "VysshayaShkola"("Higher School") Publishing House.

After the decease of N. N. Polyakhov (January 27th 1987), the direction innonholonomic mechanics, which he had established, began to be developedby his pupils: Professors of Saint Petersburg University S. A. Zegzhda andM. P. Yushkov, and Professor of Chechen State University, the head of the de-partment of mathematics and theoretical physics of the Academy of Sciencesof the Chechen Republic Sh. Kh. Soltakhanov, a graduate from Polyakhov’schair. Their collaborative work, to which they had devoted so many years,was completed in 2002 when the monograph "Equations of motion of non-holonomic systems and variational principles in mechanics"was published atSaint Petersburg University. In 2005 the Moscow Publishing House "Nau-ka"("Science") published the second revised and improved edition of thisbook "Equations of motion of nonholonomic systems and variational princi-ples in mechanics. A new class of control problems". This very book, extendedfor the English edition, is offered to readers. This book is dedicated to the100th anniversary of the birth of our teacher Professor N. N. Polyakhov.

vii

viii Preface to the English edition

The prominent specialist in nonholonomic mechanics, Professor J. G.Papastavridis (Professor of Georgia Institute of Technology, USA) has un-dertaken the work of Editor-in-Chief while editing the book in English. Itwas translated by Dr in phys.-math. sciences E. A. Gurmuzova. Gratefull ac-knowledgement is made by the authors to all those who helped in preparationof this book — E. L. Belkind, D. N. Gavrilov, D. V. Lutsiv, A. E. Mel’nikov,A. A. Nezderov, E. M. Nosova, G. A. Sinilshchikova, K. K. Tverev, S. V. Zaykov.

The authors are deeply indebted to the "Springer"Publishing House, andthe publisher of the series "Foundations of Engineering Mechanics"ProfessorV. Babitsky and the editor Dr Ch. Baumann, for their valuable advice thatcontributed greatly to improving this book. It is due to their recommenda-tions that the contents of the book has been considerably extended. Theauthors are also grateful to Ms C. Wolf and Ms V. Jessie who contributedmuch to preparing the book for the press.

While writing the book, the authors gave much attention to the role of theLagrange multipliers in analytical mechanics. Holonomic and classical non-holonomic mechanics are presented in the framework of one approach, in thiscase the properties of constrained motion which are typical of one particle(mass point) can be observed in any mechanical systems with the finite num-ber of degrees of freedom. Such an approach makes it possible to constructalso mechanics for the motion of systems with any-order constraints, whichare considered as programming ones. Reaction forces of these constraints areinterpreted as control forces that provide the motion of system under realiza-tion of the program given as an additional system of differential equations,the order of which is higher than two. Thus, a new class of control problemsis introduced. The offered theory is illustrated through solving two examplesof motion of real mechanical systems with three-order constraints imposedon their motion.

In the monograph much attention is given to studying practical problems.Along with solving a number of classical problems (for instance, problemsdealing with investigation of a car motion with possible slipping of drivingwheels and sideslip), a number of new methods for solving important practicalproblems is proposed in the book. The reader is asked to pay special attentionto the two of them (see Chapter VI).

The first method makes it possible to find natural frequencies and naturalmodes of vibration of an elastic body system in terms of the known naturalfrequencies and natural modes of the system’s separate elastic bodies (itsparts). Due to the dynamical consideration of the finite number of naturalmodes and quasistatical consideration of the higher ones of the system’s el-ements, the lower frequencies can be determined with a high accuracy froman algebraic equation.

The second method is concerned with the problems of damping the vi-bration of mechanical systems. Instead of the commonly used method thatis based on the minimization of the functional of control force squared, it isoffered to apply the generalized Gauss principle stated in the monograph for

Preface to the English edition ix

solving similar problems. As a result, the control force can be constructed asa polynomial in time. During the given time this force transfers smoothly asystem from one state to another, in particular, to the state of rest.

Especially important for the authours is the fact that their work is pub-lished in the English language, which plays today in scientific communicationthe same role as Latin in the Middle Ages. In this regard this monograph canbe useful for English scientists. It will help them to get acquainted with arather great number of works by Russian scientists.

Saint Petersburg — Grozny — Berlin, 2008

Sh. Kh. Soltakhanov, M.P. Yushkov, S.A. Zegzhda

Table of Contents

Chapter I Holonomic Systems 1

1. Equations of motion for the representation point of holonomicmechanical system . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Lagrange’s equations of the first and second kinds . . . . . . . . 43. The D’Alembert–Lagrange principle . . . . . . . . . . . . . . . . 124. Longitudinal accelerated motion of a car as an example of motion

of a holonomic system with a nonretaining constraint . . . . . 15

Chapter II Nonholonomic Systems 25

1. Nonholonomic constraint reaction . . . . . . . . . . . . . . . . . 252. Equations of motion of nonholonomic systems. Maggi’s

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283. The generation of the most usual forms of equations of motion of

nonholonomic systems from Maggi’s equations . . . . . . . . . 384. The examples of applications of different kinds equations

of nonholonomic mechanics . . . . . . . . . . . . . . . . . . . 455. The Suslov–Jourdain principle . . . . . . . . . . . . . . . . . . . 666. The definitions of virtual displacements by Chetaev . . . . . . . 74

Chapter III Linear Transformation of Forces 77

1. Some general remarks . . . . . . . . . . . . . . . . . . . . . . . . 772. Theorem on the forces providing the satisfaction of holonomic

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833. An example of the application of theorem on the forces providing

the satisfaction of holonomic constraints . . . . . . . . . . . . 884. Chetaev’s postulates and the theorem on the forces providing the

satisfaction of nonholonomic constraints . . . . . . . . . . . . 925. An example of the application of theorem on forces providing the

satisfaction of nonholonomic constraints . . . . . . . . . . . . 976. Linear transformation of forces and Gaussian principle . . . . . 100

Chapter IV Application of a Tangent Space to the Study

of Constrained Motion 105

1. The partition of tangent space into two subspaces by equationsof constraints. Ideality of constraints . . . . . . . . . . . . . . 105

2. The connection of differential variational principlesof mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xi

xii Table of Contents

3. Geometric interpretation of linear and nonlinear nonholonomicconstraints. Generalized Gaussian principle . . . . . . . . . . 113

4. The representation of equations of motion following from gener-alized Gaussian principle in Maggi’s form . . . . . . . . . . . 119

5. The representation of equations of motion following from gener-alized Gaussian principle in Appell’s form . . . . . . . . . . . 121

Chapter V The Mixed Problem of Dynamics. New Class

of Control Problems 125

1. The generalized problem of P. L. Chebyshev. A new classof control problems . . . . . . . . . . . . . . . . . . . . . . . . 125

2. A generation of a closed system of differential equations in gen-eralized coordinates and the generalized control forces . . . . 128

3. The mixed problem of dynamics and Gaussian principle . . . . . 1314. The motion of spacecraft with modulo constant acceleration

in Earth’s gravitational field . . . . . . . . . . . . . . . . . . . 1375. The satellite maneuver alternative to the Homann elliptic

motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Chapter VI Application of the Lagrange Multipliers

to the Construction of Three New Methods for the Study

of Mechanical Systems 149

1. Some remarks on the Lagrange multipliers . . . . . . . . . . . . 1502. Generalized Lagrangian coordinates of elastic body . . . . . . . 1523. The application of Lagrange’s equations of the first kind to the

study of normal oscillations of mechanical systems with dis-tributed parameters . . . . . . . . . . . . . . . . . . . . . . . 154

4. Lateral vibration of a beam with immovable supports . . . . . . 1605. The application of Lagrange’s equations of the first kind to the

determination of normal frequencies and oscillation modes ofsystem of bars . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6. Transformation of the frequency equation to a dimensionless formand determination of minimal number of parameters governinga natural frequency spectrum of the system . . . . . . . . . . 173

7. A special form of equations of the dynamics of system of rigidbodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

8. The application of special form of equations of dynamics to thestudy of certain problems of robotics . . . . . . . . . . . . . . 181

9. Application of generalized Gaussian principle to the problem ofsuppression of mechanical systems oscillations . . . . . . . . . 183

Chapter VII Equations of Motion in Quasicoordinates 193

1. The equivalence of different forms of equations of motionof nonholonomic systems . . . . . . . . . . . . . . . . . . . . . 193

Table of Contents xiii

2. The Poincare–Chetaev–Rumyantsev approach to the generationof equations of motion of nonholonomic systems . . . . . . . . 201

3. The approach of J. Papastavridis to the generation of equationsof motion of nonholonomic systems . . . . . . . . . . . . . . . 207

Appendix A The Method of Curvilinear Coordinates 213

1. The curvilinear coordinates of point. Reciprocal bases . . . . . . 2132. The relation between a reciprocal basis and gradients of scalar

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2153. Covariant and contravariant components of vector . . . . . . . . 2164. Covariant and contravariant components of velocity vector . . . 2175. Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . 2186. Covariant and contravariant components of acceleration vector.

The Lagrange operator . . . . . . . . . . . . . . . . . . . . . . 2207. The case of cylindrical system of coordinates . . . . . . . . . . . 2228. Covariant components of acceleration vector for nonstationary

basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2259. Covariant components of a derivative of vector . . . . . . . . . . 227

Appendix B Stability and Bifurcation of Steady Motions

of Nonholonomic Systems 229

Appendix C The Construction of Approximate Solutions

for Equations of Nonlinear Oscillations with the Usage

of the Gauss Principle 235

Appendix D The Motion of Nonholonomic System

without Reactions of Nonholonomic Constraints 239

1. Existence conditions for “free (unconstrained) motion” ofnonholonomic system . . . . . . . . . . . . . . . . . . . . . . . 239

2. Free motion of the Chaplygin sledge . . . . . . . . . . . . . . . . 2403. The possibility of free motion of nonholonomic system under

active forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Appendix E The Turning Movement of a Car as a Nonholonomic

Problem with Nonretaining Constraints 245

1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 2452. The turning movement of a car with retaining (bilateral)

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2463. The turning movement of a rear-drive car with nonretaining

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2494. Equations of motion of a turning front-drive car with non-retaining

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555. Calculation of motion of a certain car . . . . . . . . . . . . . . . 2586. Reasonable choice of quasivelocities . . . . . . . . . . . . . . . . 260

xiv Table of Contents

Appendix F Consideration of Reaction Forces of Holonomic

Constraints as Generalized Coordinates in Approximate

Determination of Lower Frequencies of Elastic Systems 263

Appendix G The Duffing Equation and Strange Attractor 281

References 287

Index 327

Dedicated to the 100th birthday of our teacher

Prof. Nikolai Nikolaevich Polyakhov

Introduction

This treatise is the English translation of the second edition of the book"Equations of motion of nonholonomic systems and the variational principlesof mechanics" , published by St. Petersburg University Press in 2002. Thesecond edition is considerably improved. First of all, particular attention hasbeen given to the mixed problem of dynamics (for detail, see below), whichformulates a new class of control problems in the case when the program ofmotion is given by an additional system of high-order differential equations.The theory, suggested for the solutions of such problems, is illustrated bya number of new examples. Both editions are devoted to the developmentof ideas of nonholonomic mechanics, considered before in the treatise foruniversities: "Theoretical mechanics" (Chapters "Motion with constraints"and "Variational principles of mechanics") of N. N. Polyakhov, S. A. Zegzhda,and M. P. Yushkov, which has been published by Leningrad University Pressin 1985 and reprinted by Publishing house "Higher school" in 2000 [189].The first works [185] in this direction are published by prof. N. N. Polyakhovin 1970–1974. Beginning from 1975 at the chair of theoretical and appliedmechanics of mathematical and mechanical faculty of Leningrad State Uni-versity these problems have been studied under the conduction and directparticipation of N. N. Polyakhov down to the decease of Nikolai Nikolaevichin year 1987.

In the present treatise a new definition of ideal constraint, extended to ahigh-order nonholonomic constraint, is given. Finally, the theory of genera-tion of equations of motion for a certain new class of problems is constructed.Following academician S. S. Grigoryan we shall call such problems the mixed

problems of dynamics since they have the features of both the direct andinverse problems of dynamics. Really, on the one hand the motion of me-chanical system, described by the generalized coordinates q = (q1, . . . , qs),is determined by the given generalized active forces Q = (Q1, . . . , Qs) onthe other hand it is necessary that these generalized coordinates are also thesolution of the following additional system of differential equations

fκ

n(t, q, q, . . . ,

(n)q ) = 0 , κ = 1, k , k s , (1)

where n is any integer number. The characteristics, of motion of mechanicalsystem, for the validity of which it is necessary to find the additional forcesΛ = (Λ1, . . . ,Λk), are given by equations (1).

xv

xvi Introduction

For completing the work the substantial role played the discussion inMoscow State University in the science seminar conducted by academicianV. V. Rumyantsev and prof. A. V. Karapetyan. If conditions (1) are satisfied,then, in fact, a certain control problem is solved in which the program ofmotion is given by system of differential equations (1). Following the ter-minology, accepted in the nonholonomic mechanics, these equations can becalled n-order nonholonomic constraints. However, in essence, as is pointedout above, they are a program of motion and therefore it is more rational tocall them program constraints.

The possibility of application of the generalized Gauss principle to investi-gation of problems on vibration suppression is shown, this principle is statedin the monograph and is valid for non-holonomic high-order constraints. Itturned out that solving similar problems with the help of the Pontryagin max-imum principle, which is commonly used in such cases, can be interpreted assolving a mixed problem of dynamics. The monograph gives a comparision ofthese two principles through solving the problem of vibration suppression ofmathematical pendulums that are attached to a moving trolley. The discus-sion of the suggested technique for solving problems of vibration suppressionat the Institute of problems of mechanics of RAS (Russian Academy of Sci-enes), supervised by academician F. L. Chernous’ko, has played a decisive rolein the final version of this technique.

The book consists of the introduction, the survey of main stages of devel-opment of nonholonomic mechanics, seven chapters, seven appendices, andreferences.

The survey of main stages of development of nonholonomic mechanicsgives a short description of main directions of study in nonholonomic me-chanics.

In the first chapter a notion of the point that represents a motion of aholonomic mechanical system is introduced. To generate Lagrange’s equa-tions of the first and second kinds, the approach demonstrating their unityand generality is applied. This approach permits us to write Lagrange’s equa-tions in the form, which can be used in the case of both one material pointand arbitrary mechanical system with finite or infinite numbers of degrees offreedom. The notion of ideal holonomic constraints is considered from the dif-ferent points of view. The connection of the obtained equations of motion withthe D’Alembert–Lagrange principle is analyzed. The longitudinal motion ofa car with acceleration and possible sideslip of driving wheels is considered asan example of motion of a holonomic system with a nonretaining constraint.

In the second chapter from the analog of Newton’s law the Maggi’s equa-tions, which are the most convenient equations of the nonholonomic mechan-ics, are deduced. From Maggi’s equations the most useful forms of equationsof motion of nonholonomic systems are obtained. The connection of Maggi’sequations with the Suslov–Jourdain principle is considered. The notion ofideal nonholonomic constraints is discussed. In studying nonholonomic sys-tems the approach, applied in Chapter I to analysis of motion of holonomic

Introduction xvii

systems, is employed. The role of Chetaev’s type constraints for the develop-ment of nonholonomic mechanics is considered. For the solution of a numberof nonholonomic problems, the different methods are applied.

In the third chapter the linear transformation of forces is introduced. Inthis case for holonomic systems the notion of ideal constraints and the rela-tion for virtual elementary work are used. By the transformation of forces,Lagrange’s equations of the first and second kinds are obtained. The theoremof holonomic mechanics is formulated according to which the given motionover the given curvilinear coordinate can be provided by means of an addi-tional generalized force corresponding to this coordinate. For nonholonomicsystems the linear transformation of forces is introduced by using Chetaev’spostulates. In this case with the help of generalized forces, corresponding tothe equations of constraints, the family of basic equations of nonholonomicmechanics is obtained in compact form. The theorem of holonomic mechan-ics is formulated according to which the given change of quasivelocity canbe provided by one additional force corresponding to this quasivelocity. Theapplication of the formulated theorems of the holonomic and nonholonomicmechanics is demonstrated on the example of the solution of two problemson the controllable motion connected with the flight dynamics. At the endof this chapter the linear transformation of forces is used to obtain Gauss’principle.

In the fourth chapter by means of the introduction of a tangent space, asystem of Lagrange’s equations of the second kind is represented in the vectorform. It is shown that the tangential space is partitioned by the equations ofconstraint into the direct sum of two subspaces. In one of them the componentof a vector of acceleration of system is uniquely determined by equations ofconstraints. The notion of ideality of holonomic constraints and nonholonomicconstraints of the first and second orders is analyzed. This notion is extendedto high-order constraints. The relationship and equivalence of the differentialvariational principles of mechanics are considered. It is given a geometricinterpretation of the ideality of constraints. The generalized Gauss principleis formulated. By means of this principle, for nonholonomic systems withthird-order constraints equations in Maggi’s and Appell’s forms are obtained.

In the fifth chapter the law of motion of mechanical system, represented inthe vector form, is applied to the solution of the mixed problem of dynamics.The essence of the problem is to find additional generalized forces such thatthe program constraints, given in the form of a system of differential equationsof order n 3, are satisfied. The notion of a generalized control force isintroduced. It is proved that if the number of program constraints is equalto the number of generalized control forces, then the latter can be foundas the time functions from the system of differential equations with respectto generalized coordinates and these forces. The conditions, under which thissystem of equations has a unique solution, are determined. The conditions arealso obtained under which for the constraints of any order the motion controlis realized according to Gauss’ principle. Thus, the theory is created with the

xviii Introduction

help of which a new class of control problems can be solved. This theory isused to consider two problems, connected with the dynamics of spacecraftmotion. In the first problem a radial control force, providing the motion ofspacecraft with a modulo constant acceleration, is determined as the timefunction. In the second problem we seek the law, of varying in time the radialand tangential control forces, by which a smooth passage of spacecraft fromone circular orbit to another occurs.

In the sixth chapter the Lagrange multipliers are used to construct threenew methods for the study of mechanical systems. The first of them cor-responds to the problem of determining the normal frequencies and normalforms of oscillations of elastic system, consisting of the elements with knownnormal frequencies and normal forms. In this method the conditions of con-nection of elastic bodies to one another are regarded as holonomic constraints.Their reactions equal to the Lagrange multipliers are the forces of interactionbetween the bodies of system. Using the equations of constraints, the systemof linear uniform equations with respect to the amplitudes of the Lagrangemultipliers for normal oscillations is obtained. By the solution of this systemthe normal frequencies and normal forms of complete system are expressed interms of the normal frequencies and normal forms of its elements. An approx-imate algorithm for determining the normal frequencies and normal forms,based on a quasistatic account of higher forms of its elements, is proposed.A development of this method is given in Appendix F.

The second method suggested is connected with the study of the dynamicsof system of rigid bodies. In this case the Lagrange multipliers are introducedfor the abstract constraints taking into account that the number of introducedcoordinates, by which the kinetic energy of rigid body has a simple form, isexcessive. In this case the elimination of the Lagrange multipliers leads to anew special form of equations of motion of rigid body. This form is utilizedto describe a motion of a dynamic stand, which lets us to imitate the stateof pilot in cabin in extremal situations.

The third method is applied to a problem of the vibration suppression(damping) of mechanical systems. It is based on reducing the problem given toa mixed problem of dynamics, which is solved with the help of the generalizedGauss principle.

In the seventh chapter it is shown that all known types of equations ofmotion of nonholonomic systems are equivalent since they can be obtainedfrom the invariant vector form of the law of motion of mechanical systemwith ideal constraints. The nonholonomicity of constraints, which does notallow the equations of motion to be represented in the form of Lagrange’sequations of the second kind, appears most clearly if the equations of motionof nonholonomic system are written in independent quasicoordinates. In thecase of linear constraints these equations are generated here by three dif-ferent methods. This allows us to consider the problem of nonholonomicityfrom three different points of view. In this chapter the vector form of equa-tions of dynamics, found in Chapter IV, is used to generate the equations

Introduction xix

in quasicoordinates and the equations of Poincare–Chetaev–Rumyantsev. Ageometric interpretation of the equations of Poincare–Chetaev–Rumyantsevis given. The approach of Poincare–Chetaev–Rumyantsev to the generationof equations of motion of nonholonomic systems is compared with otherapproaches.

In Appendix A the kinematics of point in curvilinear coordinates is con-sidered. The formulas obtained are extended to the motion of any mechanicalsystem. The main material of treatise is considered in the frame of the the-ory developed in the Appendix. Therefore, generally speaking, it should berecommended to read this book beginning from this Appendix.

Appendix B contains a short review of the works, devoted to the questionsof the existence, stability, and branching of steady motion of conservative non-holonomic systems. This appendix is the plenary report of A. V. Karapetyanwith the same title, which was read at the International science conferenceon mechanics "The third Polyakhov readings" (St.Petersburg, February 4-6,2003).

In Appendix C Gauss’ principle in integral form is applied to the construc-tion of approximate solutions of equations of nonlinear oscillations, in partic-ular, of the solutions, which are obtained by the Bubnov–Galerkin method.

In Appendix D the motion of nonholonomic systems in the case whenthe reactions of constraints are absent is investigated. By the Mei Fengxiangterminology such a motion is called a free (unconstrained) motion of non-holonomic system. The free motion of the Chaplygin sledge is considered.The realizing of free motion of nonholonomic systems under external forcesis discussed.

In Appendix E the possibillity of sideslip of both front and rear wheelsof a car while turning is considered. Solving this problem with nonretainingconstraints leads to necessary consideration of four possible types of the carmotion. The computational results of motion of a certain car are presented.

In Appendix F the problem of determining the lower frequencies of amechanical system that consists of elastic bodies connected to each otherby holonomic constraints is considered. It is shown that reaction forces ofthese constraints can be regarded as the generalized Lagrange coordinates.In the method suggested the equation for determining the lower frequenciesis not transcendental but an algebraic one. A number of examples illustratethat this approximate method makes it possible to define the first naturalfrequency to a high accuracy.

Appendix G is devoted to studying the possible arising of strange attrac-tors for lateral vibration of a beam with fixed supports.

The list of names in the text is the same as in the treatise published inthe "Nauka" Publishing House. The order of references is also the same. Thereferences are complemented at the end with some new items, a part of newworks is added to old items.

The theory considered is illustrated by many examples. The computa-tions, connected with the solution of problems, are due to S. V. Almazova,

xx Introduction

O. V. Almazov, Yu. A. Belousov, E. S. Bolgar, A. B. Byachkov, D. N. Gavrilov,I. N. Drozd, E. S. Drozd, T. N. Dudareva, E. Yu. Leont’eva, Lee Yang, A. E.Mel’nikov, A. A. Nezderov, Yu. L. Nikiforova, E. M. Nosova, T. N. Pogrebskaya,G. A. Sinilshchikova, A. V. Smal, N. S. Smirnova, V. P. Sysik, K.K. Tverev,A. A. Fedorov, L. G. Fedorchenko, N. G. Filippov, N. A. Khor’kova, Chen Yu,Yu. S. Sheverdin, A. E. Shevtsov, A. V. Shkondin, M. A. Yushkevich. The au-thors are grateful to all of them for their help.

As it was shown in the preface to this edition, following the advice ofthe publisher of the series "Foundations of Engineering Mehanics" of the"Springer" Publishing House Professor V. Babitsky and the editor of thePublishing House Dr Ch. Baumann the volume of the treatise has been in-creased. The work at the new text was carried out with the assistance ofS. V. Almazova (§§ 2, 6 of Chapter VI and Appendix F), A. B. Byachkov (§ 4of Chapter I and Appendix E), D. N. Gavrilov (Appendix F), Lee Yang (Ap-pendix D), A. E. Mel’nikov (§ 4 of Chapter VI and Appendix G), A. A. Nezderov(§ 4 of Chapter I and Appendix E), E. M. Nosova (§ 4 of Chapter I and Ap-pendix E), G. A. Sinilshchikova (Appendix F), Yu. A. Belousov and Chen Yu(§ 2 of Chapter II), M. A. Yushkevich (§ 4 of Chapter VI). This new text (ex-cluding §§ 2, 4 of Chapter VI and Appendices D and G) was translated byT. V. Zhil’tsova. Besides, the Preface to the English edition, § 4 of Chapter I,§ 9 of Chapter VI, Appendices E, F, and G were translated by Dr G. A.Sinilshchikova.

All new ideas of the book have undergone the careful check and kindlycritique of the scientific editor of the book, head of Chair of theoretical and ap-plied mechanics of the Mathematics and Mechanics Faculty of St.PetersburgState University, laureate of State Prize RF, honored worker of science RF,Prof. P. E. Tovstik. The authors wish to acknowledge their great indebtednessto Petr Evgen’evich Tovstik for his able assistance during the whole periodof constructing the suggested theory and writing this book.

The authors will be very grateful to readers for sending their commentson this book.

E-mail: Mikhail.Yushkov@MJ16561.spb.edu; Soltakhanov@yandex.ru

Survey of the main stages of development

of nonholonomic mechanics

The theory of motion of nonholonomic systems is of obvious scientificinterest. Already in the investigations of I. Newton, L. Euler, I. Bernoulli,J. Bernoulli, J. D’Alembert, and J. Lagrange in studying the problems on therolling of rigid bodies without slide we find elements with distinctive fea-tures of motion of systems with nonholonomic constraints. For the solutionof similar problems, S. Poisson [374. 1833] makes use of the general theo-rems of dynamics. In the book [379. 1884] E. Routh considers the problemon the rolling of rigid body without sliding on a fixed surface and reduces itto quadratures in many sophisticated cases, for example, in the case of therolling of rigid uniform ball on a cylindrical surface with cycloidal section.The motion of rolling bodies is discussed by P. Appell [266. 1899]. The inter-esting problem on the rolling without sliding of a ball with a gyroscope insideis considered by D. K. Bobylev [16. 1892]. In the case when the center of massof complete system is at the center of ball he solves this problem, havingexpressed all sought unknowns via elliptical functions. N. E. Zhukovsky [67.1893] shows that if in a spherical shell there is an additional ring and the mo-ments of inertia are chosen in a suitable way, then the study of this problemis simplified. He also gives a geometrically descriptive consideration.

All these problems were solved correctly by different methods and manyauthors. However at the end of the 19th century and early in the 20th theattempts to solve the typical nonholonomic problems, applying the methodsof holonomic mechanics, lead to many well-known errors, which played asubstantial role in the making of nonholonomic mechanics. So, in 1885 and1886 to generate equations of motion of heavy body rolling without slidingon a fixed plane, C. Neumann [366] applies usual Lagrange’s equations ofthe second kind. However he understands presently that for the solution ofsimilar problems it is necessary to use more complicated Lagrange’s equationswith multipliers [366. 1887–1888]. In 1899 C. Neumann has solved totally thisproblem [366].

A more specific problem is solved by E. Lindelof [352. 1895]. In this prob-lem he considered a body that is bounded by a surface of revolution and hasa center of inertia on the axis of revolution being a dynamic axis of symmetryof body. The forces are assumed to be conservative, in which case the forcefunction depends on the coordinates of a contact point of body only. Follow-ing the treatise of S. Poisson [374], E. Lindelof suggests to consider in placeof the general theorems of dynamics the Hamilton principle or Lagrange’sequations of the second kind, obtained from this principle. Having writtentwo equations of nonholonomic constraints, he applies them to the construc-tion of kinetic energy and assumes erroneously that the nonholonomicityof this problem is completely accounted and therefore Lagrange’s equationsof the second kind can be generated. Naturally, the system of differential

xxi

xxii Survey of the main stages of development

equations obtained turns out a more simple than the truth-value system andwas solved therefore in quadratures.

The works of E. Crescini [296. 1889] and G. Schouten [382. 1899] containalso similar errors. For a nonholonomic system E. Crescini uses erroneouslythe Hamilton–Jacobi equations and G. Schouten the Lagrange’s equations ofthe second kind. P. Molenbrock [364. 1890] and a number of other scholarsalso ignore a differential nature of nonholonomic constraints. Even one of thefuture creator of nonholonomic mechanics, L. Boltzmann, admits a similarerror [276. 1885]. He applies Lagrange’s equations to the study of a revolutionof frictional and toothed gear wheels with a nonholonomic constraint, whichgives a proportional dependence of angular rates of wheels. This error wascorrected by L. Boltzmann in 1902 [277] only.

P. Appell is interested in the elegant outwardly but untrue in essencesolution of E. Lindelof to an extent that he uses it as an example of applyingLagrange’s equations of the second kind in § 452 of the first edition of histreatise on the theoretical mechanics [265. 1896]. But in the second edition in1898, based on the investigations of J. Hadamard [311. 1894] and A. Vierkandt[398. 1892], he writes: ". . . the results of E. Lindelof are inaccurate. I havepointed out this error to E. Lindelof in 1898 and have made a correction inthe following edition of my "Traite" ".

The first to remark the substantial error, committed by E. Lindelof, andto inform the author was S. A. Chaplygin. October 25, 1895 he reported theresults on this topic on a meeting of the physics division of the Associationof lovers of natural sciences, anthropology, and ethnography. S. A. Chaplyginnotes that ". . . on the first pages of his work . . . E. Lindelof committed animportant error, by which the equations obtained turned out more simplethan the intrinsic ones and provided a seeming achievement of author". Inthis report S. A. Chaplygin presented first his equations of motion for non-holonomic systems. Two years hence he found the correct solution of theE. Lindelof problem and published the results in the paper [239].

It is of interest to note that, apparently, for the visualization of solution,S. A. Chaplygin generates the equations of motion for the E. Lindelof problem,no using his own equations but applying the principles of the motion of thecenter of mass and of conservation of angular momentum of system, in whichcase he preliminarily introduces a friction force and eliminates it then fromthe equations found. For the sake of generality, S. A. Chaplygin considers abody connected with a gyroscope and reduces the solution to quadratures,which are simplified in the case considered earlier by D. K. Bobylev [16].

After S. A. Chaplygin, the E. Lindelof problem was solved by D. Korteweg[336. 1899], using Routh’s equations with the multipliers of constraints, andby P. Appell [268. 1900] and P. V. Voronets [41. 1903] in the framework ofthe equations they suggest, and by many other scholars. Thus, the workof E. Lindelof [352] gave great impetus to the making and development ofnonholonomic mechanics. We remark that in these first problems it was ratherdifficult to take in to account correctly the fact that the constraints considered

Survey of the main stages of development xxiii

are differential. For example, in the above-mentioned work [336], D. Kortewegdescribes in detail the errors of G. Schouten, E. Lindelof, P. Molenbrock, andP. Appell but at the same time he himself commits a similar error whenhe attempts to create the theory of small oscillations with nonholonomicconstraints.

As an autonomous division of Newtonian mechanics the nonholonom-ic mechanics was formulated in the work of H. Hertz "Die Prinzipien derMechanik in neuem Zusammenhange dargestellt" [317. 1894]. The terms theholonomic and nonholonomic systems were introduced by Hertz. N. Ferrers[306. 1872], and E. Routh [379. 1877] were among the first to suggest thecorrect equations of motion with nonholonomic constraints. N. Ferrers intro-duced the expressions for Cartesian velocities in terms of generalized veloci-ties, and E. Routh wrote the equations with the Lagrange multipliers. In 1877in the third edition of his "Dynamics of a system of rigid bodies" for linearnon-holonomic constraints E. Routh introduces such form, which is usual-ly called now in literature the Lagrange equations of the second kind withmultipliers [59].

S. A. Chaplygin [239. 1895, 1897] was the first to suggest the equations ofmotion without Lagrange multipliers. He introduces the certain conditions,which are to be satisfied by the linear equations of constraints, the forces, andthe relation of kinetic energy (such systems were called then the Chaplyginsystems), and transforms the form of kinetic energy, using the equations ofconstraints. As a result, he discriminates in the left-hand side of equations ofmotion the group of addends of the type of the Lagrange operator. The restaddends characterize the nonholonomicity of system and go to zeros in thecase of integrability of differential equations of constraints. It is to be not-ed that, practically, all the considered problems of nonholonomic mechanicswere of the type of the Chaplygin systems and therefore these equations hada widespread application. In 1901 P. V. Voronets [41] generalizes Chaplygin’sequations to the cases of noncyclic holonomic coordinates and of nonstation-ary constraints.

The work of S. A. Chaplygin acted into notice many outstanding scholarsof the day. The different forms of the equations of motion for nonholonomicsystems without Lagrange multipliers were proposed. These are the equa-tions of V. Volterra [399. 1898], L. Boltzmann [277. 1902], G. Hamel [313.1904] and others. The obtained different types of the equations of motion fornonholonomic systems are generated in the quasicoordinates and have a usualstructure of Lagrange’s equations of the second kind with corrective additiveterms of nonholonomicity. We remark that in parallel with extending Chap-lygin’s equations P. V. Voronets [41] generated also the equations of motionin quasicoordinates. These investigations are represented completely in hisMaster dissertation [41. 1903]. The equations, generated by P. V. Voronets,L. Boltzmann, and G. Hamel, are highly similar in the form to each otherand are obtained almost at the same time. This explains why in the modernscientific literature different authors give them different names.

xxiv Survey of the main stages of development

For the study of the dynamics of nonholonomic systems the other formsof equations, which did not also involve the Lagrange multipliers, were sug-gested. First of all, it is Appell’s equations, which are represented shortlyin the works [267. 1899] and completely in 1900 [269]. In these equationsa notion of the energy of accelerations (the term is suggested by A. Saint-Germain [381. 1900]) is used. It is of interest for us that in the work [268.1900] Appell applies this method to the solution of the E. Lindelof problem.In 1924 J. Tzenoff [393] generates the equations of the mixed type, whichinvolve both the energy of accelerations, and the kinetic energy. J. Schouten[383. 1928] finds later the equations, possessing a contravariant structure.

Just as in the case of the equations of Boltzmann–Voronets–Hamel, theequations of the Appell type were also obtained by certain other scientists. Forexample, J. W. Gibbs [309] presented similar equations already in 1879, theanalogous equations were also obtained apparently independently of Appellby P. Jourdain [325. 1904], the same ideas were presented in the works byH. Hertz [ 317, p. 224, 371].

We remark that the equations of G. Maggi [355. 1896] were obtained,in fact, at the same time as by S. A. Chaplygin but the contemporaries didalmost not notice them. These equations, not involving the Lagrange mul-tipliers, are represented in quasicoordinates and are the linear combinationsof Lagrange’s equations of the second kind. Maggi’s equations are highlyconvenient to solve the problems of nonlinear nonholonomic mechanics [189,286, 327] and to generate the equations of motion for systems of rigid bodies[221]. However, apparently, at present they are insufficiently known. For ex-ample, utilizing the original definition of ideal constraints, V. N. Suchkov [222.1999] generates generalized Lagrange’s equations, which are coincident withMaggi’s equations with an accuracy to multipliers. In 1901 G. Maggi himselfpublished the note [356], in which he has shown that Volterra’s equationsjust as Appell’s equations can be obtained from the equations, suggested byhim already in 1896 in his book on mechanics [355]. In the treatise for univer-sities [189] from Maggi’s equations the main forms of Lagrange’s equationsof motion of nonholonomic systems are obtained. Maggi’s equations and therelations for reactions of nonholonomic constraints are discussed in the workof J. Papastavridis [370].

A new direction in obtaining equations of motion is due to H. Poincare[373. 1901]. V. V. Rumyantsev [203. 1994, p. 3] writes that "the remarkableidea of Poincare [373] that the equations of motion for holonomic mechanicalsystems can be generated by using a certain Lie transitive group of infinitelysmall transformations was extended then by Chetaev [247, 248, 292] to thecase of nonstationary constraints and dependent variables when the trans-formation group is intransitive. Chetaev transforms Poincare’s equations tothe form of canonical equations and develops the theory of integration ofthese equations". The Poincare–Chetaev theory is extended to the nonholo-nomic linear systems in the works of L. M. Markhashov, V. V. Rumyantsev,and Fam Guen [149, 203, 229]. In 1998 V. V. Rumyantsev [203] extends the

Survey of the main stages of development xxv

Poincare–Chetaev equations to nonlinear nonholonomic constraints. There-fore these equations might be called the equations of Poincare–Chetaev–

Rumyantsev. As V. V. Rumyantsev [203] remarks, these equations are generalequations of nonholonomic mechanics and the rest of types of equations ofmotion can be deduced from them. In the work [81. 2001] a geometric inter-pretation of the Poincare–Chetaev–Rumyantsev equations is given.

In parallel with the obtaining of different forms of equations of motionthe variational principles, used in the nonholonomic mechanics, were de-veloped (a detail survey, with a comprehensive bibliography, of variationalprinciples of mechanics can be found in the works of V. N. Shchelkachev andJ. Papastavridis [254, 370]). In 1894 H. Hertz in his outstanding "Die Prinzip-ien der Mechanik in neuem Zusammenhange dargestellt" [317] showed thatthe Hamilton principle in its classical formulation cannot be applied to non-holonomic systems. In the introduction to the mentioned book he explainsthis on the example of a ball, rolling mechanically without sliding. The ele-gant proof of the same fact was also given by H. Poincare [372. 1897].

The Hamilton–Ostrogradsky principle is extended first to the stationarynonholonomic systems by O. Holder [318. 1896]. A. Voss [400. 1900] extendsthis result, using curvilinear coordinates, to the case of nonstationary con-straints. Almost at the same time as by Voss the similar investigations werealso made by P. V. Voronets [41. 1901] and G. K. Suslov [219. 1901] and, whatis of interest, all these works are published in the same number of journal. Weremark that the Hamilton–Ostrogradsky principle was also extended to thecase of nonholonomic system with two free parameters by S. A. Chaplygin.The possibility of applying the integral variational principles of mechan-ics to the study of motion of nonholonomic systems is discussed in theworks of V. S. Novoselov, V. V. Rumyantsev, A. S. Sumbatov and others [171,200, 217].

When obtaining the equations of motion for nonholonomic systems, manyscholars made use of the D’Alembert–Lagrange principle but in this caseit was necessary to complete a definition of the notion of virtual displace-ments with nonholonomic constraints. For this purpose P. Appell [265] andJ. W. Gibbs [309] introduce the virtual displacements by the rules, accord-ing to which the virtual displacements coincide really with virtual velocities,what is quite natural. However P. Jourdain [326. 1908–1909] connects the cor-responding principle of nonholonomic mechanics exactly with the notion ofvirtual velocities. Note that G. K. Suslov [218. 1900] has formulated practical-ly the same principle but with somewhat modified terminology. In this con-nection the variational differential principle for nonholonomic systems mightbe called the Suslov–Jourdain principle [187]. E. Delassus [298. 1913] suggeststo call it the analytic form of the generalized D’Alembert principle. The in-vestigations, devoted to applying the Suslov–Jourdain principle, are still inprogress (see, for example, the work [285. 1993]).

In the case of applying in nonholonomic mechanics the principles ofD’Alembert–Lagrange, Jourdain, and Gauss together we need to investigate

xxvi Survey of the main stages of development

the connection between the differential variational principles of mechanics.Already early in the 20th century this question was considered (for exam-ple, in the work of R. Leitinger [343. 1913]) but a widespread study of thisproblem was begun in the work of N. G. Chetaev [245. 1932–1933] and wascompleted by the research of V. V. Rumyantsev [199. 1975–1976]. Much at-tention is being given to this direction up to day [124. 2004, 288. 1989, 387.1995].

N. G. Chetaev [245] introduces the important notion of nonholonomic me-chanics, namely the virtual displacements of system with the nonlinear non-holonomic constraints (the constraints of Chetaev’s type). A. Przeborski [375.1931–1932] introduces a similar axiom of ideality of nonholonomic constraintswhile extending Maggi’s equations to the case of nonlinear nonholonomic con-straints. Much attention to the discussion of these conditions is also given byL. Johnsen [324]. In justice to P. Appell, V. S. Novoselov calls such conditionsthe Appell–Chetaev conditions and introduces "A-displacements" term [173]for the corresponding virtual displacements. J. Papastavridis [370. 1998, 1999,2002] calls these conditions the definition of Maurer–Appell–Chetaev–Hamelfor virtual displacements with nonlinear nonholonomic constraints. Theseconditions are the main apparatus of investigation in the nonholonomic me-chanics (see, for example, the works of V. S. Novoselov [169, 170] and therecent works [348, 365]; in the works [349. 1994] the connection of the modelof Chetaev with the model of Vacco is established).

Particular attention of scholars has been given to the problems, posed bythe classics of nonholonomic mechanics. For example, the motion of heavybody of revolution in the statement of S. A. Chaplygin [239] was studied byA. S. Sumbatov [217] and A. P. Kharlamov [235] (for other similar investiga-tions, see below), the development of the theorem on reducing multiplier[242] can be found in many papers and in the treatise of Yu. I. Neimarkand N. A. Fufaev [166], the modification of Gauss’ principle, suggested byN. G. Chetaev [246] (Chetaev’s principle), was extended by V. V. Rumyantsev[198, 199], and so on. Much attention was also given to the construction ofthe new forms of equations of motion for nonholonomic systems and to theextension of the known types of equations to a more wide class of constraints:A. Przeborski [375] extended Maggi’s equations to the case of nonlinear non-holonomic constraints, V. S. Novoselov [169] suggested the equations of Chap-lygin’s type and the equations of the type of Voronets–Hamel, which allow theHamel equations with nonlinear constraints [313, 314] to be applied to nonsta-tionary and nonconservative systems, J. Papastavridis [370. 1995] extendedthe domain of applying the Boltzmann–Hamel equations, the different formsof equations were suggested by J. Nielsen [367], D. Mangeron and S. Deleanu[360], Bl. Dolaptschiew [301, 302], G. S. Pogosov [182], N. N. Polyakhov [185],M. F. Shul’gin [255.1950], I. M. Shul’gina [256], and others. In many worksthere are discussed the different forms of equations of motion (for example,[301, 302, 310, 341]). The convenient matrix form of equations of nonholo-nomic mechanics is given by Yu. G. Martynenko [146. 2000]. Ya. V. Tatarinov

Survey of the main stages of development xxvii

suggested the new form of equations of nonholonomic (and holonomic) me-chanics [225]. This form incorporates the known representations of equationsof motion, in which case many of addends is obtained by means of the formalPoisson bracket. F. Udwadia and R. Kalaba [394. 1992] obtained the equa-tions taking into account the presence of constraints that are linear withrespect to generalized accelrations.

The Kane equations [330] made themselves conspicuous especially in thewestern literature. Their geometric interpretation can be found in the workby M. Lesser [345]. By means of these equations a number of problems of non-holonomic mechanics was solved. Many studies [280, 300, 345, 363, 384, 408]showed a direct relationship between Kane’s equations, Maggi’s equations,and the Gibbs–Appell ones.

In 1906 J. Quanjel [377] obtained a canonical form of equations of mo-tion for nonholonomic systems, using Lagrange’s equations with multipliers.These results were developed by S. Dautheville, L. Johnsen and T. Poschl[324, 377]. A. J. Van der Schaft and B. M. Maschke [396] obtained the equa-tions of motion for nonholonomic systems close in form to Hamilton’s equa-tions. The Jacobi method for two curvilinear coordinates was generalized tononholonomic systems by S. A. Chaplygin [242. 1911]. A canonical form ofthe equations of nonholonomic systems was also obtained by N. N. Polyakhov[185]. The mathematical questions, related to the above problem, are dis-cussed in the foundational work of V. V. Kozlov [112]. One of theories ofthe integration of differential equations of nonholonomic mechanics was sug-gested by I. S. Arzhanykh [5]. In 1939 V. V. Dobronravov [58] extended theHamilton–Jacobi theorem to the case of canonical system of nonholonomicequations. However in the work [162. 1953] Yu. I. Neimark and N. A. Fufaev,having a bearing on their own investigations [161–163], challenge the work ofV. V. Dobronravov [58], regarding that the results obtained are valid for holo-nomic systems only. At the same time they also challenge the accuracy of gen-eration of the V. Volterra equations [399]. We remark that V. V. Dobronravovdid not agree with the said critique [58. 1952]. This discussion emphasizes thefact that the theory of motion of nonholonomic systems is rather complicated.

The works of E. A. Bolotov [18. 1904], G. K. Pozharitskii [183. 1961], V. V.Rumyantsev [196. 1961-2006], V. A. Samsonov [205. 1981-2005], A. S. Sumbatov[217. 1982], and others are devoted to the motion with nonideal constraints.

Special problems of analytic mechanics are the problems with nonretain-ing constraints. The notion of such holonomic constraints was introducedfirst by M. V. Ostrogradskii in generalizing the principle of virtual displace-ments and the D’Alembert principle to the similar systems [176]. The mo-tion with releasing constraints was also considered by G. K. Suslov [220]. Thegeneral case of the motion with nonretaining constraints is studied espe-cially completely by means of the modern mathematical tools in the worksby A. P. Ivanov and A. P. Markeev [86, 87], and is stated in the treatise ofV. F. Zhuravlev and N. A. Fufaev [72]. This book summarizes the numerousresults of different works devoted to this topic (see the bibliography at the

xxviii Survey of the main stages of development

end of the book) and permits us to make use of the powerful theory of analyt-ic mechanics for the study of a wide class of different substantial in practiceproblems: the motion of vibroimpulsive and vibratory displaced systems, therolling of different systems with possible slide and so on. Among similar prob-lems the questions of repeated collisions are completely considered, for exam-ple, in the treatise of R. F. Nagaev [160] and the possibility of lateral sliding ofcar in the book of M. A. Levin and N. A. Fufaev [130], releasing from nonholo-nomic constraints during tne turning movement of wheeled robot vehicles andcars in works [253]. A successful simulation of the motion of nonholonomicsystems with one-sided constraints is given in the work of I. I. Kossenko [122].

The classical problems of nonholonomic mechanics are the problems ona rolling of bodies on rigid surface. After the works of Kh. M. Mushtari[158. 1932] and Yu. P. Bychkov [29. 1965–1966] such problems were activelystudied by A. V. Karapetyan [96], A. S. Kuleshov [127], A. P. Markeev [141–144], V. K. Poida [184], V. V. Rumyantsev [201], V. A. Samsonov [205], Ya. V.Tatarinov [224], V. N. Tkhai [228], N. A. Fufaev [234], A. P. Kharlamov [235],E. I. Kharlamova [237], V. Ya. Yaroshchuk [263], Dong Zhiming, Yang Haix-ing [303, 410], T. Yamamoto [409] and others (for example, by L.D. Akulenkoand D. D. Leshchenko [3]). The recent results and the current status of thisquestion are given in the fundamental treatise of A. P. Markeev [143. 1992].In this book there is a comprehensive bibliography on this topic. A new ap-proach, concerning the interaction of body with a surface, is represented inthe work of V. F. Zhuravlev [70. 1998–1999].

In many problems on a motion of bodies without slide on a fixed sur-faces, particular attention has been given to the integration of a system ofdifferential equations. However the particularly many works, devoted to themathematical questions of the integrability of equations of motion for non-holonomic systems, was published beginning from the end 70s of 20th century.These are the works of A. A. Afonin, A. V. Borisov, A. A. Burov A. P. Veselov,L. E. Veselova, A. V. Karapetyan, A. A. Kilin, V. V. Kozlov, S. N. Kolesnikov,A. S. Kuleshov, I. S. Mamaev, A. P. Markeev, N. K. Moshchuk, Yu.N. Fedorov,V. A. Yaroshchuk and others [11, 20, 22, 26, 27, 37, 38, 112. 1985, 127, 142,155. 1986, 230. 1988, 263, 278, 279, 331]. Among these investigations wecan remark, in turn, the works of V. V. Kozlov [112. 1985] and A. P. Markeev[141. 1983]. Note that in many said papers together with the study of motionof the above-mentioned classical ball of Bobylev–Chaplygin the problems ofG. K. Suslov [220] and L. E. Veselova [39] are also considered. In the works ofA. V. Borisov, A. A. Kilin, I. S. Mamaev [278, 279] a possible hierarchy of thedynamics of the rolling bodies considered is suggested. A sui generis encyclo-pedia of this scientific direction is the treatise [19], which involves both thealready published and specially written works devoted to the study of thedynamics of rolling bodies.

The great difficulties over a long period moment were connected with thestudy of stability of nonholonomic systems. For example, even E. Whittaker[405], repeating the errors of F. Klein and D. Korteweg [336], regarded that the

Survey of the main stages of development xxix

differential equations of small oscillations with holonomic and nonholonomicconstraints have the same form. O. Bottema was one of the first to explaincorrectly in 1949 the influence of nonholonomicity of system on its stability(see also the recent work [281]). A detail consideration of stability of nonholo-nomic systems is given in the works of M. A. Aiserman and F. R. Gantmacher[264], I. S. Astapov [10], R. M. Bulatovich [24], D. V. Zenkov [85], A. A. Zobova[101], A. V. Karapetyan [94–97, 99, 100, 204], T. R. Kane and D. A. Levin-son [331], G. N. Knyazev [108], V. V. Kozlov [113], A. S. Kuleshov [127, 333],A. P. Markeev [143], Yu. I. Neimark and N. A. Fufaev [165, 166], A. N. Ob-morshev [175], M. Pascal [180], C. Risito [378], V. V. Rumyantsev [197, 201,204], L. N. Semenova [207], Lilong Cai [350], A. Nordmark and H. Essen [369],Zhu Haiping and Mei Fengxiang [414], J. Walker [403], P. Hagedorn [312]and others. The works of V. I. Kalenova, A. V. Karapetyan, V. M. Morozov,M. A. Salmina, E. N. Sheveleva [91–93, 423] are devoted to the questions ofstability and stabilization of a steady motion of nonholonomic systems. Itis of highly interest for us the investigations of stability of a revolution ofCeltic rattlebacks. The unusual peculiarity of their revolution was remarkedfirst by G. T. Walker still in 1895 [402]. The detailed survey (with the mainliterature on the topic considered) of modern investigations of a steady mo-tion of nonholonomic systems can be found in the work of A. V. Karapetyanand A. S. Kuleshov, published in the above-mentioned book [19] (see also Ap-pendix B in the present treatise). We remark that, basing on the above worksof I. S. Astapov, A. V. Karapetyan, A. P. Markeev, M. Pascal, on the numericalapproach in the work [351] and using computer calculations, A. V. Borisov,I. S. Mamaev, and A. A. Kilin obtained that in the motion of the Celtic rat-tlebacks the chaos and attractors [19] may occur. A good simulation of Celticrattleback is given in the work of I. I. Kossenko and M. S. Stavrovskaia [123,337]. The motion of a Celtic rattleback with friction and sliding has beenstudied in the works by T. P. Tovstik [427].

The works of V. S. Novoselov [172], V. A. Sapa [206], M. F. Shul’gin andI. M. Shul’gina [257] and many foreign scholars ([308], Luo Shaokai, Mei Fengx-iang, Qiao Yongfang, Zhang Jiefang and others) are devoted to the studyof a motion of nonholonomic systems with variable mass. A new directionin the study of stochastic nonholonomic systems is given in the works ofN. K. Moshchuk and I. N. Sinitsyn [155, 156].

The theory of motion of nonholonomic systems is successfully applied tothe solution of different technical problems, which occur in the theory ofmotion of bicycle and motor cycle (M. Bourlet, M. Boussinesq, E. D. Dikarev,S. B. Dikareva, E. Carvallo, A. M. Letov, I. I. Metelitsyn, V. K.Poida, N. A. Fu-faev [184, 282, 283, 289]), in the theory of motion of car (N. E. Zhukovsky,P. S. Lineikin, L. G. Lobas, Yu. I. Neimark, V. K. Poida, N. A. Fufaev, E. A.Chudakov [68, 72, 132, 133, 166, 184, 251, 253]), in the theory of interactionof wheel with road (V. G. Vil’ke, V. Gozdek, M. I. Esipov, A. Yu. Ishlinskii,M. V. Keldysh, I. V. Novozhilov, P. Rokar, N. A. Fufaev [40, 103, 167]), in dif-ferent machines with variators of speed (I. I. Artobolevskii, I. I. Vul’fson, Ya.

xxx Survey of the main stages of development

L. Geronimus, V. A. Zinov’ev, A. I. Kukhtenko, A. V. Mal’tsev, V. S. Novoselov,B. A. Pronin, I. I. Tartakovskii [8, 47, 128, 170, 261]), in the theory of mo-tion of electromechanical systems (A. V. Gaponov, V. A. Dievskii, O. Enge,G. Kielau, A. Yu. L’vovich, P. Maißer, Yu. G. Martynenko, F. F. Rodyukov,J. Steigenberger [45, 57, 136, 137, 145, 304, 359, 391]), and in many oth-er fields of engineering (for example, the rotor breaking-in on rigid bearing[53]). In recent years the investigations are performed which are devotedto a motion of sportsman on a skateboard and a snakeboard (Yu. G. Ispolov,B. A. Smol’nikov [320. 1996], A. S. Kuleshov [126. 2004-2007], F. Pfeiffer,M. Foery, H. Ulbrich [432. 2006]).

M. A. Levin and N. A. Fufaev have described a complicated nonholonom-ic interaction of tire with road, using a phenomenological model of rollinga deformable wheel [72, 130]. This model permits us to find the force andmoment, which are a result of interacting wheel with road when a car moves.According to the suggested approach the motion of system is described byusual Lagrange’s equations of the second kind. Such an approach was usedby E. V. Abrarova, A. A. Burov, S. Ya. Stepanov, D. P. Chevallier for the gen-eration of equations of motion, which were applied then to the investigationsof stability of a steady motion of complicated motor system, consisting of anarticulated vehicle with hitching [2].

In 1981 in the work [186] it was shown that the acceleration of systemcan be resolved into two orthogonal components, one of which is complete-ly determined by the equations of nonlinear ideal nonholonomic constraints.This result was presented in the treatise for universities [189] in 1985. In1999 Yu. F. Golubev also obtains the resolutions for nonlinear nonholonom-ic constraints [50]. Similar resolutions for linear nonholonomic constraints,were obtained in 1989 by J. Storch and S. Gates, in 1991 by H. Brauchli andV. V. Velichenko, in 1992 by W. Blajer, M. Borri, C. Bottasso, P. Mantegazza,H. Essen [31, 275, 280, 284, 305, 392]. They applied the matrix calculus andobtained the equations, which permit for a motion and reactions of holonomicand nonholonomic constraints to be found for the system of bodies connect-ed to each other. The projective method, proposed by them, is really a suigeneris form, of Maggi’s equations, convenient for computer calculations. Forapplying the methods of computer algebra to the problems of mechanics,the treatise of D. M. Klimov and V. M. Rudenko [107] is especially useful. Inthe work by F. E. Udwadia and R. E. Kalaba [394. 1992] for defining the con-straint reactions represented as linear second-order nonholonomic constraintsthe matrix calculus is used. In so doing, the partition of the whole space bythe constraint equations into two orthogonal subspaces is automatically doneby using the Moore and Penrose generalized inverse, which was already of-fered in 1920 [422]. Basing on their form of the equations of motion, theygive a new version of the Gauss principle.

Similar problems are highly actual in studying the problems of robotics.In this case it is rationally to use the treatises of G. V. Korenev [121], G. F.Moroshkin [154], D. E. Okhotsimskii and Yu. F. Golubev [178], E. P. Popov,

Survey of the main stages of development xxxi

A. F. Vereshchagin and S. L. Zenkevich [191], J. Wittenburg [406], theworks of V. A. Malyshev [139], P. Maißer [357, 358], and others. A neweffective approach to the generation of compact equations of motion for sys-tem of rigid bodies is proposed by V. A. Konoplev in the works [119, 120] andis generalized in the treatise [120. 1996]. The questions of the dynamics andcontrol by mobile wheeled robots are discussed in the works of V. I. Babitsky,A. Shipilov [435], V. N. Belotelov, V. I. Kalyonova, A. V. Karapetyan, A. I. Kob-rin, A. V. Lenskii, Yu. G. Martynenko, V. M. Morozov, D. E. Okhotsimskii,M. A. Salmina [146-148, 423].

The individual question of nonholonomic mechanics is a question on a pos-sible realization of nonholonomic constraints (the investigations of A. V. Kara-petyan, C. Caratheodori, V. V. Kozlov, I. V. Novozhilov, V. V. Kalinin, N. A.Fufaev and others [95, 115, 168, 233, 287]). Already in the early days ofnonholonomic mechanics this question is actively discussed in the works ofP. Appell, E. Delassus and others [270, 271, 298, 299]. The example of Appell–Hamel [270, 271, 315], considered from the point of view of the possibilityof mechanical constructing the nonlinear nonholonomic constraints, was ofspecial interest. Now the scholars also discuss often this example [171, 274,290, 321, 376, 408]. The incorrectness of limit passage, used by P. Appelland G. Hamel, was remarked by Yu. I. Neimark and N. A. Fufaev [164]. In thework [321] it is shown that the limit passage, applying by Appell and Hamel,reduces, in fact, the problem of rolling the disk to the study of a motion ofball. Thus, in the nonholonomic mechanics it is assumed that in the caseof motion of rigid bodies without slide and when there exist knife-edges thelinear nonholonomic constraints can occur only.

The limits of application of the theory of motion of nonholonomic sys-tems are considerably extended by the introduction of servoconstraints dueto A. Beghin and P. Appell [4, 13]. The theory of servoconstraints was active-ly developed by V. I. Kirgetov [105]. The tools of nonholonomic mechanicsturn out yet more useful for the solution of a number of control problems(see, for example, the works of S. Deneva, V. Diamandiev, V. V. Dobronravov,Yu. G. Ispolov, B. A. Smol’nikov, K. Jankowski, E. Jarzebowska, L. Steigenber-ger, Mei Fengxiang, J. Parczewski and W. Blajer, [52, 60, 89, 322, 323, 362,371]). In this case the role of nonholonomic constraints is played by a programof motion and the reaction of such constraints is a control force. The theory ofmotion of systems with program constraints and the investigations of stabilityof computational process with provision for that the equations of constraintsare satisfied approximately can be found in the works of A. S. Galiullin,I. A. Mukhametzyanov, R. G. Mukharlyamov, and V. D. Furasov [43, 157]. Weremark that the program of motion can be given in the form of differentialequations of a higher order than the first. Therefore the theory of nonholo-nomic systems with high-order constraints becomes actual.

The works of Bl. Dolaptschiew, D. Mangeron, S. Deleanu, G. Hamel,J. Nielsen, L. Nordheim, J. Tzenoff [61, 140, 238, 301, 314, 315, 360, 367, 368,393] are devoted to the motion with high-order constraints. At present, this

xxxii Survey of the main stages of development

theory is actively developed, for example, in the works of Yu. A. Gartung,V. V. Dobronravov, Do Sanh, Yu. G. Ispolov, V. I. Kirgetov, B. G. Kuznetsov,M. A. Matsura, Mei Fengxiang, B. N. Fradlin, L. D. Roshchupkin, M. A. Chuev,I. M. Shul’gina, K. Jankowski, F. Kitzka, J. Stawianowski, R. Huston and oth-ers [46, 62, 88, 105, 125, 150, 159, 177, 193, 232, 252, 256, 316, 322, 335, 353,362, 385, 389, 415]. However the numerical realization of similar theories inthe cases when they are applied to some concrete problems is missing andtherefore the validity of these theories is not tested. In this regard, examplesof ideal linear third-order nonholonomic constraints describing the motionof a spacecraft with a constant modulo acceleration and its smooth transferfrom one circular orbit to another one are of particular interest [78. 2005,79, 214].

Early in the 20th century the applying of the tensor methods to themechanics of nonholonomic systems results in the occurrence of a new fieldof geometry, namely the nonholonomic geometry. This field of geometry is de-veloped in the works of V. V. Vagner, G. Vranceanu, A. Wundheiler, Z. Gorak,A. M. Lopshits, P. K. Rashevskii, J. Synge, J. Schouten, and W. Chow [30, 134,194, 208, 223, 295, 383, 401]. The mathematical aspects of nonholonomicmechanics are considered in the works of V. I. Arnol’d, A. M. Vershik,A. P. Veselov, L. E. Veselova, V. Ya. Gershkovich, C. Godbillon, V. V. Kozlov,M. Leon, L. M. Markhashov, A. I. Neishtadt, N. N. Petrov, P. Rodrigues,D. M. Sintsov, S. Smale, L. D. Faddeev, D. P. Chevallier and others [6, 7, 33,35, 38, 48, 63, 149, 181, 209, 293, 344, 349, 354, 386, 397, 413]. Particu-lar significance for their understanding has the treatises of V. I. Arnol’d [6],A. D. Bruno [23], and B. A. Dubrovin, S. P. Novikov, A. T. Fomenko [63],C. Truesdell [418].

Note that the present survey and references do not contain unfortunatelymany highly important works. A more detailed survey of variational principlesof mechanics and equations of motion for nonholonomic systems and alsothe extensive bibliography can be found in the works of Yu. I. Neimark andN. A. Fufaev [166], J. Papastavridis [370. 1998, 2002], B. N. Fradlin [231], andV. N. Shchelkachev [254]. The interesting survey of the methods and problemsof nonholonomic mechanics is given in the work of Mei Fengxiang [362].

Ch a p t e r I

HOLONOMIC SYSTEMS

In this chapter we introduce a notion of the point that represents a motion

of mechanical system. To generate Lagrange’s equations of the first and sec-

ond kinds we make use of the approach demonstrating their unity and gen-

erality. This approach permits us to write Lagrange’s equations in the form,

which can be used both in the case of one material (mass) point and of arbi-

trary mechanical system with finite or infinite numbers of degrees of freedom.

The notion of ideal holonomic constraints is considered from the different

points of view. The connection of the obtained equations of motion with the

D’Alembert–Lagrange principle is analyzed. The longitudinal motion of a car

with acceleration is considered as an example of motion of a holonomic system

with a nonretaining constraint.

§ 1. Equations of motion for the representation point

of holonomic mechanical system

A simple and geometrically descriptive generation of equations of motionfor holonomic mechanical systems is based on applying the notion of repre-sentation point, introduced by H. Hertz. The notion of representation point,in particular, is considered in the works [25, 135, 185]. We give here theirresults.

Consider the motion of N material points with the masses mν , ν =1, N . Their position in three-dimensional space in the Cartesian coordinatesOx1x2x3 can be characterized by the radius-vectors rν = xν1i1+xν2i2+xν3i3,ν = 1, N . If on the motion of system it is imposed the holonomic constraints

fκ(t, x11, x12, x13, . . . , xN1, xN2, xN3) = 0 , κ = 1, k , (1.1)

then the vector equations of motion have the form

mν rν = Fν + R′

ν, ν = 1, N . (1.2)

Here Fν = Xν1i1 + Xν2i2 + Xν3i3 is a resultant of forces, acting on theν-th point, R′

ν= R′

ν1i1 + R′

ν2i2 + R′

ν3i3 is a constraint reaction imposedon the ν-th point. To vector equations (1.2) correspond the following scalardifferential equations

mν xνj = Xνj + R′

νj, ν = 1, N , j = 1, 2, 3 . (1.3)

1

2 I. Holonomic Systems

We use a continuous numbering for the projections of radius-vectors,forces, and reaction of constraints:

xµ = xνj , Xµ = Xνj , R′

µ= R′

νj,

µ = 3(ν − 1) + j , ν = 1, N , j = 1, 2, 3 , µ = 1, 3N .(1.4)

We also assume that

mµ = mν for µ = 3ν − 2, 3ν − 1, 3ν , ν = 1, N . (1.5)

Then equations (1.3) can be rewritten as

mµxµ = Xµ + R′

µ, µ = 1, 3N . (1.6)

Introduce the following notations

M =N∑

ν=1

mν ≡

1

3

3N∑

µ=1

mµ , mµ = mµ/M , yµ =√

mµ xµ ,

Yµ = Xµ/√

mµ , Rµ = R′

µ/√

mµ , µ = 1, 3N .

(1.7)

In this case equations (1.6) become

Myµ = Yµ + Rµ , µ = 1, 3N . (1.8)

We introduce in the 3N -dimensional Euclidean space the unit vectorsj1, . . . , j3N of Cartesian coordinates. Then scalar equations (1.8) correspondto the following vector equation

MW = Y + R , (1.9)

where one uses the 3N -dimensional vectors:

W = V = y , y = yµ jµ , Y = Yµ jµ , R = Rµ jµ , µ = 1, 3N .

Further, in the products a summation in the corresponding limits of re-peating indices is implied. The point of the mass M , the position of which inthe 3N -dimensional space is given by the radius-vector y, is called an repre-

sentation point. For this point equation (1.9) have the form of foundationallaw of mechanics for constrained motion of one point and therefore we shallcall, for short, vector equation (1.9) Newton’s second law of motion

In the 3N -dimensional Euclidean space (l = 3N−k) a set of the equationsof holonomic constraints

fκ(t, y) = 0 , y = (y1, . . . , y3N ) , κ = 1, k , (1.10)

corresponding to original equations (1.1), define an l-dimensional surface onwhich the representation point is at the moment t. Transition formulas (1.4),

1. Equations of motion for the holonomic mechanical system 3

(1.5), (1.7) permit us to define the motion of the representation point in termsof the known motion of system in three-dimensional space, and vice versa, ifthe motion of the representation point in a 3N -dimensional space is known,then the same formulas take this motion to the motion of N material pointsin usual three-dimensional space.

In the case of one point and one constraint, which is given by the equation

f1(t, y) = 0 , y = (y1, y2, y3) , yµ = xµ , µ = 1, 3 ,

the constraint reaction can be represented as

R = Λ1∇∇∇f1 + T0 = N + T0 .

Here T0 is orthogonal to the normal component N. It is important for us thatthe mathematical equation of holonomic constraint gives the direction of thevector N only. At the same time the value and direction of the vector T0

must be given by the additional characteristics of constraint, which dependon its physical realization.

The demonstrative example of the constrained motion of material pointis a spherical pendulum. Obviously, the change of the length of pendulum l

by the given law, i. e. the fact that the constraint

f1(t, y) = y21 + y2

2 + y23 − l2(t) = 0 (1.11)

holds can be provided by the force N, directed along the normal to thesphere, given at the moment of time by equation (1.11). In particular, if theconstraint is physically realized by means of the retraction of a thread, thenthe constraint reaction is the tension N. Thus, for the study of the motionof spherical pendulum it is necessary to assume T0 = 0. The holonomicconstraint, imposed on the point, is said to be ideal in the case T0 = 0.

The example of motion with nonideal constraint is the motion of the pointover a rough surface. In this case for the characteristic of T0 is often usedCoulomb’s law

T0 = −k1|N|

v

|v|, (1.12)

where k1 is a coefficient of friction.In the case of the one material point and two constraints, given by the

following equations

fκ(t, y) = 0 , κ = 1, 2 , yµ = xµ , µ = 1, 3 ,

the reaction R of these two constraints takes the form

R = Λκ∇∇∇fκ + T0 , κ = 1, 2 .

Here T0 is orthogonal to the vectors ∇∇∇fκ , κ = 1, 2. Then the point moves ina line, for example, in the case of stationary constraints in a circle (mathemat-ical pendulum). The force T0 is lacking if the reaction R has no a component,

4 I. Holonomic Systems

directed at a tangent to the line on which the point is situated at this momentof time. Such constraints are called ideal. An additional consideration of holo-nomic constraints will be continue in § 1 of the next chapter.

Using the analogy between a one material point and the point that rep-resents the motion of mechanical system, in the general case we assume that

R = Λκ∇∇∇fκ + T0 , κ = 1, k , (1.13)

and we shall say that constraints (1.10) are ideal if T0 = 0. For ideal con-straints (1.10) Newton’s second law (1.9) for the representation point can bewritten as

MW = Y + Λκ∇∇∇fκ . (1.14)

Projecting this vector equation on the axes of Cartesian coordinates, weobtain Lagrange’s equations of the first kind. If the vector equation is project-ed on the axes of curvilinear coordinates chosen specifically, then we arriveat Lagrange’s equations of the second kind. Thus, equation (1.14) shows theunity of two kinds of Lagrange’s equations. These equations are consideredin more detail in the following division

§ 2. Lagrange’s equations of the first and second kinds

Projecting vector equation (1.14) on the axes of Cartesian coordinates andrecurring from the variables yµ to the variables xµ and from the quantitiesYµ and Rµ to the quantities Xµ and R′

µ, respectively, we obtain Lagrange’s

equations of the first kind

mµxµ = Xµ + Λκ

∂fκ

∂xµ

, µ = 1, 3N , κ = 1, k . (2.1)

Equations (2.1) involve 3N +k unknowns x1, . . . , x3N , Λ1, . . . ,Λk and there-fore they can be considered together with equations of constraints (1.1). Nowwe eliminate the unknowns Λ1, . . . ,Λk from the above system. For this pur-pose we differentiate twice with respect to time the equations of constraints:

d2fκ

dt2≡

∂2fκ

∂t2+ 2

∂2fκ

∂t∂xµ

xµ +∂2fκ

∂xµ∂xµ∗

xµxµ∗ +∂fκ

∂xµ

xµ = 0 ,

κ = 1, k , µ, µ∗ = 1, 3N ,

and substitute the relations for xµ from equations (2.1). Then we have asystem of linear nonuniform algebraic equations in unknowns Λ1, . . . ,Λk. Us-ing the notion of the representation point, we represent the elements of thedeterminant of this system as

gκ∗

κ

∗= ∇∇∇fκ

∗

· ∇∇∇fκ , κ, κ∗ = 1, k .

2. Lagrange’s equations of the first and second kinds 5

Below we shall assume that holonomic constraints (1.1) are such that|gκ

∗

κ

∗| = 0. The validity of this condition permits us to find the quantities Λκ,

κ = 1, k, as the functions of the variables t, xµ, xµ, µ = 1, 3N. Note that thiscondition and the analytic relations for the functions Λκ(t, xµ, xµ), κ = 1, k,

were obtained and studied first by G. K. Suslov [220] and A. M. Lyapunov[138]. Substituting the functions Λκ(t, xµ, xµ), κ = 1, k, into formulas (2.1),we obtain 3N differential equations of the functions x1, . . . , x3N . This sys-tem is convenient for numerical integration since it is solvable for the higherderivatives xµ.

We introduce now for the representation point the curvilinear coordinatesq1, . . . , q3N . Multiplying scalarly equation (1.14) by the vectors of fundamen-tal basis eσ = ∂y/∂qσ, σ = 1, 3N , we obtain

(Myµ − Yµ)∂yµ

∂qσ= Λκ

∂fκ

∂yµ

∂yµ

∂qσ, σ, µ = 1, 3N . (2.2)

We shall regard the quantities qλ, λ = 1, l, l = 3N − k, as free independentcurvilinear coordinates. The rest of coordinates, by assumption, are the fol-lowing ql+κ = fκ(t, y), κ = 1, k. Then for ql+κ = 0, κ = 1, k, equations ofconstraints (1.10) are satisfied. In this case we have

∇∇∇fκ

· eσ =∂fκ

∂yµ

∂yµ

∂qσ=

∂ql+κ

∂yµ

∂yµ

∂qσ= δl+κ

σ=

0, σ = l + κ ,

1, σ = l + κ .

Thus, the vectors el+κ = ∇∇∇fκ , κ = 1, k, are the vectors of a reciprocal basis,which is introduced as

eσ· eτ = δσ

τ, σ, τ = 1, s .

Relations (2.2) can be represented as two systems of equations:

(Myµ − Yµ)∂yµ

∂qλ= 0 , λ = 1, l , (2.3)

(Myµ − Yµ)∂yµ

∂ql+κ

= Λκ , κ = 1, k . (2.4)

Passing in equations (2.3), (2.4) from the variables yµ to the variables xµ andfrom the quantities Yµ to the quantities Xµ, we obtain

(mµxµ − Xµ)∂xµ

∂qλ= 0 , λ = 1, l , (2.5)

(mµxµ − Xµ)∂xµ

∂ql+κ

= Λκ , κ = 1, k . (2.6)

6 I. Holonomic Systems

Taking into account the Lagrange relations

mµxµ

∂xµ

∂qσ=

d

dt

∂T

∂qσ−

∂T

∂qσ, T =

3N∑

µ=1

mµx2µ

2, σ = 1, 3N ,

and the generalized forces

Qσ = Xµ

∂xµ

∂qσ= Fν ·

∂rν

∂qσ= Y · eσ , σ = 1, 3N ,

equations (2.5) and (2.6) can be written as usual Lagrange’s equations of thesecond kind:

d

dt

∂T

∂qλ−

∂T

∂qλ= Qλ , λ = 1, l , (2.7)

d

dt

∂T

∂ql+κ

−

∂T

∂ql+κ

− Ql+κ = Λκ , κ = 1, k . (2.8)

Equations (2.7) are, in essence, the equations of motion. Then for thegiven initial data we can obtain the law of motion of the system

qλ = qλ(t) , λ = 1, l . (2.9)

Representing equations (2.8) in explicit form, assuming that ql+κ = ql+κ =ql+κ = 0, κ = 1, k, and substituting functions (2.9), we get the generalizedreactions Λκ , κ = 1, k, as time functions. If we find qλ = qλ(t, q, q) fromequations (2.7) and substitute these relations in (2.8), then we obtain thefunctions Λκ = Λκ(t, q, q).

We remark that if equation (1.14) is scalarly multiplied by the vectors eσ,σ = 1, s, we obtain

MWσ = Qσ + Λκ δl+κ

σ, σ = 1, s , κ = 1, k .

Then by (2.7) and (2.8) we have

MWσ = MW · eσ =d

dt

∂T

∂qσ−

∂T

∂qσ.

Further we shall often use this representation of covariant components of thevector MW in terms of the Lagrange operator. This implies that

MW = MWσeσ =

(d

dt

∂T

∂qσ−

∂T

∂qσ

)eσ .

The Lagrange operator involves the following quantities

pσ =∂T

∂qσ, σ = 1, s ,

2. Lagrange’s equations of the first and second kinds 7

which are called generalized impulses. We shall show that they can be re-garded as covariant components of the vector

p = MV = pσeσ ,

where V = dy/dt is a velocity of representation point. Really, we have

T =

3N∑

µ=1

mµ

x2µ

2=

3N∑

µ=1

My2µ

2=

=MV2

2=

M

2

(∂y

∂t+

∂y

∂qσqσ

)2

=M

2

(∂y

∂t+ qσ eσ

)2

,

then∂T

∂qσ= MV · eσ , eσ =

∂V

∂qσ.

Taking the derivative of the vector p = MV with time, we obtain

MW ≡

d(MV)

dt,

(d

dt

∂T

∂qτ

)eτ

−

∂T

∂qτeτ =

(d

dt

∂T

∂qτ

)eτ +

∂T

∂qτeτ .

In this case from the identity

eτ· eσ = −eτ

· eσ

it follows that∂T

∂qσ= MV ·

∂V

∂qσ= MV · eσ ,

and we have

eσ =∂V

∂qσ.

Thus, the vector of velocity V of mechanical system, its acceleration W,the fundamental basis eσ, and the derivatives of it with respect to time eσ,σ = 1, s, can be introduced by the relation for the kinetic energy of system.This will be used in Chapter IV.

So, for ideal holonomic constraints equations of motion (2.5) (or (2.7))can be separated from equations of reactions (2.6) (or (2.8)).

Lagrange’s equations of the second kind (2.7) are obtained for the in-dependent coordinates qλ, the number of which is equal to the number ofdegrees of freedom l. The left-hand sides of these equations is determinedby the relation for kinetic energy and the right-hand sides by the relationfor virtual elementary work. Taking into account that these scalar quantities(the kinetic energy and virtual elementary work) can be introduced for anymechanical system, it is natural to assume that equations (2.7) describe thelaw of motion of any mechanical system, the position of which is unique-ly given by a set of independent Lagrangian coordinates qλ, λ = 1, l. This

8 I. Holonomic Systems

generalization of Lagrange’s equations of the second kind to the case of anymechanical system, consisting of as rigid bodies as solids, can be regarded asa postulate similar to the other postulates of physics.

Notice that for the above choice of generalized coordinates the vectorsel+κ , κ = 1, k, of reciprocal basis are equal to the vectors ∇∇∇fκ such thatas equations (1.14) show, the vector of reaction for ideal constraints is de-composed into these vectors. Therefore it is convenient to consider two sub-spaces: the L-space with the basis e1, . . . , el and the K-space with the basisel+1, . . . , e3N . These two subspaces are orthogonal to each other, in whichcase in the first of them it is considered a motion of system (a subspace of

motions) and in the second subspace the generalized reactions (a subspace of

reactions).Consider now the notion of ideal holonomic constraints from another point

of view. Using the mentioned above subspaces, the acceleration of represen-tation point can be written as the sum

W = WL + WK , WL = Wλeλ , WK = Wl+κel+κ , WL · WK = 0 .

In this case by (1.13), the equation (1.9) can be replaced by two equations

MWL = YL + T0 , (2.10)

MWK = YK + Λκ∇∇∇fκ . (2.11)

HereYK = Ql+κel+κ , YL = Y − YK .

Vector equation (2.11) is equivalent to the following k scalar equations

Λκ = MWl+κ − Ql+κ , κ = 1, k . (2.12)

We shall show that the vector WK , given by the components Wl+κ, iscompletely determined by the equations of constraints, i. e. the quantitiesWl+κ can be obtained as the functions of time t, the generalized coordinatesqλ, and the generalized velocities qλ (λ = 1, l), using the equations of con-straints only. In this case relations (2.12) imply that the generalized reactionΛκ as the functions of the same variables for the given generalized forcesQl+κ can also be obtained using the equations of constraints.

Really, differentiating twice equations of constraints (1.10) with time, weobtain

∇∇∇fκ

· W = −

∂2fκ

∂t2− 2

∂2fκ

∂t∂yµ

yµ −

∂2fκ

∂yµ∂yµ∗

yµyµ∗ ,

κ = 1, k , µ, µ∗ = 1, 3N .

(2.13)

Using the transition formula (from the Cartesian coordinates to the general-ized ones)

qλ = fλ

∗(t, y) , λ = 1, l , ql+κ = fκ(t, y) , κ = 1, k ,

2. Lagrange’s equations of the first and second kinds 9

and taking into account that

ql+κ = ql+κ = 0 , κ = 1, k ,

we haveyµ = yµ(t, q) , q = (q1, . . . , ql) ,

yµ =∂yµ

∂t+

∂yµ

∂qλqλ .

Substituting these relations into the right-hand side of relations (2.13), we get

∇∇∇fκ

· W = χκ(t, q, q) , q = (q1, . . . , ql) , κ = 1, k. (2.14)

Taking into account that

W = Wl+κ el+κ + Wλeλ , ∇∇∇fκ = el+κ , λ = 1, l , κ = 1, k ,

we obtaingκ

∗

κ

∗Wl+κ

∗ = χκ(t, q, q) , κ, κ∗ = 1, k , (2.15)

wheregκ

∗

κ

∗= ∇∇∇fκ

∗

· ∇∇∇fκ , κ, κ∗ = 1, k .

Assuming, as above, that |gκ∗

κ

∗| = 0, and solving the system of linear alge-

braic equations (2.15), we obtain

Wl+κ = g∗κκ

∗χκ

∗

(t, q, q) , κ, κ∗ = 1, k . (2.16)

Here g∗κκ

∗ are elements of a matrix inverse to the matrix (gκ∗

κ

∗).

From relations (2.16) it follows that the vector WK is completely deter-mined by equations of constraints.

The effect of the equations of constraints on the vector W is given byformulas (2.14). Since ∇∇∇fκ

· WL = 0, κ = 1, k, these formulas can berewritten as

∇∇∇fκ

· WK = χκ(t, q, q) , κ = 1, k .

This implies that the constraints are satisfied for any vector WL. Thereforethe form of equations of constraints does not give any information on thevector WL. From equation (2.10) it follows that the constraints can effect onlyindirectly on the vector WL via the vector T0, which, in no way, is directlyconnection with the equations of constraints. The generalized forces Ql+κ

and the equations of constraints completely determine the normal componentN = Λκ∇

∇∇fκ of the vector of reaction R only. Therefore the ideal constraints,for which T0 = 0, can be called the constraints, which completely defined by

their analytic representations.Thus, for ideal constraints, Newton’s second law, written in the L-space,

has the same form as for free system:

MWL = YL .

10 I. Holonomic Systems

Lagrange’s equations of the first and second kinds can be applied alsoto the study of the dynamics of elastic systems with distributed parameters.Taking into account this generality of Lagrange’s equations, it is useful con-sider the case when the initial coordinates of the mechanical system of generaltype are its Lagrangian coordinates qσ, the number of which is equal to s ifthe constraints are missing. The equations of ideal holonomic constraints areassumed to be given in the form

fκ(t, q) = 0 , κ = 1, k . (2.17)

For simplicity, consider the case when the system consists of N materialpoints. The position of system is described by the curvilinear coordinates qσ,σ = 1, s, s = 3N .

We introduce now the new coordinates

qρ

∗= qρ

∗(t, q) , qσ = qσ(t, q∗) , ρ, σ = 1, s ,

and the new fundamental and reciprocal bases

e∗τ

=∂qσ

∂qτ∗

eσ , eρ

∗=

∂qρ

∗

∂qτeτ , ρ, σ, τ = 1, s . (2.18)

The coordinates q1∗, . . . , ql

∗(l = s − k) are chosen arbitrary, the rest of new

coordinates is defined by the functions ql+κ

∗= fκ(t, q), κ = 1, k. Then the

imposition of constraints (2.17) means that ql+κ

∗= 0, κ = 1, k.

From formulas (2.18) it follows that

el+κ

∗=

∂fκ

∂qτeτ = ∇∇∇fκ ,

and therefore the reaction of ideal holonomic constraints (2.17) can be rep-resented as

R = Λκel+κ

∗.

Multiplying Newton’s equation

MW = Y + Λκel+κ

∗(2.19)

scalarly by the vectors e∗1, . . . , e∗

s, we obtain

(MWσ − Qσ)∂qσ

∂qλ∗

= 0 , λ = 1, l , (2.20)

(MWσ − Qσ)∂qσ

∂ql+κ

∗

= Λκ , κ = 1, k . (2.21)

Taking into account

MWσ

∂qσ

∂qτ∗

= MW ∗

τ; Qσ

∂qσ

∂qτ∗

= Q∗

τ, σ, τ = 1, s ,

2. Lagrange’s equations of the first and second kinds 11

we haved

dt

∂T

∂qλ∗

−

∂T

∂qλ∗

= Q∗

λ, λ = 1, l ,

d

dt

∂T

∂ql+κ

∗

−

∂T

∂ql+κ

∗

− Q∗

l+κ= Λκ , κ = 1, k .

From the first system of these equations we can obtain a motion in newindependent coordinates and from the second one a generalized reaction ofconstraints (2.17). The method to find the motion and reactions is describedabove for systems (2.7), (2.8).

Multiplying scalarly equation (2.19) by the vectors of fundamental basise1, . . . , es of the initial system of coordinates q1, . . . , qs, we obtain

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ + Λκ

∂fκ

∂qσ, σ = 1, s . (2.22)

These equations involve s+k unknowns q1, . . . , qs,Λ1, . . . ,Λk, which have tobe solved together with equations of constraints (2.17). This was distinctivefor classical Lagrange’s equations of the first kind (2.1) when the Cartesian co-ordinates were used. The equations (2.22) are written in the curvilinear frame,in which case constraints (2.17) are imposed on the generalized coordinates,and therefore these equations, following N. V. Butenin and N. A. Fufaev [28],we shall call Lagrange’s equations of the first kind in generalized coordinates.In the literature equations (2.22) are also called Lagrange’s equations of the

second kind with multipliers (see, for example, [59]).V. V. Rumyantsev [202, p. 23] writes: "Lagrange notes that such cases can

occur when for the sake of simplicity of calculations it is useful to preserve agreater number of variables and to make use of indefinite multipliers". Equa-tions (2.22) permit us, as is cited above, to describe the motion of mechanicalsystems in redundant coordinates. If for the coordinates qσ (σ = 1, s) con-straints (2.17) are lacking, then they become usual Lagrange’s equations ofthe second kind. Equations (2.22) will be repeatedly used further, in partic-ular, with their help in Chapter VI some new methods to study the motionof mechanical systems will be constructed.

It is often convenient to represent equations (2.22) in explicit form butnot by the Lagrange operator. Assuming that in the coordinates qσ, σ = 1, s,

the kinetic energy of system has the form

T =M

2gαβ qα qβ , α, β = 0, s , q0 = t , q0 = 1 ,

we have

M(gσρqρ + Γσ,αβ qα qβ) = Qσ + Λκ

∂fκ

∂qσ,

Γσ,αβ =1

2

(∂gσβ

∂qα+

∂gσα

∂qβ−

∂gαβ

∂qσ

),

ρ, σ = 1, s , α, β = 0, s .

(2.23)

12 I. Holonomic Systems

Multiplying equations (2.23) by the coefficients gστ , which are the elementsof the matrix inverse to the matrix with the elements gστ and summing overσ from 1 to s, we obtain

M(qτ + Γτ

αβqα qβ) = Qτ + Λκ gστ

∂fκ

∂qσ,

Γτ

αβ= gστ Γσ,αβ , Qτ = gστ Qσ , σ, τ = 1, s , α, β = 0, s .

§ 3. The D’Alembert–Lagrange principle

Equations (2.22) involve the equations of constraints. To express the sys-tem of these equations in the form of one scalar relation only, without theequations of constraints, we consider the admissible (virtual) displacementsof mechanical system. The mathematical definition of these displacementscan be given in the following way.

We introduce the two systems of generalized Lagrangian coordinates uniq-uely determining the position of mechanical system before the holonomicconstraints are imposed on it. Suppose, these coordinates are related as

qσ = qσ(t, q∗) , qρ

∗= qρ

∗(t, q) , ρ, σ = 1, s .

In differential form these constraints between coordinates are the following

δqσ =∂qσ

∂qρ

∗

δqρ

∗, δqρ

∗=

∂qρ

∗

∂qσδqσ , ρ, σ = 1, s .

This implies that the quantities δqσ and δqρ

∗ are the partial differentials of thefunctions qσ(t, q∗) and q

ρ

∗(t, q), respectively, computed at the moment t. Theyare called admissible (virtual) displacements or the variations of coordinates.The quantities δqσ can be regarded as the contravariant components of thevector δy of virtual displacement of system. In the frame q

ρ

∗ the same vectorδy is given by a family of the quantities δq

ρ

∗ .Consider now the mechanical system with ideal holonomic constraints,

which are given by equations

fκ(t, q) = 0 , κ = 1, k .

Then, assuming

ql+κ

∗= fκ(t, q) , κ = 1, k , l = s − k ,

and making use the validity of the equations of constraints, we have

δql+κ

∗=

∂ql+κ

∗

∂qσδqσ =

∂fκ

∂qσδqσ = ∇∇∇fκ

· δy = 0 , κ = 1, k . (3.1)

3. The D’Alembert–Lagrange principle 13

Whence it follows that if we multiply each of equations (2.22) by δqσ andsummarize then over all σ, then we arrive at the relation

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 . (3.2)

This relation, generated as a consequence of equations (2.22), give theD’Alembert–Lagrange principle. We shall show how equations (2.22) can beobtained from this principle.

Multiplying each of relations (3.1) by Λκ, summarizing over all κ from 1to k, and subtracting this sum from relations (3.2), we have

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ − Λκ

∂fκ

∂qσ

)δqσ = 0 . (3.3)

Equation (3.3) is satisfied for any Λκ and any values δqσ (σ = 1, s), satisfyingequations (3.1). From system of equations (3.1) it follows that only l valuesof δqσ are independent and the rest of the variations of coordinates can beexpressed via them. Choose the quantities Λκ so that in relation (3.3) thecoefficients of dependent variations of coordinates are equal to zero. Thecoefficients of the rest of variations of coordinates are also equal to zero sincethese variations are arbitrary and independent. Thus, in relation (3.3) all thecoefficients of δqσ are equal to zero and this means that equations (2.22)follows from the D’Alembert–Lagrange principle (3.2).

In the case when the mechanical system consists of N material pointsand the initial Lagrangian coordinates qσ are Cartesian coordinates xµ (µ =1, 3N) of the points of system the D’Alembert–Lagrange principle (3.2) canbe represented in the form

(mµxµ − Xµ) δxµ = 0 , (3.4)

since in this case we have

T =mµx2

µ

2.

Relation (3.4) can also be represented as

(mν rν − Fν) · δrν = 0 ,

where the vector

δrν = δxν1i1 + δxν2i2 + δxν3i3 =∂rν

∂qλδqλ

is the vector of virtual displacement of the ν-th point. By formulas (3.1) thecoordinates δxµ, µ = 1, 3N , of the vectors δrν , ν = 1, N , satisfy the relations

∂fκ

∂xµ

δxµ = 0 , κ = 1, k .

14 I. Holonomic Systems

We consider now the notion of ideal holonomic constraints from the pointof view of the D’Alembert–Lagrange principle. This principle (3.2) can berewritten in the vector form

(MW − Y

)· δy = 0 ,

orR · δy = 0 . (3.5)

Then from equations (3.1) it follows that the reaction of ideal holonomicconstraints can be represented as

R = Λκ∇∇∇fκ ,

i. e. the vector of reaction can be decomposed into the vectors el+κ

∗= ∇∇∇fκ ,

κ = 1, k, of the basis of K-space (the subspace of reactions). The equationsof holonomic constraints give the l-dimensional surface V (t, q) on which atmoment t the representation point is to be situated. To the curvilinear co-ordinates q1

∗, . . . , ql

∗corresponds the basis e∗1, . . . , e

∗

l, situated in the plane

T (q, V ) tangent to the surface V (t, q). The vectors δy of the virtual displace-ments of representation point (the subspace of virtual displacements) lie inthis plane. Thus, the D’Alembert–Lagrange principle in the form (3.5) statesthat for ideal holonomic constraints the subspace of reactions (K-space) isorthogonal to the subspace of virtual displacements (L-space).

We show that the representation of the D’Alembertian–Lagrange prin-ciple in the form (3.5) is the generalization of usual notion of ideality ofconstraint for one material point to the case of representation point. Consid-er this condition for the system of N material points in the case when we useCartesian coordinates and the equations of constraints are represented in theform (1.10). By formulas (1.4), (1.7) condition (3.5) takes the form

R · δy = Λκ∇∇∇fκ

· δy = Λκ

∂fκ

∂yµ

δyµ = Λκ

∂fκ

∂xµ

δxµ = 0 . (3.6)

If we introduce the reaction

R′

ν= Λκ

(∂fκ

∂xν1i1 +

∂fκ

∂xν2i2 +

∂fκ

∂xν3i3

),

applied to the ν-th point in usual three-dimensional space by all constraints,then in place of formula (3.6) we have

R′

ν· δrν = 0 . (3.7)

This relation is usually regarded as a definition of ideality of holonomic con-straints, imposed on the motion of N material points. It is of hardly explain-able axiomatic nature. However in fact, as is shown above, condition (3.7) isa generalization of usual notion of constraints ideality for one material pointto the case of representation point.

4. Longitudinal accelerated motion of a car 15

§ 4. Longitudinal accelerated motion of a car

as an example of motion of a holonomic system

with a nonretaining constraint

Problem definition. Longitudinal acceleration of a car with possibleslipping of its driving (front or rear) wheels is considered. The wheels areassumed to be perfectly rigid, undeformable. The calculation model of a front-drive car is presented in Fig. I. 1. The car is supposed to consist of the bodyon the springs with shock absorbers, in which viscous drag forces of bothfront (1) and rear (2) double wheels are taken into account. C1 and C2 arethe centers of the front and rear wheels, correspondingly. B1 and B2 are theattachment points of the springs and shock absorbers to the car body.

Fig. I. 1

Let us introduce the notation: M , M1, M2 are masses of the body, thefront and rear wheels, correspondingly; J , J1, J2 are their moments of inertiaabout the centers of mass ; χ, χ1, χ2 are the coefficients of viscous drag forcesacting on the body and in the shock absorbers, correspondingly; c1, c2 arethe stiffness coefficients of the springs of front and rear wheels; kst

1 , kdyn1 (or

kst2 , k

dyn2 ) are the static and dynamic coefficients of Coulomb’s friction force

for the driving front (or rear) wheels; r1, r2 are the coefficients of rollingfriction for the front and rear wheels; R1, R2 are the radii of the correspondingwheels. Note that we consider a reduced moment of inertia, which includesrotating masses of the cardan shaft, details of the transmission gear-box, theflywheel, the crankshaft, etc., connected to the driving wheels, as the momentof inertia of driving wheels (according to Fig. I. 1 it is J1). In this case thequantity J1 depends, in general, on the relation between angular velocities ofthe driving wheels and the engine.

The engine acts on the driving wheels with some drive moment Θdr, spec-ified as a time function. In Fig. I. 1 the drive moment is applied to the drivingwheels.

16 I. Holonomic Systems

Equations of the car motion are to be obtained to determine, when slip-ping begins at the stage of acceleration and when it stops.

Rolling motion is the main mode of vehicles’ operation. When rollingwithout slipping, the instantaneous center of velocity of the wheel is locatedat the point of contact between the wheel and the road.

Let us introduce a fixed frame of reference Oxy, the y-axis being directedvertically upwards. The abscissa x = x(t) and the ordinate y = y(t) of thecentre of mass C of the car body, as well as the angle of the body rotation ϕ =ϕ(t) about the centre of mass C, will be considered as generalized coordinates.Suppose, that in the state of static equilibrium of the body we have y = 0and ϕ = 0.

The condition of rolling without slipping for a front-drive car can bewritten in the form of the first order differential linear constraint

x = ϕ1R1 , (4.1)

where ϕ1 is the angle of rotation of front driving wheels.In a similar manner, for a rear-drive car the equation of constraint shoud

be written asx = ϕ2R2 ,

where ϕ2 = ϕ2(t) is the angle of rotation of rear driving wheels.The constraint (4.1) is holonomic: integrating expression (4.1) yields

x = ϕ1R1 (x = ϕ2R2) .

The constraint (4.1) is nonretaining, the car is released from it when thedriving wheels begin to slip. When slipping occurs, instead of the condition(4.1) the following inequality

ϕ1 >x

R1

begins to be fulfilled. Note, that when the car is accelerating, this inequal-ity can not be opposite in sign. So, the constraint under consideration isunilateral.

The slipping occurs, when the horizontal reaction force of the road towardsthe driving wheels of the car reaches some "limit"value, which is related withthe static Coulomb friction force. Let us start analysis with the case whenthe constraint (4.1) is satisfied.

Motion without slipping. In the absence of slipping of the drivingwheels the kinetic energy of the car T takes the form

T =1

2

((M + M1 + M2 +

J1

R21

+J2

R22

)x2 + My2 + Jϕ2

).

The system potential energy Π is equal to the potential energy of the springsdeformation. We shall measure it from the position of static equilibrium. Let

4. Longitudinal accelerated motion of a car 17

us introduce auxiliary coordinates for each of the springs: y1 = y1(t) andy2 = y2(t), which are the vertical upward displacements of the points B1 andB2 from the position of static equilibrium. By expressing y1 and y2 in termsof y and ϕ, we obtain

Π =1

2(c1y

21 + c2y

22) =

=1

2(c1(y + L1ϕ)2 + c2(y − L2ϕ)2) .

(4.2)

Here Lk (k = 1, 2) are the horizontal distances from the point C to the pointsBk (see Fig. I. 1).

The Rayleigh scattering function corresponding to the resistance forcestakes the form

R =1

2(χx2 + χ1y

21 + χ2y

22) =

=1

2(χx2 + χ1(y + L1ϕ)2 + χ2(y − L2ϕ)2) .

(4.3)

Let us take into consideration the drive moment Θdr and the momentof rolling friction of the wheels. The elementary work of generalized forcesapplied to the driving front wheels is the product of the drive moment Θdr

by the angular displacement δϕ1 and the moment of rolling friction by theangular displacement δϕ1. The elementary work of generalized forces appliedto the driven rear wheels is the product of the moment of rolling friction bythe angular displacement δϕ2. As a result, we obtain

δAx = Θdrδϕ1 − N1r1δϕ1 − N2r2δϕ2 .

Here N1 and N2 are the vertical components of the road reaction towards thefirst and the second pair of wheels, correspondingly. Expressing the angulardisplacements δϕ1 and δϕ2 in terms of the elementary displacement δx implies

δAx =

(Θdr

R1− N1

r1

R1− N2

r2

R2

)δx .

Now determine the reactions N1 and N2. They depend on gravity forces of thewheels, the statical reactions of the body towards the wheels N st

1 and N st2 , and

the dynamical corrections to them, which are due to the vertical movementof the body and its rotation. Besides these, the damping forces, which aregenerated by shock absorbers when the body is oscillating, should be alsotaken into account. These forces are calculated by the Rayleigh scatteringfunction (4.3) and applied to the body at the points B1 and B2. At the sametime, according to the third Newton law they act from the shock absorbersto the axles of front and rear wheels. Thus, we have

N1 = M1g + N st1 − c1(y + L1ϕ) − χ1(y + L1ϕ) , N st

1 =MgL2

L1 + L2,

N2 = M2g + N st2 − c2(y − L2ϕ) − χ2(y − L2ϕ) , N st

2 =MgL1

L1 + L2.

18 I. Holonomic Systems

Finally, the influence of the drive moment on the body should be takeninto consideration. This moment is opposite in direction to the drive momentΘdr and is the same as the drive moment in value. The elementary work ofthis moment is

δAϕ = Θdrδϕ . (4.4)

Note that taking into consideration the influence of this moment on thebody rotation is of principal importance. Unfortunately, it is not taken intoaccount in a number of studies.

Using the written above expressions for the kinetic and potential ener-gies, Rayleigh’s function, and the elementary work, we write the Lagrangeequations of the second kind

(M + M∗

1 + M∗

2 ) x =Θdr

R1

−

(M1 +

ML2

L1 + L2

)r1g

R1

−

(M2 +

ML1

L1 + L2

)r2g

R2

+

+

(c1r1

R1

+c2r2

R2

)y +

(c1r1L1

R1

−

c2r2L2

R2

)ϕ − χx+

+

(χ1r1

R1

+χ2r2

R2

)y +

(χ1r1L1

R1

−

χ2r2L2

R2

)ϕ , (4.5)

M y = −(c1 + c2)y − (c1L1 − c2L2)ϕ−

− (χ1 + χ2)y − (χ1L1 − χ2L2)ϕ ,

J ϕ = Θdr − (c1L1 − c2L2)y − (c1L2

1 + c2L2

2)ϕ−

− (χ1L1 − χ2L2)y − (χ1L2

1 + χ2L2

2)ϕ ,

where M∗

1 = M1 +J1

R21

, M∗

2 = M2 +J2

R22

.

The system of differential equations (4.5) describes the motion of a front-drive car in the absence of driving wheels slipping. The analogous equationsof motion for a rear-drive car differ from system (4.5) in that the expressionΘdr/R1 should be replaced with Θdr/R2 in them.

Note that the second and the third equations of system (4.5) do notcontain the variable x and can be integrated apart from the first equation.

Motion with slipping. From kinematics standpoint, slipping of the driv-ing front wheels occurs when the point K1 of contact with the road ceasesto be an instantaneous velocity center of the driving wheels, that is whenϕ1 = x(t)/R1. It is obvious that in this case one should introduce a new "in-dependent"generalized coordinate ϕ1, which defines a rotation angle of thedriving front wheels relative to the initial position. As a result, the expressionfor the system kinetic energy takes the form

T =1

2

((M + M1 + M2 +

J2

R22

)x2 + My2 + Jϕ2 + J1ϕ

21

). (4.6)

4. Longitudinal accelerated motion of a car 19

Expressions (4.2) and (4.3) for the potential energy Π and Rayleigh’sfunction R, as well as the elementary work (4.4) are the same as in the caseof motion without slipping. The elementary work δA′

x, that is done in moving

through displacement of the car body δx = δϕ2R2 by Coulomb’s friction forcek

dyn1 N1 and by the rolling friction moment of the driven wheels, is

δA′

x= k

dyn1 N1δx − r2N2δϕ2 =

(k

dyn1 N1 −

r2N2

R2

)δx . (4.7)

When slipping occurs, the driving front wheels are affected by the slippingfriction force k

dyn1 N1, the rolling friction moment, and the drive moment Θdr.

Therefore

δAϕ1= (Θdr − k

dyn1 N1R1 − r1N1)δϕ1 . (4.8)

From expressions (4.6), (4.2), (4.3), (4.4), (4.7), (4.8) it follows that the sys-tem of the Lagrange equations for a front-drive car moving with slippingtakes the form

(M + M1 + M∗

2 ) x =

(M1 +

ML2

L1 + L2

)k

dyn1 g −

(M2 +

ML1

L1 + L2

)r2g

R2+

+

(c2r2

R2− c1k

dyn1

)y −

(c1k

dyn1 L1 +

c2r2L2

R2

)ϕ − χx+

+

(χ2r2

R2− χ1k

dyn1

)y −

(χ1k

dyn1 L1 +

χ2r2L2

R2

)ϕ ,

M y = −(c1 + c2)y − (c1L1 − c2L2)ϕ − (χ1 + χ2)y − (χ1L1 − χ2L2)ϕ ,

J ϕ = Θdr − (c1L1 − c2L2)y − (c1L21 + c2L

22)ϕ − (4.9)

− (χ1L1 − χ2L2)y − (χ1L21 + χ2L

22)ϕ ,

J1 ϕ1 = Θdr −

(M1 +

ML2

L1 + L2

)(r1 + k

dyn1 R1)g+

+ c1(r1 + kdyn1 R1)y + c1(r1 + k

dyn1 R1)L1ϕ+

+ χ1(r1 + kdyn1 R1)y + χ1(r1 + k

dyn1 R1)L1ϕ .

20 I. Holonomic Systems

Analogous equations for a rear-drive car appear as

(M + M∗

1 + M2) x =

(M2 +

ML1

L1 + L2

)k

dyn2 g −

(M1 +

ML2

L1 + L2

)r1g

R1+

+

(c1r1

R1− c2k

dyn2

)y +

(c2k

dyn2 L2 +

c1r1L1

R1

)ϕ − χx+

+

(χ1r1

R1− χ2k

dyn2

)y +

(χ2k

dyn2 L2 +

χ1r1L1

R1

)ϕ ,

M y = −(c1 + c2)y − (c1L1 − c2L2)ϕ − (χ1 + χ2)y

− (χ1L1 − χ2L2)ϕ ,

J ϕ = Θdr − (c1L1 − c2L2)y − (c1L21 + c2L

22)ϕ−

− (χ1L1 − χ2L2)y − (χ1L21 + χ2L

22)ϕ , (4.10)

J2 ϕ2 = Θdr −

(M2 +

ML1

L1 + L2

)(r2 + k

dyn2 R2)g+

+ c2(r2 + kdyn2 R2)y − c2(r2 + k

dyn2 R2)L2ϕ+

+ χ2(r2 + kdyn2 R2)y − χ2(r2 + k

dyn2 R2)L2ϕ .

Note that the second and third equations of systems (4.9) and (4.10) areunchanged in comparison with the case of motion without slipping. They canbe also integrated apart from the first equations.

The conditions of occurrence and termination of slipping. Thelast equation of system (4.9) for a front-drive car can be rewritten as

J1ϕ1 = Θdr − r1N1 − kdyn1 N1R1 .

In the absence of slipping we should set ϕ1 = x/R1 in this equation, and

the quantity kdyn1 N1 should be replaced by the friction force Fdr. Note that

this friction force Fdr is a driving force in the problem concerned, that is whyit has a subscript "dr". The force Fdr is less than the static Coulomb frictionforce kst

1 N1. Therefore, the dynamic condition of motion of the driving fontwheels without slipping takes the form

Θdr

R1−

r1N1

R1−

J1x

R21

= Fdr < kst1 N1 . (4.11)

4. Longitudinal accelerated motion of a car 21

Hence, the system of equations (4.5) can be used as long as the valuesof N1 and x, calculated from it, satisfy inequality (4.11). Recall that forcalculating the reaction N1 the quantities y, ϕ, and their derivatives shouldbe known.

If at some instant t1 inequality (4.11) is violated, and the slipping of thefront wheels begins, then system (4.9) should be integrated. Now the driving

wheels are affected by the dynamic Coulomb friction force kdyn1 N1. As stated

at the beginning of the section, the non-holonomic constraint imposed on thedriving wheels is nonretaining and unilateral. Therefore, if when integratingsystem (4.9) at some instant t2 the equality ϕ1 = x/R1 is fulfilled, then itmeans that the slipping of driving wheels has terminated and constraint (4.1)is restored. Starting with the time instant t2, one should come to integrat-ing the system of differential equations (4.5). So, the condition of slippingtermination is the fulfillment of the equality

ϕ1(t) =x(t)

R1. (4.12)

The analogous condition of motion without slipping for a rear-drive carand the condition of slipping termination are

Θdr

R2−

r1N1

R2−

J2x

R22

= Fdr < kst2 N2 ,

ϕ2 =x

R2.

The example of problem solving. As an example consider the accel-eration of a hypothetical front-wheel drive car.

Let us assume that the period of time under consideration is equal to 50seconds. At the initial instant of time (t0 = 0) the car is immovable and startsto accelerate under the action of the drive moment (the moment is measuredin N · m, time t is measured in seconds):

Θdr =

750 sinπt

40, t 37 ,

750 sin37π

40, t > 37 .

(4.13)

The values of parameters used in calculations are: M = 955 kg (includingmasses of the front and rear axles: 515 kg and 440 kg, correspondingly);J = 1394.2 kg·m2; χ = 10 N·s·m−1; L1 = 1.17 m; L2 = 1.307 m; M1 =M2 = 14 kg; J1 = 21.52 kg·m2; J2 = 1.076 kg·m2; R1 = R2 = 0.392 m;c1 = c2 = 20000 N/m; χ1 = χ2 = 1000 N·s·m−1; r1 = r2 = 0.0024 m;

kst1 = 0.3; k

dyn1 = 0.25; the gravitational acceleration is g = 9.8 m/s

2.

The passage from system (4.5) to system (4.9) and then back again tosystem (4.5) is performed on the basis of realization of conditions (4.11)

22 I. Holonomic Systems

Fig. I. 2

and (4.12). It is explained by Fig. I. 2, corresponding to the numerical dataspecified above. In this figure the static Coulomb friction force F st

fr = kst1 N1

and the dynamic Coulomb friction force Fdynfr = k

dyn1 N1 are shown by solid

lines, and the driving friction force

Fdr =Θdr

R1−

r1N1

R1−

J1x

R21

. (4.14)

is shown by a dashed line As follows from the figure, the driving friction forceincreases from the instant of time t0 = 0 till the instant t1 = 14.004 s, butinequality (4.11) is fulfilled. Therefore, system (4.5) is to be integrated.

Starting with the instant of time t1 = 14.004 s, there comes the secondstage of the car acceleration — motion with slipping. It is described by system(4.9), the initial data for which are found from the values of functions thatare the solution of system (4.5) for t1 = 14.004 s.

When integrating system (4.9), we determine the instant of time t2, whenequality (4.12) is fulfilled. In the example given t2 = 38.747 s. At this momentof time t2 constraint (4.12) is imposed instantly, and the third stage of the caracceleration begins — the resumption (recovering) of motion without slipping.It is described by system (4.5), the initial data for which are found from thevalues of functions that are the solution of system (4.9) for t2 = 38.747 s.

The passage from system (4.9) to system (4.5) when t2 = 38.747 s, causedby instantaneous imposition of constraint (4.12), is followed by the jump ofacceleration x (see Fig. I. 3): for t1 < t < t2 the acceleration x is found fromsystem (4.9), and for t > t2 it is found from system (4.5), in which caseϕ1 = x/R1. Therefore, for t = t2 we have

x(t2 − 0) = x(t2 + 0) .

4. Longitudinal accelerated motion of a car 23

Fig. I. 3

This acceleration jump is connected with decrease of the value of traction(force of cohesion) of the front wheels with the road from the value k

dyn1 N1

down to the value Fdr, defined by formula (4.14), in which x is calculatedfrom system (4.5) with the initial data taken from the end of motion describedby system (4.9). This jump of traction of the front wheels with the road ischaracterized by the segment A3A4 in Fig. I. 2. It is an interesting feature ofthe car acceleration in the presence of the driving wheels slipping.

Fig. I. 4

24 I. Holonomic Systems

Fig. I. 5

The further check of condition (4.11) allows us to make a conclusion thatunder the given law of variation of the drive moment (4.13), the new slippingof the driving wheels does not occur till the 50th second.

Therefore, in Fig. I. 2 the change of the driving force acting on the drivingwheels is characterized by the arc OA1 when time changes from t0 till t1; atthe instant t1 we have the force jump A1A2 caused by the difference betweenthe static and dynamic coefficients of Coulomb friction (traction coefficients);when time changes from t1 till t2 the driving force is described by the arcA2A3, which is almost horizontal. At the instant of time t2 the jump A3A4 ofthe driving force occurs; and the force Fdr changes according to the segmentA4A5 when t > t2.

The change of the generalized coordinates y, ϕ is shown in Fig. I. 4, andthat of the coordinates x, ϕ1 is shown in Fig. I. 5. We see that at the stage ofthe driving wheels slipping, the coordinate ϕ1 rapidly increases.

Note that, if tyres are compliant and the traction coefficient depends con-tinuously on the speed, the curves in figures I. 2 and I. 3 will be replaced withcontinuous ones. However, a sharp change of the driving force and accelera-tion at the instants t1 and t2 remains.

C h a p t e r II

NONHOLONOMIC SYSTEMS

From the analog of Newton’s law, Maggi’s equations are deduced which are

the most convenient equations of the nonholonomic mechanics. From Mag-

gi’s equations the most useful forms of equations of motion of nonholonom-

ic systems are obtained. The connection between Maggi’s equations and the

Suslov–Jourdain principle is considered. The notion of ideal nonholonomic

constraints is discussed. In studying nonholonomic systems the approach, ap-

plied in Chapter I to analysis of the motion of holonomic systems, is employed.

The role of of Chetaev’s type constraints for the development of nonholonom-

ic mechanics is considered. For the solution of a number of nonholonomic

problems, the different methods are applied.

§ 1. Nonholonomic constraint reaction

Consider the Cartesian coordinates Ox1x2x3 with the unit vectors i1,i2, i3. If on the motion of material point of the mass m it is imposed thenonholonomic constraint

ϕ(t, x, x) = 0 , x = (x1, x2, x3) , (1.1)

then the second Newton’s law can be represented as

mw = F + R′ , (1.2)

where F = (X1, X2, X3) is an active force, acting on the point, and R′ =(R′

1, R′

2, R′

3) is constraint reaction (1.1).Consider the vector R′. We differentiate equation of constraint (1.1) with

respect to time:

ϕ ≡∂ϕ

∂t+

∂ϕ

∂xk

xk +∂ϕ

∂xk

xk = 0 , k = 1, 2, 3 . (1.3)

Together with the usual vector ∇∇∇ϕ =∂ϕ

∂xk

ik we introduce the new vector

∇∇∇′ϕ proposed by N. N. Polyakhov [185]:

∇∇∇′ϕ =∂ϕ

∂xk

ik .

Then equation (1.3) can be rewritten as

∂ϕ

∂t+∇∇∇ϕ · v +∇∇∇′ϕ · w = 0 . (1.4)

25

26 II. Nonholonomic Systems

Multiplying scalarly equation (1.2) by∇∇∇′ϕ and equation (1.4) by m, we obtain

R′ · ∇∇∇′ϕ = −m(∂ϕ

∂t+∇∇∇ϕ · v

)− F · ∇∇∇′ϕ .

This implies that the vector R′ can be represented in the form

R′ = Λ∇∇∇′ϕ + T0 = N + T0 ,

Λ = −m

∂ϕ

∂t+ m∇∇∇ϕ · v + F · ∇∇∇′ϕ

|∇∇∇′ϕ|2, T0 · N = 0 .

(1.5)

Note that the only component N of constraint reaction depends on (1.1),in which case by formulas (1.5) it is defined as a certain function of t, x, x.In particular, equations (1.1) and (1.2) are also valid for T0 = 0. The non-holonomic constraints of such type we shall called ideal. If T0 = 0, then theconstruction of the vector T0 should be described separately, based on theadditional characteristics of the physical realization of constraint (1.1). As arule, T0 essentially depends on the quantities |N| and, in lesser degree, ont, x, x.

Consider the partial case of holonomic constraint, namely

f(t, x) = 0 . (1.6)

Represent it in the form of (1.1):

ϕ ≡ f =∂f

∂t+

∂f

∂xk

xk = 0 .

Then we have∂ϕ

∂xk

=∂f

∂xk

,

and therefore for holonomic constraint (1.6) the vector∇∇∇′ϕ, introduced above,coincides with the usual vector ∇∇∇f . Here, as is shown in Chapter I, the vec-tor N is directed along a normal to the surface, given by equation (1.6),and the vector T0 lies in the plane tangential to this surface. In particular,if equation (1.6) gives a certain material surface, on which the point mustmove, then for the ideally burnished surface (for ideal holonomic constraint)we have T0 = 0. Otherwise we need to point out a rule for construction ofthe vector T0, for example, to give Coulomb’s law (1.12) from Chapter I.

Assume now that on the motion of material point it is imposed two non-holonomic constraints

ϕκ(t, x, x) = 0 , x = (x1, x2, x3) , κ = 1, 2 .

Arguing as above, we obtain

∂ϕκ

∂t+∇∇∇ϕκ · v +∇∇∇′ϕκ · w = 0 , κ = 1, 2 .

1. Nonholonomic constraint reaction 27

The differential equation of motion has, as before, the form of (1.2). This lawpermits us to eliminate the vector w from the previous relations and to writethem as

R′ · ∇∇∇′ϕκ ≡ R′κ = −(m

∂ϕκ

∂t+ m∇∇∇ϕκ · v + F · ∇∇∇′ϕκ

), κ = 1, 2 .

This implies that if we represent the vector R′ as the sum

R′ = Λκ∇∇∇′ϕκ + T0 , (1.7)

where T0 is a certain unknown vector orthogonal to the vectors ∇∇∇′ϕκ , thenthe coefficients Λκ can be found from the following system of equations

Λ1|∇∇∇′ϕ1|2 + Λ2∇∇∇

′ϕ1 · ∇∇∇′ϕ2 = R′1 ,

Λ1∇∇∇′ϕ1 · ∇∇∇′ϕ2 + Λ2|∇∇∇

′ϕ2|2 = R′2 .

Thus, the components Λκ∇∇∇′ϕκ of the vector R′ are uniquely defined by

equations of constraints and the force F.We remark that a similar reasoning can also be used for two holonomic

constraints since in this case we have ∇∇∇′ϕκ = ∇∇∇fκ . Therefore if there arethe two holonomic constraints, then the reaction can be represented as

R′ = Λκ∇∇∇fκ + T0 , κ = 1, 2 .

We consider now the motion of representation point under the conditionthat there exist k nonholonomic constraints:

ϕκ(t, y, y) = 0 , κ = 1, k . (1.8)

Then like the motion of one material point we can write

MW = Y + R , (1.9)

which has the form of the second Newton’s law in the vector form. In thesequel relation (1.9) is called the second Newton’s law just as in Chapter I.

Using formula (1.7) in the case of representation point, we have

R = N + T0 , N = Λκ∇∇∇′ϕκ , ∇∇∇′ϕκ =

∂ϕκ

∂yµ

jµ , T0 · N = 0 . (1.10)

In Chapter IV the notion of a tangent space to the manifold of all pos-sible configurations of a mechanical system will be introduced. The set ofthe Lagrange equations of the second kind describing a motion of the uncon-strained mechanical system is written in this space as a single vector-valuedequality which has a form of the second Newton law. This makes it possibleto generalize formulas (1.5), (1.7) not only to mechanical systems, consistingof the finite number of particles, but also to arbitrary mechanical systemswith the finite number of degrees of freedom.

28 II. Nonholonomic Systems

Thus, the conclusion on the structure of the constraint reaction obtainedfor a single particle is of general nature. It is fundamental. Expressions (1.5),(1.7), (1.10) for the reaction force were obtained by N. N. Polyakhov in 1974[185]. Later these results were included into the treatise (manual for univer-sities) [189]. In 2001 O. M. O’Reilly and A. R. Srinivasa [416] devoted theirwork to deriving and discussion of expressions (1.5).

§ 2. Equations of motion of nonholonomic systems.

Maggi’s equations

Assume that the nonlinear nonholonomic constraints, imposed on a mo-tion of system, in the curvilinear coordinates q = (q1, . . . , qs) have the form

ϕκ(t, q, q) = 0 , κ = 1, k . (2.1)

In the case of the motion of system of N material points s = 3N .Now we pass from the variables q = (q1, . . . , qs) to the new nonholonomic

variables v∗ = (v1∗, . . . , vs

∗) by formulas

vρ

∗= vρ

∗(t, q, q) , ρ = 1, s . (2.2)

If the solvability conditions are satisfied, then we can write the inverse trans-formation

qσ = qσ(t, q, v∗) , σ = 1, s . (2.3)

Assuming that the derivatives of functions (2.2), (2.3) are continuous, we canintroduce the two systems of linearly independent vectors:

εεετ =∂qσ

∂vτ∗

eσ , εεερ =∂v

ρ

∗

∂qτeτ , ρ, τ = 1, s . (2.4)

Since the product is as follows

εεερ · εεετ =∂v

ρ

∗

∂qσ

∂qσ

∂vτ∗

= δρ

τ=

0 , ρ = τ ,

1 , ρ = τ ,

vectors (2.4) can be regarded as the vectors of the fundamental and reciprocalbases. We shall say that bases (2.4) are nonholonomic.

By assumption, the equations of constrains (2.1) are such that

|∇∇∇′ϕκ · ∇∇∇′ϕκ∗

| = 0 , κ, κ∗ = 1, k .

In this case in transition formulas (2.2) the last functions can be given in thefollowing way

vl+κ

∗= ϕκ(t, q, q) , l = s − k , κ = 1, k . (2.5)

2. Equations of motion of nonholonomic systems Maggi’s equations 29

Therefore if constraint (2.1) is satisfied, then we have vl+κ

∗= 0. Then by

formulas (2.4) we have

εεεl+κ =∂ϕκ

∂qτeτ ≡∇∇∇′ϕκ , κ = 1, k .

We introduce two subspaces orthogonal to each other with the nonholo-nomic bases εεε1, . . . , εεεl and εεεl+1, . . . , εεεs and call them L-space and K-space, respectively. Decompose the acceleration vector into the following twocomponents

W = WL + WK , WL = Wλεεελ , WK = Wl+κεεεl+κ , WL · WK = 0 .

Here the wavy sign denotes that the components of acceleration vector aretaken for nonholonomic bases (2.4) but not for the usual fundamental andreciprocal bases.

The second Newton’s law (1.9) is replaced then by the following twoequations:

MWL = YL + RL , (2.6)

MWK = YK + RK . (2.7)

Differentiating the equations of constraints (2.1) with respect to time andtaking into account that the vector W can be represented as

W = (qσ + Γσ

αβqαqβ)eσ , σ = 1, s , α, β = 0, s ,

we obtainεεεl+κ · W = χκ

1 (t, q, q) ,

χκ

1 (t, q, q) = −∂ϕκ

∂t−

∂ϕκ

∂qσqσ +

∂ϕκ

∂qσΓσ

αβqαqβ ,

κ = 1, k , α, β = 0, s .

These equations are similar to equations (2.14) of Chapter I. This impliesthat the vector WK as the function of t, qσ, qσ, σ = 1, s is uniquely de-termined by constraints equations. By equation (2.7) for the given forceYK the vector WK can be obtained by means of the constraint reactionRK = N = Λκ∇∇∇

′ϕκ . Unlike the above the component WL is independent ofthe mathematical definition of the equations of constraints. It can be deter-mined from equation (2.6) for any vector RL, in particular, for RL ≡ T0 = 0if in L-space the equation of properly motion has the form

MWL = YL .

It is naturally to call nonholonomic constraints (2.1), which do not influencethe vector WL, ideal. For these constraints the vector of reaction is as follows

R = RK = N = Λκ∇∇∇′ϕκ . (2.8)

30 II. Nonholonomic Systems

By formulas (1.9) and (2.8) the second Newton’s law for ideal nonholonomicconstraints has the form

MW = Y + Λκ∇∇∇′ϕκ . (2.9)

Multiplying this equation by the vectors εεελ, λ = 1, l, we obtain Maggi’s

equations(MWσ − Qσ

)∂qσ

∂vλ∗

= 0 , λ = 1, l , (2.10)

where

MWσ − Qσ =d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ , σ = 1, s .

For linear nonholonomic constraints these equations have been obtained byMaggi in 1896 [355]. Later for nonlinear nonholonomic constraints, by meansof the generalized principle of D’Alembert–Lagrange they have been generat-ed by A. Przeborski [375]. Integrating equations (2.1), (2.10) with the giveninitial data, we can find the law of motion of the system

qσ = qσ(t) , σ = 1, s . (2.11)

Multiplying equation (2.9) by the vectors εεεl+κ, κ = 1, k, we obtain the secondgroup of Maggi’s equations

(MWσ − Qσ

) ∂qσ

∂vl+κ

∗

= Λκ , κ = 1, k . (2.12)

For the given law of motion of system (2.11), the generalized reactions ofnonholonomic constraints (2.1) can be determined as the time functions fromthe above equations. Formulas (2.12) do not give directly the quantities Λκ

as the functions of t, q, q. They can be found from the following equations

εεεl+κ · W = χκ

1 (t, q, q), WK =1

M(YK + Λκ∇∇∇

′ϕκ) .

Thus, for nonholonomic constraints the introduction of nonholonomicbases (2.4) permits us to obtain the two subspaces K and L. These sub-spaces turn out orthogonal to each other and in studying the problems inthese subspaces it is convenient to make use of Maggi’s equations (2.10)and (2.12).

Maggi’s equations are highly convenient to consider the motion of non-holonomic systems. It is to be noted that they are valid for any nonholonom-ic constraints, including the nonlinear ones. Most of the well-known formsof equations, describing the motion of nonholonomic systems, can be ob-tained from these equations (for detail, see the next section), for example,Chaplygin’s equations

d

dt

∂(T )

∂qλ−

∂(T )

∂qλ+

∂T

∂ql+κ

(∂bl+κ

λ∗

∂qλ−

∂bl+κ

λ

∂qλ∗

)qλ

∗

= Qλ ,

λ, λ∗ = 1, l , κ = 1, k ,

(2.13)

2. Equations of motion of nonholonomic systems Maggi’s equations 31

if the equations of constraints (2.1) are represented as

ql+κ = bl+κ

λ(q1, . . . , ql)qλ , λ = 1, l , κ = 1, k , (2.14)

or the Hamel–Boltzmann equations

d

dt

∂T ∗

∂πλ−

∂T ∗

∂πλ+

∂T ∗

∂πl+κ

γλ(l+κ)λ∗ πλ∗

= Qλ ,

λ, λ∗ = 1, l, , κ = 1, k , l = s − k ,

(2.15)

for the nonholonomic constraints of the form

ϕκ(t, q, q) ≡ al+κ

σ(q)qσ = 0, κ = 1, k , σ = 1, s , (2.16)

if in place of formulas (2.2), (2.3) there are introduced the following relationsbetween the time derivatives of the generalized coordinates q1, . . . , qs and ofthe quasicoordinates π1, . . . , πs:

πρ = aρ

σ(q)qσ , qσ = bσ

ρ(q)πρ , ρ, σ = 1, s . (2.17)

In Chaplygin’s equations the symbol (T ) denotes, as usual [59], the kineticenergy in which the generalized velocities ql+κ, κ = 1, k, are replaced byrelations (2.14). Similarly, in the Hamel–Boltzmann equations T ∗ denotesthe kinetic energy if in it the quantities qσ, σ = 1, s, are replaced by theirrelations in unknowns πρ, ρ = 1, s. Recall that the analytic representationsof nonholonomic constraints (2.16) give k last quasivelocities πl+1, . . . , πs in

formulas (2.17). Besides, equations (2.15) involves the generalized forces Qλ,

which correspond to the quasivelocities πλ(λ = 1, l) :

Qλ = Qσ

∂qσ

∂πλ, λ = 1, l , σ = 1, s , (2.18)

and the objects of nonholonomicity γλ(l+κ)λ∗

γλ(l+κ)λ∗ = bσ

λbτ

λ∗

(∂al+κ

σ

∂qτ−

∂al+κ

τ

∂qσ

),

λ, λ∗ = 1, l , κ = 1, k , σ, τ = 1, s .

(2.19)

The derivatives ∂T ∗/∂πλ are computed by formulas

∂T ∗

∂πλ=

∂T ∗

∂qσ

∂qσ

∂πλ, λ = 1, l , σ = 1, s . (2.20)

The following equations [169, 314]

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ−

∂T

∂qσ

(d

dt

∂qσ

∂vλ∗

−∂qσ

∂πλ

)= Qλ , λ = 1, l

are more general than Chaplygin’s equations. Whence it follows that in thecase of linear stationary transformations (2.17) with respect to quasivelocities

32 II. Nonholonomic Systems

and generalized velocities we can obtain Chaplygin’s equations. ThereforeV. S. Novoselov calls the above equations the equations of Chaplygin’s type.

Similarly, the equations more general than the Hamel–Boltzmann onesare the equations of Hamel–Novoselov [169, 314]

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+

∂T ∗

∂vρ

∗

∂qσ

∂vλ∗

(d

dt

∂vρ

∗

∂qσ−

∂vρ

∗

∂qσ

)= Qλ , λ = 1, l ,

which are obtained also for nonlinear constraints (2.1). V. S. Novoselov callsthese equations the equations of the Voronets–Hamel type (for detail, see § 1Chapter VII).

In the equations of Chaplygin, Hamel–Boltzmann, and those similar tothem, the authors have made an attempt to discriminate the Lagrange oper-ator. Then the rest of addends in the left-hand sides of equations characterizethe nonholonomicity of system. Therefore in the case of the integrability ofconstraints the differential equations become usual Lagrange’s equations ofthe second kind of holonomic mechanics. Equations (2.13), (2.15), and thesimilar ones are generated for the concrete forms of usually linear nonholo-nomic constraints of the type (2.14), (2.16) and therefore they are usefulfor solving the corresponding problems. As a rule, such equations make itpossible to obtain the minimal number of equations of motion. For example,the left-hand sides of Chaplygin’s equations (2.13) involve only the unknownq1, . . . , ql and after integration of these equations the rest of the coordinatesql+1, . . . , qs can be found from equations of constraints (2.14).

As distinct from this, Maggi’s equations are valid, as is mentioned above,for any of nonholonomic constraints, including the constraints nonlinear ingeneralized velocities. An important point is that for generating differentialequations of motion (2.10) we need to apply a single equitype techniqueto all problems: after the choice of the generalized coordinates q1, . . . , qs

the expressions for the left-hand sides of usual Lagrange’s equations of thesecond kind are generated; the transition formulas (2.2) to nonholonomicvariables are introduced, in which case the last of them take account of therelations of nonholonomic constraints by means of formulas (2.5); the inversetransformation (2.3) is found and after differentiating it with respect to newnonholonomic variables, the equations of motion (2.10) are generated. Heretwo remarks can be given which are useful from the computational point ofview.

Firstly, for numerical integrating system (2.10) together with constraints(2.1) it is necessary previously to differentiate the latter with respect to timeand to obtain the equations linear in generalized accelerations. These equa-tions and Maggi’s ones are the system of linear nonuniform algebraic equa-tions in unknown q1, . . . , qs. Solving them, we obtain the system of differentialequations convenient for numerical integration.

Secondly, in the case of nonlinear nonholonomic constraints (2.1) the ob-taining of inverse transformations (2.3) from formulas (2.2) may turn out diffi-cult. To avoid this we need to to compile the matrix of derivatives (∂v

ρ

∗/∂qσ),

2. Equations of motion of nonholonomic systems Maggi’s equations 33

ρ, σ = 1, s, using formulas (2.2), and to find then the inverse matrix (∂qσ/∂vρ

∗),ρ, σ = 1, s, the elements of which are used for generating Maggi’s equations.

Consider one more type of equations of nonholonomic mechanics. In thecase of ideal constraints (2.1) equation (2.9) can be represented as

MW = Y + Λκ

∂ϕκ

∂qτeτ . (2.21)

Multiplying scalarly equation (2.21) by the vectors of fundamental basis eσ,

σ = 1, s, of the original system of curvilinear coordinates, we obtain thefollowing relation

MWσ = Qσ + Λκ

∂ϕκ

∂qσ, σ = 1, s ,

which can be rearranged in the form

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ + Λκ

∂ϕκ

∂qσ, σ = 1, s . (2.22)

These equations are usually called Lagrange’s equations of the second kind

with undetermined multipliers [59]. Together with the equations of nonholo-nomic constraints (2.1) they give a system of s + k differential equations ins + k unknowns qσ, σ = 1, s, Λκ , κ = 1, k. This is the reason why, likeequations (2.22) of Chapter I, they can be called Lagrange’s equations of the

first kind in generalized coordinates for nonholonomic systems [28].If the original system of coordinates are Cartesian, then we have

qσ = yσ , eσ = eσ = jσ , σ = 1, s ,

ϕκ(t, y, y) = 0 , κ = 1, k ,

and equations (2.22) take the form

Myσ = Yσ + Λκ

∂ϕκ

∂yσ

, σ = 1, s . (2.23)

Equations (2.23) are usual Lagrange’s equations of the first kind for nonholo-nomic constraints rearranged for representation point.

F. Udwadia and R. Kalaba [394. 1992] derived the equations of dynamicsin the matrix form taking into consideration the presence of nonholonomicconstraints up to the second order with the help of the Moore and Penrosegeneralized inverse. In their opinion "the equations of motion obtained in thispaper appear to be the simplest and most comprehensive so far discovered".

Note that the partition of the whole s-dimensional space into the di-rect sum of the K-space and L-space by means of constraint equations (2.1)actually corresponds to the application of Moore and Penrose generalized in-verse. (A more general case for the constraints that are linear in generalizedaccelerations is considered in Chapter IV). This partition led to expressions

34 II. Nonholonomic Systems

(2.12) for generalized reactions. Substituting them into equations (2.22),we obtain

Aστ (t, q, q) qτ = Bσ(t, q, q) ,

Aστ = M

(gστ − gσ∗τ

∂qσ∗

∂vl+κ

∗

∂ϕκ

∂qσ

),

Bσ = Qσ − Qσ∗

∂qσ∗

∂vl+κ

∗

∂ϕκ

∂qσ+ M Γσ∗,αβ qαqβ

∂qσ∗

∂vl+κ

∗

∂ϕκ

∂qσ−

−M Γσ,αβ qαqβ , σ, σ∗, τ = 1, s , α, β = 0, s , κ = 1, k .

These formulae do imply the Udwadia–Kalaba equations

qτ = Aτσ(t, q, q)Bσ(t, q, q) , σ, τ = 1, s ,

where Aτσ are elements of the matrix inverse to the matrix (Aστ ). We notethat these equations can be also derived with the help of the linear forcetransformation, which will be introduced in the next chapter, and eliminationof generalized reaction forces Λκ, κ = 1, k, from equations (2.22) in a simillarway as it was described for holonomic systems in § 2 of Chapter I.

V. V. Rumyantsev [203] thinks that the most general equations of nonholo-nomic mechanics are the Poincare–Chetaev equations. They were introducedby H. Poincare [373] and N. G. Chetaev [247, 248, 292] for holonomic sys-tems. The mathematical problems, concerning their structure, and their placein the new theory of Hamiltonian systems were considered by V. I. Arnol’d,V. V. Kozlov, A. I. Neishtadt [7], L. M. Markhashov [149], and others. In thesequel they were generalized to nonholonomic systems due to the works ofL. M. Markhashov [149], V. V. Rumyantsev [203], and Fam Guen [229]. As isshown by V. V. Rumyantsev [203], all the rest of types of the equations of mo-tion for nonholonomic mechanics with the linear nonholonomic constraintsof the first kind can be obtained from the Poincare–Chetaev equations. Weassume that these constraints have the form

vl+κ

∗= al+κ

σ(t, q)qσ + al+κ

0 (t, q) = 0 , κ = 1, k , σ = 1, s . (2.24)

Arguing as in the work [203], we supplement equations (2.24) with the fol-lowing relations

vλ

∗= aλ

σ(t, q)qσ + aλ

0 (t, q) , λ = 1, l , l = s − k , σ = 1, s ,

which implies that the generalized velocities can uniquely be represented as

qσ = bσ

τ(t, q)vτ

∗+ bσ

0 (t, q) , σ, τ = 1, s .

Introducing, for short, the notions [203]

q0 = t , q0 = 1 , v0∗

= 1 , a0α

= b0α

= δ0α

, α = 0, s ,

2. Equations of motion of nonholonomic systems Maggi’s equations 35

we havevα

∗= aα

βqβ , qβ = bβ

αvα

∗, α, β = 0, s .

Denote the Lagrange function L = T − Π, which was found as the functionof variables t, qσ, vσ

∗, σ = 1, s, by L∗(t, q, v∗). In these notions the Poincare–

Chetaev equations for nonholonomic systems with constraints (2.24) are thefollowing [203]:

d

dt

∂L∗

∂vλ∗

= cρ

µλvµ

∗

∂L∗

∂vρ

∗

+ cρ

0λ

∂L∗

∂vρ

∗

+ bσ

λ

∂L∗

∂qσ+ Qλ ,

λ, µ = 1, l, , ρ, σ = 1, s .

(2.25)

Here Qλ = bσ

λQσ are generalized nonpotentional forces, corresponding to the

quasivelocities vλ∗, λ = 1, l, and cρ

µαand c

ρ

0λare the coefficients, given in the

form

cρ

αβ= aρ

γ

(∂b

γ

β

∂qδbδ

α−

∂bγα

∂qδbδ

β

)=

(∂aρ

γ

∂qδ−

∂aρ

δ

∂qγ

)bγ

αbδ

β,

α, β, γ, δ = 0, s , ρ = 1, s .

(2.26)

As V. V. Rumyantsev emphasizes [203], the function L∗, entering into equa-tions (2.25), depends, generally speaking, on all quasivelocities v

ρ

∗ , ρ = 1, s,and therefore the equations of constraints (2.24) vl+κ

∗= 0, κ = 1, k, should

be used only after the generation of equations (2.25). By (2.26) for holonomicconstraints we have cl+κ

αβ=0, κ = 1, k, α, β = 0, s, and in this case the above

remark does not refer to holonomic systems.Equations (2.25), supplemented with the equations

qσ = bσ

λ(t, q)vλ

∗+ bσ

0 (t, q) , λ = 1, l , σ = 1, s ,

are the closed system of equations in unknowns qσ, σ = 1, s, and vλ∗, λ = 1, l.

The number of sought independent variables is minimal and the differentialequations in unknowns as qσ as vλ

∗are of the first kind. This is the advantage

of equations (2.25) in contrast with Maggi’s equations.The Hamel–Novoselov and Poincare–Chetaev equations are also consid-

ered in Chapter VII, where they are obtained by three different approaches.Finally, we give some important remarks.In studying the motion of rigid body linear nonholonomic constraint (2.24)

occurs, in particular, in the case when the projection of the velocity v ofthe point M of rigid body on the direction of the unit vector j of body isequal to zero by virtue of its interaction with another body. This exampleof nonholonomic constraint is the most routine one. Therefore we considerit in more detail. We shall show that the assumption that the consideredconstraint is ideal means that the force, applied to the point M of body inthe result of its interaction with another body, is equal to Λj if the equationof constraint is as follows

ϕ = v · j = aσ(t, q)qσ + a0(t, q) = 0 , σ = 1, s , s 6 .

36 II. Nonholonomic Systems

Here v is the velocity of the point M and qσ,σ = 1, s, are generalized coordi-nates of rigid body. By assumption, the constraint is linear and therefore theunit vector j can only depend on the time t and on the generalized coordinatesqσ, σ = 1, s, but not on the generalized velocities qσ, σ = 1, s.

Equations (2.22) implies that for the proof of such assertion it is sufficientto show that the generalized forces Rσ, corresponding to the force Λj, can berepresented in the form

Rσ = Λ∂ϕ

∂qσ.

Really, by definition, we have

Rσ = Λj ·∂r

∂qσ,

where r = r(t, q) is the radius vector of the point M . The velocity v of thepoint M is as follows

v =∂r

∂t+

∂r

∂qσqσ .

Hence∂r

∂qσ=

∂v

∂qσ, σ = 1, s ,

and we have

Rσ = Λj ·∂v

∂qσ, σ = 1, s .

The vector j is independent of the variables qσ, σ = 1, s. In this case thequantities Rσ, σ = 1, s, can be represented as

Rσ = Λ∂(v · j)

∂qσ= Λ

∂ϕ

∂qσ, σ = 1, s ,

which was to be proved.Consider another example from the dynamics of rigid body, related to the

problem on the controllable motion of rigid body but not the problem on itsrolling or sliding motion. We assume that it is necessary to realize the freemotion, of rigid body, such that the projection of the vector of instantaneousangular velocity ωωω on the fixed axis j is a given function of time t and thegeneralized coordinates qσ, σ = 1, 6. Thus equation of linear nonholonomicconstraint (2.24) is assumed to be given in the form

ϕ = ωωω · j + a0(t, q) = 0 . (2.27)

We shall show that from equations (2.22) it follows that the "ideal"realizationof such program of motion is possible by means of an additional system of

2. Equations of motion of nonholonomic systems Maggi’s equations 37

forces such that its force resultant is equal to zero and the resultant momentabout the center of mass is equal to Λj.

Suppose, ρρρν are the radius-vectors of the points of application of theadditional forces Fν , the number of which is equal to n. By definition, wehave

Rσ =

n∑

ν=1

Fν ·∂ρρρν

∂qσ=

n∑

ν=1

Fν ·∂vν

∂qσ=

n∑

ν=1

Fν ·∂

∂qσ(vC + ωωω × rν) . (2.28)

Here vC is a velocity of the center of mass and rν is a radius vector of thepoint of application of the additional force Fν relative to the center of mass.

By virtue of the problem setting we have

n∑

ν=1

Fν = 0 ,

n∑

ν=1

rν × Fν = Λj . (2.29)

Therefore relations (2.27) and (2.28) yield

Rσ =n∑

ν=1

Fν ·∂(ωωω × rν)

∂qσ=

n∑

ν=1

Fν ·∂ωωω

∂qσ× rν =

=n∑

ν=1

(rν × Fν) ·∂ωωω

∂qσ= Λj ·

∂ωωω

∂qσ= Λ

∂ϕ

∂qσ.

In the above proof the fact that the unit vector j is that of fixed frameis not used. We need only in the fact that this vector is independent of thegeneralized velocities qσ, σ = 1, 6. It can be a vector, which is of a givendependence of time and generalized coordinates, and therefore it can, in par-ticular, be the unit vector, rigidly connected with body.

The essential distinction between the considered problem and the problemon the rolling or sliding motion of rigid body is that the validity of constraint(2.27) can be provided by different families of the additional forces Fν ,ν =1, n, satisfying condition (2.29) while in the problem on the rolling or slidingmotion the validity of constraint is provided by the one additional force Λj,applied to the contact point M . It is also important that the generation of thisforce as the function of the variables t, q, qσ, σ = 1, s, is given by the contactinteraction of two bodies. This force can be eliminated and the motion canbe found from the equations of constraints and, for example, from Maggi’sequations, which do not involve the Lagrange multipliers. For controllablemotion, the generation of the moment Λj is realized by the control systemand only after applying the found control moment Λj the motion can satisfyequation (2.27).

38 II. Nonholonomic Systems

§ 3. The generation of the most usual

forms of equations of motion

of nonholonomic systems from Maggi’s equations

We obtain now the mentioned above forms of equations of motion ofnonholonomic systems from Maggi’s equations.

Chaplygin’s and Voronets’ equations. Suppose that on the systemare imposed the stationary linear nonholonomic constraints, the equations ofwhich take the form

ql+κ = βl+κ

λ(q)qλ, λ = 1, l, κ = 1, k . (3.1)

Then, assuming

vλ

∗= qλ, λ = 1, l,

vl+κ

∗= ql+κ − βl+κ

λ(q)qλ, κ = 1, k ,

we have∂qµ

∂vλ∗

= δµ

λ=

1, µ = λ ,

0, µ = λ ,λ, µ = 1, l ,

∂ql+κ

∂vλ∗

= βl+κ

λ, λ = 1, l, κ = 1, k .

From these relations it follows that for nonholonomic constraints, givenby (3.1), Maggi’s equations (2.10) can be represented as

Mwλ + Mwl+κβl+κ

λ= Qλ + Ql+κβl+κ

λ,

λ = 1, l, κ = 1, k .(3.2)

Suppose that the kinetic energy T is independent of the generalized co-ordinates ql+κ and Ql+κ = 0 (κ = 1, k). Then equations (3.2) have theform

d

dt

∂T

∂qλ−

∂T

∂qλ+ βl+κ

λ

d

dt

∂T

∂ql+κ

= Qλ, λ = 1, l . (3.3)

Transform equation (3.3). By means of equation of constraints (3.1), we elim-inate all velocities ql+κ from the relation for the kinetic energy T , and denoteby T∗ the obtained expression for kinetic energy. In this case the relationshold

∂T∗

∂qλ=

∂T

∂qλ+

∂T

∂ql+κ

∂ql+κ

∂qλ=

∂T

∂qλ+

∂T

∂ql+κ

βl+κ

λ, (3.4)

∂T∗

∂qλ=

∂T

∂qλ+

∂T

∂ql+κ

∂ql+κ

∂qλ=

∂T

∂qλ+

∂T

∂ql+κ

∂βl+κ

µ

∂qλqµ ,

λ, µ = 1, l .

(3.5)

3. The generation of the most usual forms of equations 39

We assume that the coefficients βl+κ

λare independent of ql+κ, κ = 1, k.

Then, differentiating in time relation (3.4), we obtain

d

dt

∂T∗

∂qλ=

d

dt

∂T

∂qλ+ βl+κ

λ

d

dt

∂T

∂ql+κ

+∂T

∂ql+κ

dβl+κ

λ

dt=

=d

dt

∂T

∂qλ+ βl+κ

λ

d

dt

∂T

∂ql+κ

+∂T

∂ql+κ

∂βl+κ

λ

∂qµqµ ,

λ, µ = 1, l .

(3.6)

Computing the quantities d(∂T/∂qλ)/dt and ∂T/∂qλ by formulas (3.6) and(3.5) and substituting them into equations (3.3), we get

d

dt

∂T∗

∂qλ−

∂T∗

∂qλ−

∂T

∂ql+κ

(∂βl+κ

λ

∂qµ−

∂βl+κ

µ

∂qλ

)qµ = Qλ ,

κ = 1, k , λ, µ = 1, l .

(3.7)

These equations were obtained by S.A.Chaplygin [239].Now we eliminate the dependent velocities ql+1, ql+2, ..., ql+k from the ex-

pressions ∂T/∂ql+κ in equations (3.7), using equations of constraints (3.1).Then we get the system of l equations in unknown functions q1, q2, ..., ql.Thus, Chaplygin’s equations permit us to determine q1(t), q2(t), ..., ql(t) in-dependently of constraints (3.1) and to find then the rest of ql+1(t), ql+2(t), ...,ql+k(t) from equations (3.1).

Suppose, the coefficients βl+κ

λsatisfy the following conditions

∂βl+κ

µ

∂qλ−

∂βl+κ

λ

∂qµ= 0 , κ = 1, k , λ, µ = 1, l . (3.8)

According to the assumption that the coefficients βl+κ

λare independent of

ql+κ (κ = 1, k) this implies that they take the form

βl+κ

λ=

∂U l+κ

∂qλ, λ = 1, l , κ = 1, k . (3.9)

Here U l+κ are the functions of coordinates q1, q2, ..., ql. Substituting relations(3.9) into equations (3.1), we obtain

ql+κ = U l+κ(q1, q2, ..., ql) , κ = 1, k .

Thus, the coordinates ql+κ result from the rest. Therefore if conditions(3.8) are satisfied the motion is described by usual Lagrange’s equations.

Now we generate the equations of motion, obtained by P. V. Voronets [41.1901]. Consider a mechanical system with constraints given in the form (3.1)without the additional assumptions, which arrive to Chaplygin’s equations.

40 II. Nonholonomic Systems

In the case when the kinetic energy T depends on all coordinates Maggi’sequations (3.2) are the following

d

dt

∂T

∂qλ−

∂T

∂qλ+

(d

dt

∂T

∂ql+κ

−∂T

∂ql+κ

)βl+κ

λ= Qλ + Ql+κβl+κ

λ,

κ = 1, k , λ = 1, l .

(3.10)

Arguing as above, we reduce these equations to the Voronets ones. Relations(3.5) preserve their form. In accordance with that the coefficients βl+κ

λdepend

now on all qσ, relations (3.6) take the form

d

dt

∂T∗

∂qλ=

d

dt

∂T

∂qλ+ βl+κ

λ

d

dt

∂T

∂ql+κ

+∂T

∂ql+κ

∂βl+κ

λ

∂qµqµ+

+∂T

∂ql+κ

∂βl+κ

λ

∂ql+νβl+ν

µqµ , κ, ν = 1, k , λ, µ = 1, l .

(3.11)

In the considered case, together with (3.5) and (3.11) we need to accountfor the following relations

βl+κ

λ

∂T∗

∂ql+κ

= βl+κ

λ

(∂T

∂ql+κ

+∂T

∂ql+ν

∂βl+νµ

∂ql+κ

qµ

).

This relation, together with relations (3.5) and (3.11), permits us to representequations (3.10) as

d

dt

∂T∗

∂qλ−

∂T∗

∂qλ− βl+κ

λ

∂T∗

∂ql+κ

−∂T

∂ql+κ

βl+κ

λµqµ =

= Qλ + Ql+κβl+κ

λ, λ, µ = 1, l , κ = 1, k ,

(3.12)

where

βl+κ

λµ=

∂βl+κ

λ

∂qµ−

∂βl+κ

µ

∂qλ+

∂βl+κ

λ

∂ql+νβl+ν

µ−

∂βl+κ

µ

∂ql+νβl+ν

λ.

Equations (3.12) are called Voronets’ equations. The equations of motion(3.12) together with equations of constraints (3.1) are the system of differen-tial equations for obtaining the functions qσ(t), σ = 1, s.

In the case of constrained motion of system acted by forces, which havea potential, equations (3.12) take the form

d

dt

∂T∗

∂qλ−

∂(T∗ + U)

∂qλ− βl+κ

λ

∂(T∗ + U)

∂ql+κ

−∂T

∂ql+κ

βl+κ

λµqµ = 0 ,

λ, µ = 1, l , κ = 1, k .

In the partial case when the coordinates ql+1, ql+2, ..., ql+k, correspondingto the eliminated velocities, do not enter into the relations for kinetic andpotential energies in explicit form and also into the equations of constraints,Voronets’ equations (3.12) coincide with Chaplygin’s equations (3.7).

3. The generation of the most usual forms of equations 41

The equations in quasicoordinates (the Hamel–Novoselov,

Voronets–Hamel, and Poincare–Chetaev equations). As is known, theprojections of the vector of instantaneous angular velocity ωωω on the fixed ax-es cannot be regarded as the derivatives with respect to certain new angles,which uniquely determine the position of rigid body. Similarly, it may turnout that the quantities v

ρ

∗ , which are a one-to-one function of the generalizedvelocities qσ, cannot be regarded as derivatives with respect to the certainnew coordinates q

ρ

∗ . Therefore the quantities vρ

∗ are called quasivelocities andthe variables πρ, given by formulas

πρ =

∫t0

t

vρ

∗dt ,

are called quasicoordinates.In the relation for the kinetic energy T the generalized velocities qσ are

changed by the quasivelocities vρ

∗ . The function thus obtained is denoted byT ∗. Consider, which form can have Maggi’s equations, represented as

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vλ∗

= 0 , σ = 1, s , λ = 1, l , (3.13)

when used the function T ∗.Taking into account the relations

∂T ∗

∂vλ∗

=∂T

∂qσ

∂qσ

∂vλ∗

,∂T ∗

∂qσ=

∂T

∂qσ+

∂T

∂qρ

∂qρ

∂qσ,

ρ, σ = 1, s , λ = 1, l ,

we have (d

dt

∂T

∂qσ

)∂qσ

∂vλ∗

=d

dt

(∂T

∂qσ

∂qσ

∂vλ∗

)−

−∂T

∂qσ

d

dt

∂qσ

∂vλ∗

=d

dt

∂T ∗

∂vλ∗

−∂T

∂qσ

d

dt

∂qσ

∂vλ∗

,

(3.14)

∂T

∂qσ

∂qσ

∂vλ∗

=∂qσ

∂vλ∗

(∂T ∗

∂qσ−

∂T

∂qρ

∂qρ

∂qσ

)=

=∂qσ

∂vλ∗

∂T ∗

∂qσ−

∂T

∂qρ

∂qρ

∂qσ

∂qσ

∂vλ∗

.

(3.15)

In the double sum in the right-hand side of relation (3.15) we exchange theindices of summing ρ and σ. As a result we have

∂T

∂qσ

∂qσ

∂vλ∗

=∂qσ

∂vλ∗

∂T ∗

∂qσ−

∂T

∂qσ

∂qσ

∂qρ

∂qρ

∂vλ∗

. (3.16)

Consider the operator

∂

∂πρ=

∂qσ

∂vρ

∗

∂

∂qσ, ρ, σ = 1, s . (3.17)

42 II. Nonholonomic Systems

Under the assumption vρ

∗ = πρ = qρ

∗ it passes into the operator of partialderivative with respect to the new coordinate q

ρ

∗ since we have

∂qσ

∂vρ

∗

∂

∂qσ=

∂qσ

∂qρ

∗

∂

∂qσ=

∂qσ

∂qρ

∗

∂

∂qσ=

∂

∂qρ

∗

.

By (3.17) relation (3.16) takes the form

∂T

∂qσ

∂qσ

∂vλ∗

=∂T ∗

∂πλ−

∂T

∂qσ

∂qσ

∂πλ.

According to relation (3.14) this implies that Maggi’s equations (3.13) takethe form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ−

∂T

∂qσ

(d

dt

∂qσ

∂vλ∗

−∂qσ

∂πλ

)= Q∗

λ,

σ = 1, s , λ = 1, l .

(3.18)

Here

Q∗

λ= Qσ

∂qσ

∂vλ∗

. (3.19)

Equations (3.18) are called, sometimes, Chaplygin’s type equations [169].Consider the partial case when the generalized velocities qσ and the qua-

sivelocities vρ

∗ are related by the following linear uniform stationary relations

vρ

∗= αρ

σ(q)qσ , qσ = βσ

ρ(q)vρ

∗,

ρ, σ = 1, s ,(3.20)

and the equations of constraints are the following

vl+κ

∗≡ αl+κ

σ(q)qσ = 0 , κ = 1, k . (3.21)

In this case, using relations (3.20) and operator (3.17) and taking into accountthat after performing the operations of differentiation it can be assumed thatvl+κ

∗= 0 (κ = 1, k), we have

d

dt

∂qσ

∂vλ∗

=d

dtβσ

λ(q) =

∂βσ

λ

∂qρqρ =

∂βσ

λ

∂qρβρ

µvµ

∗=

= vµ

∗

∂qρ

∂vµ

∗

∂βσ

λ

∂qρ= vµ

∗

∂βσ

λ

∂πµ, ρ, σ = 1, s , λ, µ = 1, l ;

∂qσ

∂πλ=

∂qρ

∂vλ∗

∂qσ

∂qρ=

∂qρ

∂vλ∗

∂βσµ

∂qρvµ

∗=

= vµ

∗

∂βσµ

∂πλ, ρ, σ = 1, s , λ, µ = 1, l .

Then equations (3.18) take the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ−

∂T

∂qσ

(∂βσ

λ

∂πµ−

∂βσµ

∂πλ

)vµ

∗= Q∗

λ,

σ = 1, s , λ, µ = 1, l .

(3.22)

3. The generation of the most usual forms of equations 43

These equations are usually called Chaplygin’s equations in quasicoordinates

[166, 169]. Note that equations (3.18) and (3.22) should be considered to-gether with the equations of nonholonomic constraints.

Equations (3.18) and (3.22) involve as the function T ∗ as the functionT . We reduce now Maggi’s equations (3.13) to the form that involves thefunction T ∗ only. The following relations

∂T

∂qσ=

∂T ∗

∂vρ

∗

∂vρ

∗

∂qσ, ρ, σ = 1, s ,

yield the relation(

d

dt

∂T

∂qσ

)∂qσ

∂vλ∗

=∂qσ

∂vλ∗

d

dt

(∂T ∗

∂vρ

∗

∂vρ

∗

∂qσ

)=

=

(d

dt

∂T ∗

∂vρ

∗

)∂v

ρ

∗

∂qσ

∂qσ

∂vλ∗

+∂T ∗

∂vρ

∗

∂qσ

∂vλ∗

d

dt

∂vρ

∗

∂qσ.

Since∂v

ρ

∗

∂qσ

∂qσ

∂vλ∗

= δρ

λ=

1 , ρ = λ ,

0 , ρ = λ ,

we have (d

dt

∂T

∂qσ

)∂qσ

∂vλ∗

=d

dt

∂T ∗

∂vλ∗

+∂T ∗

∂vρ

∗

∂qσ

∂vλ∗

d

dt

∂vρ

∗

∂qσ. (3.23)

Taking into account the relations

∂T

∂qσ=

∂T ∗

∂qσ+

∂T ∗

∂vρ

∗

∂vρ

∗

∂qσ

and operator (3.17), we obtain

∂T

∂qσ

∂qσ

∂vλ∗

=∂T ∗

∂πλ+

∂T ∗

∂vρ

∗

∂qσ

∂vλ∗

∂vρ

∗

∂qσ.

Then from the above and formulas (3.19) and (3.23) it follows that Maggi’sequations (3.13) can be represented in the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+

∂T ∗

∂vρ

∗

∂qσ

∂vλ∗

(d

dt

∂vρ

∗

∂qσ−

∂vρ

∗

∂qσ

)= Q∗

λ,

ρ, σ = 1, s , λ = 1, l .

(3.24)

Equations (3.18) and (3.24) can be applied to both holonomic and non-holonomic systems with as the linear with respect to velocities ideal con-straints as the nonlinear ones. In the case when the time does not enter intothe kinetic energy and the equations of constraints in explicit form, equations(3.18) and (3.24) were obtained by G. Hamel [314] and in the general caseby V. S. Novoselov [169]. Therefore we shall call these equations the Hamel–

Novoselov ones.

44 II. Nonholonomic Systems

In the case when the quasivelocities are defined by formulas (3.20) andthe constraints are given by equations (3.21) we have

∂qσ

∂vλ∗

d

dt

∂vρ

∗

∂qσ= βσ

λ

dαρσ

dt= βσ

λ

∂αρσ

∂qτqτ = βσ

λβτ

µ

∂αρσ

∂qτvµ

∗,

∂qσ

∂vλ∗

∂vρ

∗

∂qσ= βσ

λ

∂αρτ

∂qσqτ = βσ

λβτ

µ

∂αρτ

∂qσvµ

∗,

ρ, σ, τ = 1, s , λ, µ = 1, l .

Then (3.24) takes the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+ c

ρ

λµvµ

∗

∂T ∗

∂vρ

∗

= Q∗

λ,

cρ

λµ=

(∂αρ

σ

∂qτ−

∂αρτ

∂qσ

)βσ

λβτ

µ,

ρ, σ, τ = 1, s , λ, µ = 1, l .

(3.25)

In the case l = s these equations and the relations for the coefficientscρστ

were obtained first by P. V. Voronets in 1901 [41]. In 1904 for l < s theseresults were obtained once more by G. Hamel [313]. Therefore these equationsare usually called the Voronets–Hamel equations but G. Hamel itself calledthem the Euler–Lagrange equations. We remark that in the literature theyare also called the Hamel–Boltzmann equations.

Together with the works of P. V. Voronets, H. Poincare obtains [373] theequations, which are highly close to equations (3.25). Poincare’s equations

correspond to the case when in equations (3.25) for l = s the coefficients cρστ

are constant and the forces are expressed via the forcing function U :

Q∗

τ= βσ

τ

∂U

∂qσ, σ, τ = 1, s .

In this case equations (3.25) can be represented in the form, suggested byH. Poincare:

d

dt

∂L∗

∂vτ∗

= cρ

στvσ

∗

∂L∗

∂vρ

∗

+ βσ

τ

∂L∗

∂qσ, L∗(q, v∗) = T ∗ + U ,

ρ, σ, τ = 1, s .

(3.26)

When generated equations of motion (3.26) H. Poincare made use of thegroup theory. The approach of Poincare was developed then in the works ofN. G. Chetaev, L. M. Markhashov, V. V. Rumyantsev, and Fam Guen. Theygeneralized Poincare’s equations to the case when the coefficients cρ

στare not

constant and the motion is acted by as potential as nonpotential forces. Be-sides, V. V. Rumyantsev considered the case of nonlinear first-order nonholo-nomic constraints. These equations, describing the motion of nonholonomicsystems, are called the Poincare–Chetaev–Rumyantsev equations. For detail,see Chapter VII.

4. The examples of applications 45

§ 4. The examples of applications

of different kinds equations

of nonholonomic mechanics

E x a m p l e II .1 . The motion of a double-mass system with holonomic

and nonholonomic constraints (the application of Maggi’s equations). Con-sider in the horizontal plane Oxy the motion of two points M1(x1, y1) andM2(x2, y2) with masses m. They are connected by a rigid rod whose mass isignored and length is 2l (Fig. II. 1,a). The other similar examples of the sys-tems of finite numbers of material particles with nonholonomic constraintsare considered in the work [84]. The short runner with clinches (a skate) ishorizontally fixed at the middle point C of the rod at right angles. The run-ner has a knife-edge and therefore it allows the displacement without frictionalong the knife-edge but it does not allow the motion in perpendicular direc-tion. We assume that since the runner is of sufficiently small length and hasclinches, the system can freely rotate about its center.

The following holonomic constraint

(x2 − x1)2 + (y2 − y1)

2 = (2l)2

is imposed on the motion of points. Then the position of system is uniquelydefined by three parameters. As the generalized coordinates we shall regardthe Cartesian coordinates x, y of the middle point of rod and the angle θ

between the direction of the rod and the axis Oz:

q1 = x , q2 = θ , q3 = y . (4.1)

Then we havex1 = x + θl sin θ , y1 = y − θl cos θ ,

x2 = x − θl sin θ , y2 = y + θl cos θ .(4.2)

Now we obtain the equation of nonholonomic constraint. Since the runneris at the point C of the middle of rod, this point may have only the velocityperpendicular to the axis of rod. The projections of velocity of any of twopoints of rigid body on the straight line, passing through these points, areequal. Since there exists a skate, the velocity of middle point of rod v hasno a projection on the axis of rod and therefore the velocities v1 and v2 ofthe points M1 and M2 also have no this projection. This condition can berepresented as

x1

x2=

y1

y2.

With (4.2) this implies that

θ(x cos θ + y sin θ) = 0 .

The above equation is satisfied for θ = 0 or under the condition

x cos θ + y sin θ = 0 . (4.3)

46 II. Nonholonomic Systems

In the case θ = 0 the angle θ is constant and therefore we have a translationalmotion for the linear displacement of the point C. Such a motion is realizedfor a long runner, which opposes the rotation of system round the pointC. Since we consider the case of short runner, the nonholonomic constraintis given by (4.3). The represented in Fig. II. 1,a system with nonholonomicconstraint (4.3) may explain, in particular, the motion, on one skate, of thevertically standing figure-skater and in the case θ = 0 the motion of the skateron racing skates.

Note that constraint (4.3) is satisfied for θ = 0 and for θ = 0. In this casewe cannot assume that the equation θ(x cos θ + y sin θ) = 0 is more generalthan equation (4.3) and therefore we cannot treat its as the example of non-linear nonholonomic constraint. Similarly, the functional relations, obtainedin the other examples of the work [84], cannot also be regarded as nonlinearnonholonomic constraints.

For generation of equations of motion we obtain first the relation for thekinetic energy T . By (4.2) we have

T =m

2

(x2

1 + y21 + x2

2 + y22

)= m

(x2 + y2 + θ2l2

).

This implies the relation

MW1 = 2mx , MW2 = 2ml2θ , MW3 = 2my , (4.4)

where M = 2m is a mass of representation point.In accordance with the general theory the new velocities v1

∗, v2

∗, v3

∗are

introduced by formulas v1∗

= q1 ≡ x, v2∗

= q2 ≡ θ, v3∗

= x cos θ + y sin θ.

Then we obtain

x ≡ q1 = v1∗, θ ≡ q2 = v2

∗, y ≡ q3 =

v3∗− v1

∗cos θ

sin θ. (4.5)

By (4.4), (4.5) Maggi’s equations (2.10) take the form

2mx − Q1 + (2my − Q3)(− ctg θ) = 0 ,

2ml2θ − Q2 = 0 .(4.6)

We remark that the second equation coincides with usual Lagrange’s equationof the second kind, which corresponds to the generalized coordinate θ sincein the equation of nonholonomic constraint (4.3) is lacking the velocity θ.

System of equations (4.6) must be supplemented by the equation of con-straint (4.3). Differentiating it in time, we obtain

x cos θ − xθ sin θ + y sin θ + yθ cos θ = 0 . (4.7)

Solving the system of equations (4.6) and (4.7) as the system of algebraiclinear nonhomogeneous equations in unknowns x, y, θ and representing the

4. The examples of applications 47

obtained results as the system of six first-order differential equations, wehave

x = vx , y = vy , θ = ωz ,

vx = ωz(vx sin θ − vy cos θ) cos θ + (Q1 sin θ − Q3 cos θ) sin θ/(2m) ,

vy = ωz(vx sin θ − vy cos θ) sin θ − (Q1 sin θ − Q3 cos θ) cos θ/(2m) ,

ωz = Q2/(2ml2) .

This normal form of the system of differential equations is convenient to usethe numerical integration methods.

For the computation of generalized reaction of nonholonomic constraintby formula (2.12) we have now the following relation

Λ = (2my − Q3)/ sin θ .

Consider the motion of system acted by the force F = Fxi+Fyj, imposedat the point C, and when there exists the moment N = Nzk. Besides, wealso take into account the resistance forces Fresist

1 = −µv1, Fresist2 = −µv2

(µ = const), applied at the points M1,M2 (Fig. II. 1,a), in which case togeneralized coordinates (4.1) correspond the following generalized forces:

Q1 ≡ Qx = Fx − 2µx , Q2 ≡ Qθ = Nz − 2µl2θ , Q3 ≡ Qy = Fy − 2µy .

For concrete computation we assumed that m = 7 kg, l = 1 m, µ =0.6 N·s/m, Fx = Fy = 2 N. In Fig. II. 1,b are given three trajectories, whichthe point C traces in the time 15 s for Nz1 = 1 N·m, Nz2 = 0.65 N·m,Nz3 = 0.3 N·m. The initial data are zero.

E x a m p l e II .2 . The motion of figure-skater (the application of Chap-lygin’s equations). Now we apply Chaplygin’s equations to the solution of the

Fig. II. 1

48 II. Nonholonomic Systems

Fig. II. 2

following problem: find the motion of lop-sided figure-skater on the short skateA (Fig. II. 2).

We introduce the moving and stationary coordinates; Axyz and Oξηζ,respectively,. The motion is acted by the resistance force Fresist = −κ1vC

and the drag torque Nresist = −κ2ωωω, C is a center of mass of figure-skater.Since the figure-skater can move only along a skate going round at a time,

the constraint, imposed on the system considered, consists in that the velocityof the point A is always directed along the moving axis Ax, i. e. its projectionvAy on the axis Ay is equal to zero at each moment of time. Denote the unitvectors of stationary coordinates Oξηζ by i1, j1, k1 and the coordinates ofa center of gravity in the stationary coordinates by ξC , ηC . The coordinatesof a center of gravity in the moving coordinates Axyz are assumed to be thefollowing: xC = α, yC = β.

As the generalized coordinates of system we regard the coordinates of thepoint A and the angle between the axes Ax and Oξ, namely

q1 = ξ , q3 = η , q2 = θ .

We obtain now the equation of constraint. Represent the constraint in termsof projections of the vector vA on the fixed axis Oξη, taking into accountthat

vA = vAξi1 + vAηj1 = ξi1 + ηj1 .

The projection of the vector vA on the axis Ay has the form

vAy = −ξ sin θ + η cos θ .

Then the equation of constraint vAy = 0 can be written as

ϕ(t, q1, q2, q3, q1, q2, q3) ≡ −ξ sin θ + η cos θ = 0 . (4.8)

The kinetic energy is determined by the Konig theorem:

T =1

2M

[(ξ− θ(α sin θ+β cos θ))2 +(η+ θ(α cos θ−β sin θ))2 +k2

Cθ2

], (4.9)

4. The examples of applications 49

where kC is a radius of inertia of body relative to the axis, passing throughthe center of gravity and perpendicularly to the plane of motion, M is a massof system.

After the transformation in accordance with the equation of constraint,the relation for kinetic energy takes the form

(T )=1

2M

[(ξ−θ(α sin θ + β cos θ))2+(ξ tg θ+θ(α cos θ−β sin θ))2+k2

Cθ2

].

Now we write Chaplygin’s equation in unknown coordinate ξ:

d

dt

∂(T )

∂ξ−

∂(T )

∂ξ+

∂T

∂η

[(∂b3

1

∂ξ−

∂b31

∂ξ

)ξ +

(∂b3

2

∂ξ−

∂b31

∂θ

)θ

]= Qξ . (4.10)

By the above notions Chaplygin’s equation for constraint takes the form

q3 = b31q

1 + b32q

2 , b31 = tg θ , b3

2 = 0 .

In this case equation (4.10) can be represented as

d

dt

∂(T )

∂ξ+

∂T

∂η

(−

∂b31

∂θ

)θ = Qξ .

Using the relation for kinetic energy, we obtain

ξ + ξθ tg θ − θβ cos θ − θ2α cos θ =Qξ cos2 θ

M.

We generate now the equation of motion over the coordinate θ. Performingsimilar numerical computation, we obtain

γ2 cos2 θθ − β cos θξ + (α cos θ − β sin θ)ξθ =Qθ cos2 θ

M,

where γ2 = α2 + β2 + k2C

.The generalized forces, acting on the system, are the following

Qξ = −κ1ξ , Qη = −κ1η , Qθ = −κ2θ . (4.11)

Finally, we obtain a system of differential equations in Chaplygin’s form,describing the motion of a figure-skater in the case when its center of masslies not above the skate:

ξ + ξθ tg θ − θβ cos θ − θ2α cos θ = −κ1ξ cos2 θ/M ,

γ2 cos2 θθ − β cos θξ + (α cos θ − β sin θ)ξθ = −κ2θ cos2 θ/M ,

η = ξ tg θ .

(4.12)

Note that the considered motion of a figure-skater is one of possible in-terpretations of the motion of Chaplygin’s sledge. One more problem, relatedto Chaplygin’s sledge, is considered in Appendix D.

50 II. Nonholonomic Systems

E x a m p l e II .3 . The motion of figure-skater (application of Maggi’sequations). We generate Maggi’s equations for the problem considered inExample II. 2. The frames of reference and the generalized coordinates areintroduced as above. Then the relations for the kinetic energy T and thecovariant components of generalized forces Qξ, Qθ, Qη are given by formulas(4.9) and (4.11). The equation of constraint (4.8) has the form

ξ tg θ − η = 0 . (4.13)

We introduce the new nonholonomic variables in the following way:

v1∗

= ξ , v2∗

= θ , v3∗

= ξ tg θ − η .

Having performed the change of the old variables to the new ones, we obtainthe following inverse transformation

ξ = v1∗, θ = v2

∗, η = v1

∗tg θ − v3

∗.

Using these formulas, we can compute the derivatives:

∂q1

∂v1∗

= 1 ,∂q2

∂v1∗

= 0 ,∂q3

∂v1∗

= tg θ ,

∂q1

∂v2∗

= 0 ,∂q2

∂v2∗

= 1 ,∂q3

∂v2∗

= 0 ,

∂q1

∂v3∗

= 0 ,∂q2

∂v3∗

= 0 ,∂q3

∂v3∗

= 1 .

Using the computed coefficients in Maggi’s equations (2.10) and performingsome simplifications, we obtain the differential equations of motion for thesystem

ξ + η tg θ − θβ

cos θ− θ2 α

cos θ= −

κ1

M(ξ + η tg θ) ,

γ2θ + η(α tg θ − β sin θ) − ξ(α sin θ + β cos θ) = −κ2

Mθ .

(4.14)

These equations should be integrated together with equation of constraint(4.13).

Compare the obtained results with those, get in Example II. 2. Using themethod of Chaplygin, we change in system (4.14) the quantities η and η totheir expressions from the equation of nonholonomic constraint (4.13). Thenwe have

ξ + tg θ

(ξ tg θ + ξθ

1

cos2 θ

)− θ

β

cos θ− θ2 α

cos θ= −

κ1

M(ξ + ξ tg2 θ) ,

γ2θ +

(ξ tg θ + ξθ

1

cos2 θ

)(α cos θ − β sin θ)−

−ξ(α sin θ + β cos θ) = −κ2

Mθ .

4. The examples of applications 51

After the transformations we arrive at the system

ξ1

cos2 θ+ ξθ

tg θ

cos2 θ− θ

β

cos θ− θ2 α

cos θ= −

κ1

Mξ ,

γ2θ − ξβ

cos θ+ ξθ

(α cos θ − β sin θ)

cos2 θ= −

κ2

Mθ .

It is easily remarked that multiplying these equations by cos2 θ, we obtainChaplygin’s equations (4.12), generated in Example II. 2.

Thus, Maggi’s equations give a more simple method to find the equationsof motion than Chaplygin’s equations, in which case it is not required thatthe mechanical system satisfies additional conditions. It is sufficient only togenerate the relations for the kinetic energy and generalized forces, to choicerationally new nonholonomic variables, to find the derivatives of inverse trans-formation, and to construct the linear combinations of the Lagrange opera-tors. Besides, by equation (2.12) we can easily write the relations for gener-alized reactions of nonholonomic constraints. For the considered problem weobtain

Λ

M= ξ tg θ + ξθ

1

cos2 θ+ θ(α cos θ− β sin θ)− θ2(α sin θ + β cos θ) +

κ1

Mξ tg θ .

In Fig. II. 3 are given the results of numerical integrating the system ofdifferential equations for 10 s. Here we assumed that

γ2 = 0.07 m2 , κ1/M = 1 s−1 , κ2/M = 0.02 m2 · s−1 ,

ξ(0) = 0 , ξ(0) = 5 m · s−1 , η(0) = 0 , η(0) = 0 ,

θ(0) = 0 , θ(0) = 12.5 s−1 , α = 0 , β = 0 .

Fig. II. 3

52 II. Nonholonomic Systems

Fig. II. 4

E x a m p l e II .4 . The motion of car in a sweep (the application of theHamel–Boltzmann equations). Consider a motion of car (Fig. II. 4), consistingof a body of the mass M1 and a front axis with the mass M2. Suppose, theyhave the moments of inertia J1 and J2 about the vertical axes through theircenters of mass, respectively. The front axis can rotate about their verticalaxis through its center of mass. The masses of wheels and backward axis,regarded as separate parts, are assumed to be negligible. The motion of caris subject to the force F1(t), acting along its longitudinal axis Cx, and tothe moment L1(t), rotating the front axis. In this case F1(t), L1(t) are thegiven functions of time. In addition, we take into account the resistance forceF2(vC), acting in the direction opposite to the direction of the velocity vC

of the center of mass C of body, the drag torque L2(θ), which is applied tothe front axis and is opposite to the angular velocity of its rotating, and therighting moment L3(θ). A similar scheme was considered in the work [132] asthe simplified mathematical model for the car motion on a sweep. At presentit can be of interest when studying wheeled robot vehicles [146-148, 423].

We generate the Hamel–Boltzmann equations for the study of the motionof this system.

The motion of car in the horizontal plane is considered in the fixed coordi-nates Oξηζ. The car position is given by the following generalized coordinates:q1 = ϕ is an angle between the longitudinal axis of car Cx and the axis Oξ,q2 = θ is an angle between the front axis and the perpendicular to the axisCx, and q3 = ξC , q4 = ηC are coordinates of the point C.

On the motion of car the two nonholonomic constraints are imposed whichexpress that the sideways sliding motion of the backward and front axes ofcar is missed. Their equations can be written similarly to formula (4.8) fromExample II. 2:

−ξB sin ϕ + ηB cos ϕ = 0 ,

−ξA sin(ϕ + θ) + ηA cos(ϕ + θ) = 0 .(4.15)

4. The examples of applications 53

Here ξA,ηA,ξB ,ηB are the coordinates of the centers of mass for the front andbackward axes of car. We assume that the distances between the centers ofmass of these axes and the center of gravity of the body of car are equal to l1and l2, respectively. Then the equations of nonholonomic constraints (4.15)take the form

ϕ1 ≡ −ξC sin ϕ + ηC cos ϕ − l2ϕ = 0 ,

ϕ2 ≡ −ξC sin(ϕ + θ) + ηC cos(ϕ + θ) + l1ϕ cos θ = 0 .(4.16)

We introduce the quasivelocities by formulas

π1 = ϕ , π2 = θ ,

π3 = −ξC sin ϕ + ηC cos ϕ − l2ϕ ,

π4 = −ξC sin(ϕ + θ) + ηC cos(ϕ + θ) + l1ϕ cos θ ,

(4.17)

i. e. in formulas (2.17) the coefficients aρσ(q), ρ, σ = 1, 4, have the form

a11 = 1 , a2

2 = 1 , a31 = −l2 , a3

3 = − sin ϕ , a34 = cos ϕ ,

a42 = l1 cos θ , a4

3 = − sin(ϕ + θ) , a44 = cos(ϕ + θ) .

To formulas (4.17) corresponds the inverse transformation

q1 ≡ ϕ = π1 , q2 ≡ θ = π2 ,

q3 ≡ ξC = b31π

1 + b33π

3 + b34π

4 ,

q4 ≡ ηC = b41π

1 + b43π

3 + b44π

4 ,

(4.18)

whereb31 = (l1 cos ϕ cos θ + l2 cos(ϕ + θ)/ sin θ ,

b33 = cos(ϕ + θ)/ sin θ , b3

4 = − cos ϕ/ sin θ ,

b41 = (l1 sin ϕ cos θ + l2 sin(ϕ + θ))/ sin θ ,

b43 = sin(ϕ + θ)/ sin θ , b4

4 = − sin ϕ/ sin θ .

The rest of coefficients aρσ

and bσρ, are equal to zero.

Thus, in transformation (2.17) the matrices (aρσ) and (bσ

ρ) are obtained.

Now we can compute the coefficients of nonholonomicity by formulas (2.19).We have

γ133 = −γ331 = b33 cos ϕ + b4

3 sin ϕ ,

γ134 = −γ431 = b34 cos ϕ + b4

4 sin ϕ ,

γ241 = −γ142 = l1 sin θ + b31 cos(ϕ + θ) + b4

1 sin(ϕ + θ) ,

γ143 = −γ341 = γ243 = −γ342 = b33 cos(ϕ + θ) + b4

3 sin(ϕ + θ) ,

γ144 = −γ441 = γ244 = −γ442 = b34 cos(ϕ + θ) + b4

4 sin(ϕ + θ) .

(4.19)

The rest of quantities γλ(l+κ)λ∗ , are equal to zero.

54 II. Nonholonomic Systems

The kinetic energy of system consists of the kinetic energies of the bodyand the front axis and is computed by formula

2T = M∗(ξ2C

+ η2C

) + J∗ϕ2 + J2θ2 + 2J2ϕθ + 2M2l1ϕ(−ξC sin ϕ + ηC cos ϕ) ,

M∗ = M1 + M2 , J∗ = J1 + J2 + M2l21 .

(4.20)The generalized forces, acting on the car, can be represented as

Q1 ≡ Qϕ = 0 ,

Q2 ≡ Qθ = L1(t) − L2(θ) − L3(θ) ,

Q3 ≡ QξC= F1(t) cos ϕ − F2(vC)ξC/vC ,

Q4 ≡ QηC= F1(t) sin ϕ − F2(vC)ηC/vC , vC =

√ξ2C

+ η2C

.

(4.21)

Then by (2.18) we have

Q1 = (F1(t) cos ϕ − F2(vC)ξC/vC)b31 + (F1(t) sin ϕ − F2(vC)ηC/vC)b4

1 ,

Q2 = L1 − L2 − L3 ,

Q3 = (F1(t) cos ϕ − F2(vC)ξC/vC)b33 + (F1(t) sin ϕ − F2(vC)ηC/vC)b4

3 ,

Q4 = (F1(t) cos ϕ − F2(vC)ξC/vC)b34 + (F1(t) sin ϕ − F2(vC)ηC/vC)b4

4 .

(4.22)Using formulas (4.18) and (4.20) we can construct the relation for T ∗:

2T ∗ = π21

(M∗

((β3

1)2 + (β41)2

)+

+ J∗ + M2l21 + 2M2l1

(β4

1 cos ϕ − β31 sin ϕ

))+

+ π22J2 + π2

3

(M∗

((β3

3)2 + (β43)2

))+ π2

4

(M∗

((β3

4)2 + (β44)2

))+

+ π1π22J2 + π1π3

(2M∗

(β3

1β33 + β4

1β43

)+ 2M2l1

(β4

3 cos ϕ − β33 sin ϕ

))+

+ π1π4

(2M∗

(β3

1β34 + β4

1β44

)+ 2M2l1

(β4

4 cos ϕ − β34 sin ϕ

))+

+ π3π4

(2M∗

(β3

3β34 − β4

3β44

)).

Omitting some tedious calculations and using formulas (2.20), (4.19), and(4.22), we obtain the Hamel–Boltzmann equations (2.15) for the problemconsidered:

4. The examples of applications 55

[J∗ + M2l

21 + 2M2l1l2 + M∗

(l22 + l21 cos2 θ + 2l1l2 cos2 θ

sin2 θ

)]ϕ + J2θ−

−(l1 + l2)

2M∗ cos θ

sin3 θϕθ =

=

(F1(t) cos ϕ −

F2(vC)ξC

vC

)b31 +

(F1(t) sin ϕ −

F2(vC)ηC

vC

)b41 ,

J2(ϕ + θ) = L1(t) − L2(θ) − L3(θ) .

(4.23)Note that we need to solve this system together with equations of con-straints (4.16).

As an example, we consider the motion of the hypothetical compact motorcar with

M1 = 1000 kg , M2 = 110 kg , J1 = 1500 kg · m2 ,

J2 = 30 kg · m2 , l1 = 0.75 m , l2 = 1.65 m

and with the following power characteristics:

F1(t) = 2500 N , F2(vC) = κ2vC , κ2 = 100 N · s · m−1 ,

L1(t) = 15 N · m , L2(θ) = κ1θ , κ1 = 0.5 N · m · s ,

L3(θ) = κ3θ , κ3 = 100 N · m .

The results of the numerical solution of nonlinear system of differentialequations (4.16), (4.23) are given in Fig. II. 5. In computing we make use ofthe following initial data

ϕ(0) = 0 , ϕ(0) = 0 , θ(0) = π/180 rad , θ(0) = 0 , ξC(0) = 0 ,

ξC(0) = 0.00176856 m · s−1, ηC(0) = 0 , ηC(0) = 0.000018008 m · s−1 .

Fig. II. 5

56 II. Nonholonomic Systems

E x a m p l e II .5 . The turning movement of a car (the application ofMaggi’s equations and the Lagrange equations of the first kind). Considernow the motion of car from Example II. 4 by means of Maggi’s equations.We make use of the same curvilinear coordinates. Then the equations ofconstraints take the form (4.16) and the kinetic energy and the generalizedforces are given by formulas (4.20) and (4.21), respectively.

We introduce the following new nonholonomic variables

v1∗

= ϕ , v2∗

= θ ,

v3∗

= −l2ϕ − ξC sin ϕ + ηC cos ϕ ,

v4∗

= l1ϕ cos θ − ξC sin(ϕ + θ) + ηC cos(ϕ + θ)

and write the inverse transformation

q1 ≡ ϕ = v1∗, q2 ≡ θ = v2

∗,

q3 ≡ ξC = β31v1

∗+ β3

3v3∗

+ β34v4

∗,

q4 ≡ ηC = β41v1

∗+ β4

3v3∗

+ β44v4

∗,

(4.24)

whereβ3

1 = (l1 cos ϕ cos θ + l2 cos(ϕ + θ)/ sin θ ,

β33 = cos(ϕ + θ)/ sin θ , β3

4 = − cos ϕ/ sin θ ,

β41 = (l1 sin ϕ cos θ + l2 sin(ϕ + θ))/ sin θ ,

β43 = sin(ϕ + θ)/ sin θ , β4

4 = − sin ϕ/ sin θ .

(4.25)

In this case the first Maggi’s equation has the form

(MW1 − Q1)∂q1

∂v1∗

+ (MW3 − Q3)∂q3

∂v1∗

+ (MW4 − Q4)∂q4

∂v1∗

= 0 . (4.26)

Since the equations of constraints do not involve the velocity θ, the secondMaggi’s equation is Lagrange’s equation of the second kind:

MW2 − Q2 = 0 . (4.27)

The relations MWσ can be computed in terms of the kinetic energy byformulas

MWσ =d

dt

∂T

∂qσ−

∂T

∂qσ, σ = 1, 4 .

Then by (4.20), (4.21), (4.24), (4.25) equations of motion of car (4.26), (4.27)can be represented as

[J∗+M2l1(l1−β3

1 sin ϕ+β41 cos ϕ)

]ϕ+J2θ + (M∗β3

1−M2l1 sin ϕ)ξC+

+(M∗β41 + M2l1 cos ϕ)ηC = M2l1ϕ

2(β31 cos ϕ + β4

1 sin ϕ)+

+[F1(t) cos ϕ − F2(vC)ξC/vC

]β3

1 +[F1(t) sin ϕ − F2(vC)ηC/vC

]β4

1 ,

J2(θ + ϕ) = L1(t) − L2(θ) − L3(θ) .

(4.28)

4. The examples of applications 57

If the initial data and the analytic representations of the functions F1(t),F2(vC), L1(t), L2(θ), L3(θ) are given, then after numerical integrating non-linear system of differential equations (4.16), (4.28), we find the law of motionof car

ϕ = ϕ(t) , θ = θ(t) , ξC = ξC(t), ηC = ηC(t) . (4.29)

Now we can determine generalized reactions. For the second group ofMaggi’s equations we have

Λ1 = (MW3 − Q3)∂q3

∂v3∗

+ (MW4 − Q4)∂q4

∂v3∗

,

Λ2 = (MW3 − Q3)∂q3

∂v4∗

+ (MW4 − Q4)∂q4

∂v4∗

,

or the same in the expanded form:

Λ1 =[M∗ξC−M2l1(ϕ sin ϕ+ϕ2 cos ϕ)−F1(t) cos ϕ+F2(vC)ξC/vC ]β33+

+[M∗ηC +M2l1(ϕ cos ϕ−ϕ2 sin ϕ)−F1(t) sin ϕ+F2(vC)ηC/vC ]β43 ,

Λ2 =[M∗ξC−M2l1(ϕ sin ϕ+ϕ2 cos ϕ)−F1(t) cos ϕ+F2(vC)ξC/vC ]β34+

+[M∗ηC +M2l1(ϕ cos ϕ−ϕ2 sin ϕ)−F1(t) sin ϕ+F2(vC)ηC/vC ]β44 .

Taking into account (4.29), we obtain the law of variation of the general-ized reactions Λi = Λi(t), i = 1, 2. Using these functions we can study theconditions, under which nonholonomic constraints (4.16) are satisfied. If thereaction forces are equal to the Coulomb friction forces, then these constraintsneed not be satisfied and the car can begin to slide along their axes. Notethat in Appendix D the motion of nonholonomic systems in the absence ofreaction forces of nonholonomic constraints is considered, and in AppendixE the turning movement of a car with possible side slipping of its wheels isstudied.

Thus, Maggi’s equations can be generated in the same easy manner asLagrange’s equations of the second kind. For ideal nonholonomic constraints,Maggi’s equations decompose into two groups. The first group, together withthe equations of constraints, permits one to find the law of motion of a non-holonomic system and then the generalized reactions can be determined fromthe second group. We notice that for generating the Hamel–Boltzmann equa-tions it is required much greater calculations than for Maggi’s equations.

It is of interest to compare the obtained Maggi’s equations (4.28) with theHamel–Boltzmann equations (4.23). The second equations of these systemsare coincide. If we obtain the relations ξC and ηC from the equations ofconstraints and substitute them into the first equation of system (4.28), thenwe obtain the first equation of system (4.23).

We could also write the Lagrange equations of the first kind in curvilinearcoordinates (see equations (2.22) of the present Chapter). In the problemconsidered they have a form:

58 II. Nonholonomic Systems

J∗ϕ + J2θ − M2l1ξC sin ϕ + M2l1ηC cos ϕ = −Λ1l2 + Λ2l1 cos θ ,

J2(θ + ϕ) = L1(t) − L2(θ) − L3(θ) ,

M∗ξC − M2l1ϕ sin ϕ − M2l1ϕ2 cos ϕ =

= F1 cos ϕ − L2(θ) − Λ1 sin ϕ − Λ2 sin(ϕ + θ) ,

M∗ηC + M2l1ϕ cos ϕ − M2l1ϕ2 sin ϕ =

= F1 sin ϕ − k2ηC + Λ1 cos ϕ + Λ2 cos(ϕ + θ) .

The four equations given include four unknown generalized coordinates andtwo unknown Lagrange multipliers, so they have to be solved together withthe equations of constraints (4.16). This is characteristic (peculiar) for theLagrange equations of the first kind. If we differentiate in time the equa-tions of constraints and with the help of them eliminate generalized reactionforces, then we get Maggi’s equations of motion (4.28), as well as formulasfor defining Λ1 и Λ2.

E x a m p l e II .6 . The rolling of ellipsoid on a rough plane (the genera-tion of Maggi’s equations). It is to be noted that the specific form of Maggi’sequations depends essentially on the choice of the variables v

ρ

∗ . By successfulchoice we can considerably simplify calculations, concerning the reduction ofthe problem to the system of differential equations, represented in normalform.

As an example, consider the rolling of homogeneous rigid body of ellip-soidal form on the fixed plane. The center of ellipsoid coincident with centroidis taken as the origin of the moving coordinates Cxyz, the axes of which arerigidly fixed with the axes of body (Fig. II. 6). We assume that the plane π,on which the ellipsoid rolls, coincides with the plane Oξη of the fixed sys-tem of coordinates Oξηζ. Denote by ξ, η, ζ the coordinates of the center of

Fig. II. 6

4. The examples of applications 59

ellipsoid relative in the fixed frame. The velocity of the contact point p canbe computed by formula

vp = vC + ωωω ×−−→CP.

For the rolling without sliding the velocity of the point p is equal to zeroand therefore the equation of constraint can be represented as

vC + ωωω ×−−→CP = ξiξ + ηiη + ζiζ +

∣∣∣∣∣∣

iξ iη iζωξ ωη ωζ

ξ0 η0 ζ0

∣∣∣∣∣∣= 0 . (4.30)

Here ξ0, η0, ζ0 are the coordinates of the points P in the frame with ξ1η1ζ1,the axes of which ξ1, η1, ζ1 are parallel to the axes ξ, η, ζ of the fixed coordi-nates, respectively. It can be shown that the quantities ξ0, η0, ζ0 are computedby formulas

− ξ0ζ = (a2 − b2) sin θ cos ψ sin ϕ cos ϕ+

+ (c2 − a2 sin2 ϕ − b2 cos2 ϕ) sin ψ cos θ sin θ ,

− η0ζ = (a2 − b2) sin ψ sin θ sin ϕ cos ϕ+

+ (a2 sin2 ϕ + b2 cos2 ϕ − c2) cos ψ cos θ sin θ ,

ζ0 = −ζ = −

√a2 sin2 θ sin2 ϕ + b2 sin2 θ cos2 ϕ + c2 cos2 θ .

Here a, b, c are the semiaxis of ellipsoid; ψ, θ, ϕ are the Euler angles, givingthe orientation of the system of coordinates Cxyz with respect to the framewith ξ1η1ζ1.

Vector equation (4.30) is equivalent to three scalar equations for nonholo-nomic constraints:

ϕ1 ≡ ξ + ωηζ0 − ωζη0 = 0 ,

ϕ2 ≡ η + ωζξ0 − ωξζ0 = 0 ,

ϕ3 ≡ ζ + ωξη0 − ωηξ0 = 0 .

(4.31)

As generalized Lagrangian coordinates we regard the coordinates ξ, η, ζ ofthe center of mass and the Euler angles ψ, θ, ϕ. The numerical computationof the kinetic energy of ellipsoid in these coordinates is based on applyingthe Konig theorem. Then we have

T =M

2(ξ2 + η2 + ζ2) +

Jωω2

2.

The quantity Jωω2 can be represented as

Jωω2 = Aω2x

+ Bω2y

+ Cω2z,

where A,B,C are the moments of inertia of the ellipsoid about the axes x, y, z,

respectively. Since the ellipsoid, by assumption, is a homogeneous rigid body,we get

A =M(b2 + c2)

5, B =

M(c2 + a2)

5, C =

M(a2 + b2)

5.

60 II. Nonholonomic Systems

The projections ωx, ωy, ωz of the vector ωωω on the axes of moving coordinatesCxyz are the following:

ωx = ψ sin θ sin ϕ + θ cos ϕ ,

ωy = ψ sin θ cos ϕ − θ sin ϕ ,

ωz = ψ cos θ + ϕ .

These formulas allow one to compute the covariant components of thevector MW

MWξ = Mξ , MWη = Mη , MWζ = Mζ ,

MWϕ =d

dt

∂T

∂ϕ−

∂T

∂ϕ, MWψ =

d

dt

∂T

∂ψ−

∂T

∂ψ, MWθ =

d

dt

∂T

∂θ−

∂T

∂θ.

Since the explicit relations for Wϕ, Wψ and Wθ are lengthy, they are omitted.The quantities ωξ, ωη, ωζ , entering into the equations of constraints (4.31),

take the formωξ = ϕ sin ψ sin θ + θ cos ψ ,

ωη = −ϕ cos ψ sin θ + θ sin ψ ,

ωζ = ϕ cos θ + ψ .

In this case if we assume that v1∗

= ξ, v2∗

= η, v3∗

= ζ, v3+κ

∗= ϕκ , κ = 1, 3,

then by reason of the complicated dependence of the functions ϕκ on thevelocities qσ the relations ∂qσ/∂vλ

∗turn out highly awkward and therefore

final Maggi’s equations are also complicated. The task turn out more simpleif as the free variables vλ

∗we choose the angular velocities ωξ, ωη, ωζ . It can

be shown that if the quasivelocities vρ

∗ are given by formulas

v1∗

= ωξ , v2∗

= ωη , v3∗

= ωζ ,

v4∗

= ξ + ωηζ0 − ωζη0 , v5∗

= η + ωζξ0 − ωξζ0 , v6∗

= ζ + ωξη0 − ωηξ0 ,

then we get∂ξ

∂ωξ

= 0 ,∂η

∂ωξ

= ζ0 ,∂ζ

∂ωξ

= −η0 ,

∂ϕ

∂ωξ

=sin ψ

sin θ,

∂ψ

∂ωξ

= −sin ψ cos θ

sin θ,

∂θ

∂ωξ

= cos ψ ,

∂ξ

∂ωη

= −ζ0 ,∂η

∂ωη

= 0 ,∂ζ

∂ωη

= ξ0 ,

∂ϕ

∂ωη

= −cos ψ

sin θ,

∂ψ

∂ωη

=cos ψ cos θ

sin θ,

∂θ

∂ωη

= sin ψ ,

∂ξ

∂ωζ

= η0 ,∂η

∂ωζ

= −ξ0 ,∂ζ

∂ωζ

= 0 ,

4. The examples of applications 61

∂ϕ

∂ωζ

= 0 ,∂ψ

∂ωζ

= 1 ,∂θ

∂ωζ

= 0 .

Substituting these relations into Maggi’s equations, we obtain them in explicitform.

This example demonstrates that the problems on the rolling of one bodyon the surface of the other are complicated even under the assumption thatthe constraint, given by equation (4.30), is ideal. The dynamics of the rigidbody, contacting with a rigid surface, is considered in the treatise ofA. P. Markeev [143]. The new theory of the interaction of a rolling rigid bodywith a deformable surface is supposed by V. F. Zhuravlev [70].

E x a m p l e II .7 . Equations of motion and the technical realization

of the Appell–Hamel problem (the generation of Maggi’s equations and La-grange’s equations of the first kind in the case of nonlinear nonholonomicconstraints).

The example of P. Appell [270, 271] on the motion of a special nonholo-nomic system (Fig. II. 7, a) was of fundamental importance for the develop-ment of Analytic mechanics. This problem has intensively been considered,especially in journal "Rendiconti del circolo matematico di palermo"(1911–1912). Some works were due to E. Delassus. He considered the example ofAppell in more detail in the work [298] and in his book [299]. This problemwas also studied by G. Hamel [315, p. 502–505]. Until the present time theproblem of Appell–Hamel is of interest for scientists (see, for example, theworks [274, 376, 408]).

In the example of Appell–Hamel the motion of a disk with knife-edge onthe horizontal plane Oξη is considered. The horizontal axis of disk throughits center C is fixed in a weightless frame, the stubs of which can slide onthe plane without friction (Fig. II. 7, a). The frame prevents the overturn ofdisk. The disk is rigidly fixed with a coaxial cylinder. The nonstretchablethread, which is turned over the two blocks fixed with the frame, is windedon the cylinder. To the end of thread the mass m is hung, the descent ofwhich results in the rolling of disk. The axis of the mass descend is spaced atρ from the contact point B of the disk with horizontal plane. We also assumethat the frame is fixed with the parallel to BC smooth rail, preventing theswinging of mass. The disk and cylinder have the radii a and b, respectively.

Denote the angle between the plane of the rolling of disk and the axis Oξ

by θ, the angle of rotation of the wheel about its axis by ϕ, the coordinatesof the mass m by x, y, z, the coordinates of the point B by ξ, η. Obviously,the coordinates are related as

x = ξ + ρ cos θ , y = η + ρ sin θ . (4.32)

On the motion of system it is imposed the linear nonholonomic constraints

ξ = aϕ cos θ , η = aϕ sin θ , (4.33)

z = −bϕ . (4.34)

62 II. Nonholonomic Systems

Fig. II. 7

Taking into account constraints (4.33) and (4.34), G. Hamel generates theequations of motion for the considered system [315]. Further, he analyzes thelimiting case as ρ → 0. In this case it is necessary to consider only the changeof the coordinates x, y, z of the mass m, in which case the following nonlinearnonholonomic constraint occurs:

ϕ1 ≡ x2 + y2 −a2

b2z2 = 0 . (4.35)

P. Appell also considered a similar passage to the limit, introducing the pa-rameter α, which is the quotient of the moment of inertia of the disk aboutits diameter to the quantity ρ.

The problem of Appell–Hamel was considered most completely and inmore detail by Yu. I. Neimark and N. A. Fufaev in the paper [164] entering intothe book [166], which became the classical monograph on the nonholonomicmechanics. They notice [166, p. 227, 228] that ". . . the system, consideredby P. Appell and G. Hamel, with nonlinear nonholonomic constraints can beobtained from the nonholonomic system with linear constraints by meansof the passage to the limit as ρ → 0. However in this case we have thedepression of the system of differential equations, i. e. their degeneracy andtherefore it is not known in advance wether the motions of the limit system(ρ = 0) coincide with the limit motions of nondegenerate system as ρ → 0.Therefore the question remains open: to what extent the equations of motionfor degenerated system describe properly the motion of the original systemwith the vanishingly small ρ?". The authors performed the "investigation,which was based on the study of the motions of nondegenerate system for

4. The examples of applications 63

Fig. II. 8

ρ > 0 and ρ < 0, the limit motions of nondegenerate system as |ρ| → 0,and the motions of degenerated system. This investigation implies that themotions of degenerated system differs essentially from the limit motions andtherefore the example of nonholonomic system with nonlinear nonholonomicconstraints is incorrect".

Thus, when used the mentioned above passage to the limit, P. Appell andG. Hamel in place of the study of the original system investigated the degen-erated system. We regard the motion of the obtained degenerated system asthe separate problem of mechanics: there is the mass m with the coordinatesx, y, z, on the motion of which is imposed nonlinear nonholonomic constraint(4.35). Note that in the model of P. Appell and in the corresponding modelof V. S. Novoselov [171] when the mass is connected with the disk by the setof inertialess pinions (Fig. II. 7, b) the case ρ = 0 can easily be carried outtechnically (Fig. II. 8, a, b). However for ρ = 0 in the mentioned above modelsthe satisfaction of constraints (4.33) remains essential. In this case the con-straints yield the following relation, imposed on the projections of velocitiesof the mass m:

y = x tg θ . (4.36)

Here we take into account that for ρ = 0 by virtue of formulas (4.32) we havex = ξ, y = η. In studying the degenerated system constraint (4.36) is notaccounted and constraint (4.35) is only introduced such that, by assumption,the velocity of the center of disk can have any direction. This means that, byusing constraints (4.35) only, we replace the motion of disk by the motion ofball.

64 II. Nonholonomic Systems

Thus, in studying the degenerated system it should be required that con-straint (4.36) is satisfied, i. e., together with the coordinates x, y, z, it is nec-essary to look to the change of the variable θ. The neglect of the masses ofdisk, frame, and blocks results in the degeneracy of system and therefore thevariable θ turns out to be the “massless” coordinate. If this coordinate is ig-nored, then it is impossible to describe the motion of massless ball by meansof the motion of massless disk.

The technical realization is obviously hard in the case when the connec-tion between the velocity of descending the load and the velocity of the centerof ball is provided by means of the nonstretchable thread or the system ofpinions. However it is possible to study the motion of the mass m with thecoordinates x, y, z under the condition that constraint (4.35) is satisfied only.For this purpose we consider the following control problem: the motion of ma-terial point with the mass m is realized in such a way that by virtue of (4.35)its vertical velocity is varied proportionally to the velocity of the motion ofits trace in horizontal plane. The realization of such problem can be providedby means of the new technical means. We generate Maggi’s equations andLagrange’s equations of the first kind, using exactly such problem setting.

So, we consider the problem on the space motion of material point withnonlinear nonholonomic constraint (4.35). In this case the generalized coor-dinates are the following

q1 = x , q2 = y , q3 = z . (4.37)

We introduce the following new nonholonomic variables:

v1∗

= x , v2∗

= y , v3∗

= x2 + y2 −a2

b2z2 . (4.38)

In the Appell–Hamel problem Maggi’s equations take the form

(MW1 − Q1)∂q1

∂v1∗

+ (MW2 − Q2)∂q2

∂v1∗

+ (MW3 − Q3)∂q3

∂v1∗

= 0 ,

(MW1 − Q1)∂q1

∂v2∗

+ (MW2 − Q2)∂q2

∂v2∗

+ (MW3 − Q3)∂q3

∂v2∗

= 0 ,

(MW1 − Q1)∂q1

∂v3∗

+ (MW2 − Q2)∂q2

∂v3∗

+ (MW3 − Q3)∂q3

∂v3∗

= Λ .

(4.39)

These relations involve the derivatives∂qσ

∂vρ

∗

, σ, ρ = 1, 3, for numerical com-

putation of which it is required the transformation inverse to transforma-tion (4.38). However, the obtaining of such transformation is a difficult tasksince nonholonomic constraint (4.35) is nonlinear. Therefore we obtain thesederivatives in the following way. Compute the matrix

(αρ

σ) =

(∂v

ρ

∗

∂qσ

), σ, ρ = 1, 3 .

4. The examples of applications 65

By (4.38) we have

α11 = 1 , α1

2 = 0 , α13 = 0 ,

α21 = 0 , α2

2 = 1 , α23 = 0 ,

α31 = 2x , α3

2 = 2y , α33 = −2a2b−2z .

Find the matrix (βσρ) inverse to the matrix (αρ

σ). Then we obtain

β11 = 1 , β1

2 = 0 , β13 = 0 ,

β21 = 0 , β2

2 = 1 , β23 = 0 ,

β31 = h2x/z , β3

2 = h2y/z , β33 = −h2/(2z) , h2 = b2/a2 .

(4.40)

It is important for us that (βσρ) =

(∂qσ

∂vρ

∗

).

For the considered problem we have

T = m(x2 + y2 + z2)/2 , Π = mgz .

Then, using formulas (4.40), we can generate Maggi’s equations (4.39):

x + (mz + mg)(h2x/z) = 0 , (4.41)

my + (mz + mg)(h2y/z) = 0 , (4.42)

(mz + mg)(−h2/(2z)) = Λ . (4.43)

If we now consider Lagrange’s equations of the first kind

mx = Λ2x , my = Λ2y , mz = −mg + Λ(−2z/h2) , (4.44)

then it is easily seen that they are the linear combination of equations (4.41)–(4.43).

We need to solve Lagrange’s equations of the first kind (4.44) togetherwith the equation of constraint (4.35), in which case the fact that the un-knowns involve the reaction Λ makes to be rather sophisticate the solution.At the same time the determining of the motion itself from equations (4.35),(4.41), (4.42) turns out more simple since the reaction can be found fromequation (4.43). In addition, using Maggi’s equation, the form of reactioncan be obtained also in original curvilinear coordinates (4.37).

In fact, in the case of ideal nonholonomic constraint (4.35) we have

R = N = Λ∇∇∇′ϕ1 = Λ∂ϕ1

∂qτeτ = (mz +mg)(−h2/z)(xi+ yj−h−2zk) . (4.45)

By (4.45) the direction of horizontal constraint (4.35) is opposed to the hor-izontal component of velocity of the mass m, what is distinctive feature of

66 II. Nonholonomic Systems

the motion of ball. Consider also nonholonomic constraint (4.36), which isconvenient to rewrite as

ϕ2 ≡ y − x tg θ = 0 . (4.46)

Then, by reason of the existence of constraint (4.46), together with the reac-tion R given by (4.45), we also need to consider the reaction

R∗ = Λ∗∇∇∇′ϕ2 = Λ∗(− tg θi + j) .

The latter provides the satisfaction of constraint (4.46) and is distinctivefeature of the motion of disk.

In the case of the massless coordinate θ the value of this angle for ρ = 0does not enter into the system of equations of motion and therefore it isnatural to consider the rolling of the ball in place of the disk. For ρ = 0 wemight consider the rolling of the massless disk but in this case there mustexists the mechanism for the disk to be oriented in the corresponding way.The indefiniteness, which occur in determining the angle θ, can be put outif either ρ = 0 is satisfied for the massless disk, frame, and blocks, either forρ = 0 with due regard any of these masses. In place of the account of theirmasses we can evidently consider not the material point of the mass m butthe body of the same mass, which, descending in the rail mentioned above,rotates together with the frame about the axis BC.

The examples of employing the Hamel–Novoselov equations are given,in particular, in the work [65] and the examples of the application of thePoincare–Chetaev equations in the works [149, 203, 229].

§ 5. The Suslov–Jourdain principle

Consider the vector

V = vσeσ , vσ ≡ qσ , σ = 1, s . (5.1)

In the general case this vector V differs from the velocity of representationpoint since

V =∂y

∂t+

∂y

∂qσqσ =

∂y

∂t+ vσeσ .

Above the new variables vρ

∗ were introduced by formulas (2.2) in place of thevariables vσ ≡ qσ. By assumption, there is also inverse transformation (2.3).We emphasize that in the mentioned above transformations the time t andthe coordinates qσ are regarded as parameters. Introduce the variations δ′vσ

and δ′vρ

∗ of the variables vσ and vρ

∗ , defining them as the partial differentialsof these functions, related as

δ′vσ =∂vσ

∂vρ

∗

δ′vρ

∗, δ′vρ

∗=

∂vρ

∗

∂vσδ′vσ , ρ, σ = 1, s . (5.2)

5. The Suslov–Jourdain principle 67

Recall that in formulas (2.2) we make use of relations (2.5) and therefore

δ′vl+κ

∗=

∂ϕκ

∂vσδ′vσ = 0 , σ = 1, s , κ = 1, k , (5.3)

and formulas (5.2) take the form

δ′vσ =∂vσ

∂vλ∗

δ′vλ

∗, δ′vλ

∗=

∂vλ∗

∂vσδ′vσ , σ = 1, s , λ = 1, l . (5.4)

Consider the vector

δ′V = δ′vσeσ =∂vσ

∂vλ∗

δ′vλ

∗eσ = δ′vλ

∗εεελ , (5.5)

and construct together with the vector V, given by relation (5.1), the newvector

V = V + δ′V = (vσ + δ′vσ)eσ = (qσ + δ′vσ)eσ .

We substitute the coordinates qσ +δ′vσ of the generalized velocity V into theequations of constraints (2.1) and expand the functions ϕκ (as the functionof the variables qσ only) in the Taylor series in the neighborhood of the pointwith the coordinates (q1, . . . , qs), corresponding to time t:

ϕκ(t, q, q + δ′v) = ϕκ(t, q, q) +∇∇∇′ϕκ · δ′V + o(|δ′V|) , κ = 1, k . (5.6)

These relations imply that if for the point with the coordinates (q1, . . . , qs)

at time t the generalized velocity V is kinematically possible, then with anaccuracy to the first order the velocity V = V + δ′V is also kinematicallypossible under the condition

∇∇∇′ϕκ · δ′V = 0 , κ = 1, k . (5.7)

Thus, a set of the vectors δ′V, satisfying equation (5.7), describes the

kinematically possible changes, of the generalized velocity V, allowed byconstraints at time t when the system is in the position (q1, . . . , qs). The

arbitrary vector δ′V, satisfying relations (5.7), is called a variation of the

generalized velocity V.Since the variations δ′vλ

∗are linear independent, the family of Maggi’s

equations (2.10) is equivalent to one equation

(MWσ − Qσ

)∂qσ

∂vλ∗

δ′vλ

∗= 0 ,

which by virtue of formulas (5.4) can be represented as

(MWσ − Qσ

)δ′vσ = 0 (5.8)

68 II. Nonholonomic Systems

or by (5.5) in the vector form:

(MW − Y

)· δ′V = 0 . (5.9)

It is important for us that these equations are independent of the choice of freevariables vλ

∗. They are obtained as a sequence of equations of motion (2.10)

and therefore as a sequence of Newton’s equation (2.9), written for idealnonholonomic constraints with reaction (2.8). We remark that by formulas(5.9), (2.8), (2.9) we have

R · δ′V = 0 , (5.10)

i. e. the reaction of ideal nonholonomic constraints is orthogonal to the vectorof variation of generalized velocity.

We obtain now Maggi’s equations from relation (5.9) being regarded as

the initial one. Since the vector δ′V has the form (5.5), then scalar product(5.9) is as follows

(MWσ − Qσ

)∂qσ

∂vλ∗

δ′vλ

∗= 0 ,

which implies in accordance with linear independence of variations that δ′vλ∗,

λ = 1, l. So, we arrive at Maggi’s equations (2.10).Thus, relation (5.9) can be regarded as a differential variational principle

of mechanics, according to which for systems with ideal retaining nonholo-nomic constraints the scalar product of the vector of constraint reaction bythe variation of generalized velocity is equal to zero. This principle has beenformulated in 1908–1909 by P. Jourdain [326] and in 1900 by G. K. Suslov [218],who named it a universal equation of mechanics. That is why it is reasonableto refer this principle as the Suslov–Jourdain principle.

E x a m p l e II .8 . The equations of motion for Novoselov’s reducer (thegeneration of equations of motion by means of the Suslov–Jourdain princi-ple). We generate the equations of motion for friction reducer, which wasconsidered first by V. S. Novoselov [170]. The reducer (Fig. II. 9) transmitsthe rotation from shaft 1 to shaft 2 and consists of disk A, rigidly fixed withshaft 1, wheel B, freely rotating on shaft 3, shaft 2 with cylinder C, and acentrifugal regulator by the masses K and N and a spring with the deflectionrate c1. The motion of muff D of regulator with the help of a cable, turnedover fixed blocks O1 and O2, and a spring with the deflection rate c2 resultsin the displacement of shaft 3 with wheel B and changes the distance ρ

between the average circle of wheel B and the axis of shaft 1. Wheel B hasthe radius a. The sizes given are the following: PN = NL = LK = KP = l.

The position of the friction reducer is given by the following generalizedcoordinates: the rotation angles of shafts q1 =ϕ1 and q2 =ϕ2 and the distanceq3 =x of muff D from joint L. As is shown in Fig. II. 9, the distance ρ is relatedto x as

x − ρ = c ≡ const .

5. The Suslov–Jourdain principle 69

Fig. II. 9

We consider a system with the nonholonomic constraint

ϕ(t, q1, q2, q3, q1, q2, q3) ≡ (x − c)ϕ1 − Rϕ2 = 0. (5.11)

If the sliding is missed, then constraint (5.11) gives the condition that therotational velocities of the points of contact of wheel B with disk A andcylinder C are equal.

The kinetic and potential energies are defined as

T =1

2

[JAϕ2

2+JC ϕ22+mDx2+mB ρ2+JB

R2

a2ϕ2

2+

+2mN

((l2−

x2

4

)ϕ2

2+l2x2

4l2−x2

)],

Π =1

2c1(δ1 + x − x0)

2 +1

2c2(δ2 + x0 − x)2 .

Here δ1, δ2 are the static deformations of springs with the deflection rates c1

and c2, respectively, x0 is the static declination of muff D from joint L.For this system the Suslov–Jourdain principle is as follows

(MW1 − Q1)δ′ϕ1 + (MW2 − Q2)δ

′ϕ2 + (MW3 − Q3)δ′x = 0 . (5.12)

The relation between the variations of velocities have the form

∂ϕ

∂ϕ1δ′ϕ1 +

∂ϕ

∂ϕ2δ′ϕ2 = 0 . (5.13)

Therefore in equation (5.12) the variations δ′ϕ2 and δ′x are independent.Express from relation (5.13) the variation δ′ϕ1 in terms of δ′ϕ2 and then

70 II. Nonholonomic Systems

from equation (5.12) we obtain

(MW1 − Q1)R

x − c+ (MW2 − Q2) = 0 , (5.14)

MW3 − Q3 = 0 . (5.15)

Here Q1 = M1, Q2 = −M2 are the moments of forces, impressed upon shafts1 and 2, respectively, and Q3 = −∂Π/∂x.

From the general theory it follows that the equations obtained coin-cide with Maggi’s equations. We remark that the second of them is usualLagrange’s equation of the second kind since the coordinate x is holonomic.

Since

MWσ =d

dt

∂T

∂qσ−

∂T

∂qσ, σ = 1, 3 ,

then equations (5.14) and (5.15) can be rewritten as

JA

R

x − cϕ1 + J(x)ϕ2 − mNxxϕ2 = M1

R

x − c− M2 ,

m(x)x+1

2mNxϕ2

2++2l2x

(4l2+x2)2mN x2 = c1(−δ1−x+x0)+c2(−x+x0+δ2) .

(5.16)Here

J(x) = JC + JB

R2

a2+

1

2mN (4l2 − x2) ,

m(x) = mB + mD +2mN l2

4l2 − x2.

The equations of motion (5.16) together with equation of constraint (5.11)give a closed system for determining the functions ϕ1(t), ϕ2(t), x(t).

It is to be noted that if we substitute equation of constraint (5.11), dif-ferentiated with respect to time, into the first equation of system (5.16),then the equations can be written in the Appell form. Such equations werealso generated by A. I. Lur’e [135]. This example was also considered byYa. L. Geronimus [47]. His results coincide with equations (5.16).

E x a m p l e II .9 . The motion of mechanical system with a fluid fly-

wheel (the generation of the equations of motion for nonholonomic systemswith the help of the Suslov–Jourdain principle and Lagrange’s equation ofthe first kind in generalized coordinates). The fluid flywheel consists of twocentrifugal wheels filled by oil: a driving torus and a turbine. The drivingtorus is fixed with the motor shaft and while rotating, by means of bladesand centrifugal force, it speeds up the oil, which falls with a great velocity onthe blades of turbine, setting the latter in motion. The turbine is placed onthe shaft of consumer and therefore by means of the fluid flywheel the rota-tion is imparted from the leading shaft to the driven one, in which case theconnection between them turns out nonrigid. At present the fluid flywheelsgain a wide application in different powerful transmissions, in the starters of

5. The Suslov–Jourdain principle 71

Fig. II. 10

gas turbines, the drives of pumps, the conveyors of hoisting machines, and soon. The study of transient processes in similar arrangements is of importancesince the transient regime averages about sixty percents of their operationtime.

Consider one of possible approaches to the study of transient processesin systems with fluid flywheel or with fluid converter, which differs from thefluid flywheel by an additional wheel (of reactor). We shall regard mechanicalsystems with hydrotransmissions as first-order nonholonomic systems. Thispermits us to eliminate the reaction and to obtain the equation of motion,which is to be integrated together with the equation of constraint.

Denote by ω1 and L1 an angle velocity and a moment generated by amotor, respectively, by J1 a moment of inertia of impeller and driving partsof motor, by ω2 and L2 an angle velocity and a drag torque produced bycustomer, respectively, by J2 a moment of inertia of a turbine and drivenparts of device.

Suppose that for the racing of system the following characteristics of themotor L1 = L1(ω1) and the customer L2 = L2(ω2) hold which taken offfor the regimes being steady-state (see. Fig. II. 10; these and all subsequentnumerical data are taken from the work [106]). The quantities L1, L2 aregiven in Newton-meters, (N·m), ω1, and ω2 in the seconds in the minus firstdegree (s−1), t in seconds (s).

The process of the racing of system can be partitioned into three stage. Atthe first stage after a starting of motor its moment L1, applied to the drivingparts of device, is used for their racing and for the racing of fluid in the fluidflywheel. When the flow occurs in the work chamber of fluid flywheel themoment L occurs on its fixed driving torus. At the end of the first stage thevalue of the moment L becomes sufficient for the initiation of driven parts(L2 for ω2 = 0) and we have the second stage when the turbine begins torotate with the ascending angle velocity ω2.

The racing of system at the third stage is characterized by increasing theangle velocity ω2 with deacceleration of flow, in which case the moment L,generated by the turbine, is greater than the moment, which the driving torus

72 II. Nonholonomic Systems

Fig. II. 11

imparts to a flow. At this stage the moment L generally decreases. Whenits value for the certain angle velocity ω2 becomes equal to the moment ofcustomer L2, the racing is ended and the system operates in the steady-stateregime for L1 = L = L2.

The analysis of experimental and computed investigations of racing pro-cesses for the systems with different relative moments of inertia J = J1/J2

and with different characteristics of a motor and a customer shows that inview of a great power of motor the angle velocity of driving shaft variesslightly and the angle velocity of driven shaft varies essentially at the initialperiod and tends asymptotically to a certain constant value as the mode ofoperation tends to the steady-state regime when we have ω1 = ω2 + const.For the racing of system with hydrotransmission the graphs of the functionsω1 = ω1(t) and ω2 = ω2(t) have a specific form shown in Fig. II. 11. Thesegraphs demonstrate that for as nonstationary as stationary regimes we haveω1 = ω2. Therefore between the angle velocities of the driving and drivenshafts there exists a certain functional relation, which can be regarded as anonholonomic constraint. Since from the graphs of the functions ω1 = ω1(t)and ω2 = ω2(t) we can obtain the relation for the angle velocities as a functionof time, then the equation of nonholonomic constraint can be represented as

ϕ(t, ω1, ω2) ≡ ω2 − i(t)ω1 = 0 . (5.17)

The kinetic energy of system is defined by the following relation

T =J1ω

21

2+

J2ω22

2.

Let us generate the equations of motion.The Suslov–Jourdain principle for this system takes the form

(MW1 − Q1)δ′ω1 + (MW2 − Q2)δ

′ω2 = 0 . (5.18)

The relation between the variations of angle velocities is given by the followingformula

∂ϕ

∂ω1δ′ω1 +

∂ϕ

∂ω2δ′ω2 = 0 . (5.19)

5. The Suslov–Jourdain principle 73

Hence from equation (5.18) with relations (5.17) and (5.19) we have

J1dω1

dt+ i(t)J2

dω2

dt= L1 − i(t)L2 . (5.20)

Equation (5.20) together with equation of constraint (5.17) give the closedsystem for determining the functions ω1(t) and ω2(t).

Equation of motion (5.20) can also be obtained in a different way. Wewrite Lagrange’s equations of the first kind in generalized coordinates fornonholonomic system (2.22):

d

dt

∂T

∂ω1−

∂T

∂ϕ1= L1 + R1 ,

d

dt

∂T

∂ω2−

∂T

∂ϕ2= −L2 + R2 .

(5.21)

Here the generalized reactions are defined as

R1 = Λ∂ϕ

∂ω1= −i(t)Λ , R2 = Λ

∂ϕ

∂ω2= Λ .

Eliminating Λ from system (5.21), we obtain equation of motion (5.20).When passing to the stationary regime, the relation ω1 = ω2 + const is

satisfied, in which case we have

R1|ω1=ω2+const = −Λ∂ϕ

∂ω1

∣∣∣ω1=ω2+const

= −Λ ,

i. e. −R1 = R2. The model suggested coincides with the model considered inthe work [106] since in the equations of this work

L1 = (J1 + J∗)dω1

dt+ L , L = J2

dω2

dt+ L2 ,

the correction J∗ε1, which accounts for the moment, initiating the racing offlow in rotor blades, is equal to zero. In the above equations R1 = −R2 = −L.In the work [106] the quantity J∗ and the moment L, transmitted by a fluidflywheel, are accounted empirically.

Having solved the system of equations (5.17), (5.20), we can find thereaction and, by that, the moment developed by the fluid flywheel.

The system of equations (5.17), (5.20) was numerically integrated by com-puter. The computing permits us to obtain the following relations: the behav-ior of the angle velocities of the driving and driven shafts in time, the changeof the moments on the driving and driven shafts, the behavior of the moment,transmitted by fluid flywheel, in time. The certain results of computing arerepresented in Fig. II. 12.

Thus, the method suggested permits us, using the experimental dataω1 = ω1(t), ω2 = ω2(t), to describe nonstationary processes in systems with

74 II. Nonholonomic Systems

Fig. II. 12

hydrotransmission. In this case we take into account an error of experimentonly while in the other methods, moreover, the additional inaccuracy occursby reason of using the approximate theory for the account of hydrodynamicprocesses.

§ 6. The definitions of virtual

displacements by Chetaev

As is noted in the review of the main stages of the development of non-holonomic mechanics, the definition of nonholonomic system was introducedfirst by Hertz [317] in 1894. He was the first to pay attention to the pos-sible existence of such kinematical constraints, which do not impose somerestrictions on the possibility of the passing of system from one position toanother. The development of nonholonomic mechanics was essentially dueto the work of E. Lindelof [352], in which with the help of usual methods ofholonomic mechanics there were obtained the incorrect equations of motionof nonholonomic system. In particular, S. A. Chaplygin points to this errorand suggests his own method [239] to obtain the equations of motion. Theerror similar to Lindelof’s error was also made by C. Neumann [366], whatwas repeatedly remarked in the literature (see, for example, [41]). Further, in1899, C. Neumann gives already the certainly valid equations of motion [366].

For the description of motions of nonholonomic systems, P. Appell,L. Boltzmann, V. Volterra, P. V. Voronets, G. Hamel, J. Gibbs,Bl. Dolaptschiew, G. A. Maggi, L. M. Markhashov, J. Nielsen, V. S. Novoselov,G. S. Pogosov, A. Przeborski, V. V. Rumyantsev, J. Schouten, Fam Guen,N. Ferrers, J. Tzenoff, S. A. Chaplygin, M. F. Shul’gin, and another authorssuggest a number of different methods to generate the differential equationsof motion. Some of them are considered, for example, in the treatises [59,166]. At the time of intensive development of nonholonomic mechanics manyscholars often obtained similar results and therefore the same forms of equa-tions of motion have different names. The investigations, connected with the

6. The definitions of virtual 75

possible applications of these equations to more wide classes of nonholonomicconstraints, are being continued to the present (see, for example, [370]).

In generating the equations of motion of nonholonomic systems, most au-thors made use of the D’Alembert–Lagrange principle that they extended tothe case of the system under consideration. In this case they need to clari-fy what should be regarded as a virtual displacement for the given type ofconstraint. V. V. Kozlov [114, p. 60, 61] notes that ". . . for such method forconstructing the dynamics, the hypotheses must involve the definition of vir-tual displacements"and "even in the simplest case of stationary integralableconstraint the definition of virtual displacements is the independent hypoth-esis of dynamics". Exactly such hypothesis (see below (6.3)) was brightlyformulated by N. G. Chetaev. He aims [245, p. 68] ". . . at the introduction ofthe notion of virtual displacement for nonlinear constraints in such a waythat to save as D’Alembert’s, as Gaussian principles . . . ".

For the sake of generality, we consider an arbitrary mechanical system,the position of which is given by the generalized coordinates qσ, σ = 1, s.Suppose that on this system is imposed the following nonlinear nonholonomicconstraints

ϕκ(t, q, q) = 0 , κ = 1, k , k < s . (6.1)

We remark that the nonholonomicity of these constraints makes itselfevident in the fact that in spite of their existence the passage of systemfrom any its position with the coordinates qσ

0 , σ = 1, s, to another with thecoordinates qσ

1 , σ = 1, s, is kinematically possible.According to N. G. Chetaev for the real motion of the system considered

the D’Alembert–Lagrange principle

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 (6.2)

must be satisfied. We assume that the kinetic energy T has the form

T =M

2gαβ(t, q)qαqβ , α, β = 0, s , q0 = t , q0 = 1

and the generalized forces Qσ are given in terms of the functions of the time t,the coordinates qσ, and the generalized velocities qσ (σ = 1, s).

N. G. Chetaev also assume that the quantities δqσ, entering into theD’Alembert–Lagrange principle (6.2), satisfy the following conditions

∂ϕκ

∂qσδqσ = 0 , κ = 1, k . (6.3)

Nonlinear nonholonomic constraints (6.1) such that conditions (6.3) are as-sumed to be valid, were named the constraints of the Chetaev type.

As is shown in the previous section, the general principle of nonholo-nomic mechanics is the Suslov–Jourdain principle (5.9) or (5.8) in which thevariations of velocity must satisfy conditions (5.3). The virtual displacements,

76 II. Nonholonomic Systems

allowed by the constraints of the Chetaev type, are satisfied exactly such con-ditions (6.3). Therefore the generalized principle of D’Alembert–Lagrange,which in the case of Chetaev’s postulate permits us to apply the usualD’Alembert–Lagrange principle (6.2) to the study of nonholonomic systems,coincides with the Suslov–Jourdain principle. The comparison of formulas(6.3) and (5.3) implies, in turn, that the virtual displacements (δq1, . . . , δqs),introduced by Chetaev for nonlinear nonholonomic constraints, coincide withthe variations of the generalized velocity (δ′v1, . . . , δ′vs). Like the holonomicproblems from formula (5.10) it follows that the reaction of constraints ofthe Chetaev type is orthogonal to the virtual displacements satisfying condi-tions (6.3).

Differential forms (6.3) show that the scalar products of the vectors δy =δqσeσ by the vectors ∇∇∇′ϕκ ≡ εεεl+κ, κ = 1, k, are equal to zero. Formulas (5.7)stress even this orthogonality.

Thus, conditions (6.3) suggested by N. G. Chetaev, which give the ax-iomatic definition of virtual displacements with nonlinear nonholonomic con-straints, show the possibility of the transition, in the nonholonomic mechan-ics, from the vectors given on the manifold of possible positions of mechanicalsystem to the vectors given on the manifold of possible velocities of system.

Relations (6.3) have played an important role in the development of non-holonomic mechanics. The different forms of conditions of type (6.3) werealso introduced by the other famous scholars, for example, by J. W. Gibbs[309], P. Appell [265], and A. Przeborski [375]. J. Papastavridis [370. 1997,2002] refers to conditions (6.3) as the Maurer–Appell–Chetaev–Hamel defi-nition. Note, that much attention to this definition of virtual displacementswith nonlinear nonholonomic constraints is paid in the works of Norwegianscientist L. Johnsen [324], which are not too famous.

We shall repeatedly remark further the role of Chetaev’s type constraintsfor obtaining the cited results.

C h a p t e r III

LINEAR TRANSFORMATION OF FORCES

In this Chapter the linear transformation of forces is introduced. In this

case for holonomic systems the notion of ideal constraints and the relation

for virtual elementary work are applied. By the transformation of forces, La-

grange’s equations of the first and second kinds are obtained. The theorem of

holonomic mechanics is formulated by which the given motion over the giv-

en curvilinear coordinate can be provided by an additional generalized force

corresponding to this coordinate. For nonholonomic systems the linear trans-

formation of forces is introduced applying Chetaev’s postulates. In this case

with the help of generalized forces, corresponding to the equations of con-

straints, the family of fundamental equations of the nonholonomic mechanics

is obtained in compact form. The theorem of holonomic mechanics is for-

mulated according to which the given change of quasivelocity can be provided

by one additional force corresponding to this quasivelocity. The application

of the formulated theorems of the holonomic and nonholonomic mechanics is

demonstrated on the example of the solution of two problems on the control-

lable motion connected with the flight dynamics. At the end of this chapter

the linear transformation of forces is used to obtain the Gauss principle.

§ 1. Some general remarks

In § 1 of Chapter I for studying the motion of representation point underconstraints (1.10)

fκ(t, y) = 0 , y = (y1, . . . , y3N ) , κ = 1, k , (1.1)

the following differential equations (1.8) are used:

Myµ = Yµ + Rµ , µ = 1, 3N . (1.2)

In equations (1.1) and (1.2), the general number of which is equal to 3N +k,the unknowns are the functions yµ and the reactions Rµ, µ = 1, 3N . Thus,the number of unknowns exceeds the number of equations by 3N − k. Therearises the natural question wether the present problem can be solved and wecan attain that the solution of problem is unique and the number of equationsis equal to the number of unknown. It turns out that for the answer to thatimportant question of analytic mechanics the type of constraints: holonomicor nonholonomic, is nonessential. Therefore for the sake of generality, weassume that the constraints can be given by either the equations

ϕκ(t, y, y) = 0 , κ = 1, k , (1.3)

77

78 III. Linear Transformation of Forces

or the equations

ψκ(t, y, y, y) = al+κ

µ(t, y, y) yµ + al+κ

0 (t, y, y) = 0 ,

κ = 1, k , µ = 1, 3N .(1.4)

Note that the functions ϕκ(t, y, y) can also depend nonlinearly on velocities.Differentiating of equations of constraints (1.1) twice and equations (1.3)

once with respect to time, we can represent all types of equations of con-straints in uniform differential form (1.4). Then for holonomic constraints(1.1) we obtain

al+κ

µ=

∂fκ

∂yµ

, al+κ

0 =∂2fκ

∂t2+ 2

∂2fκ

∂t ∂yµ

yµ +∂2fκ

∂yµ ∂yµ∗

yµyµ∗ ,

κ = 1, k , µ, µ∗ = 1, 3N

and for nonholonomic constraints (1.3) we have

al+κ

µ=

∂ϕκ

∂yµ

, al+κ

0 =∂ϕκ

∂t+

∂ϕκ

∂yµ

yµ ,

κ = 1, k , µ = 1, 3N .

As is known [194, 295], given nonholonomic constraints, it is impossible,in principle, to introduce the less number of new variables via which thecoordinates yµ, µ = 1, 3N , can uniquely be expressed. Therefore we can usethe equations of constraints only for that to change from the unknowns Rµ

to the new unknowns Λκ , the number of which must be equal to the numberof constraints and via which the reactions Rµ can uniquely be expressed.

This problem of analytic mechanics can be solved in the following way. Atfirst we suppose that the constraints are lacking. Then the vector equationof motion is as follows

MW = Y .

For concrete initial data it permits us to find uniquely a subsequent motionof mechanical system when the force Y is a given function of time, a positionof system, and its velocities.

Suppose now that the force Y is lacking and the system moves mechan-ically. At time t = t0 when the system is in a position with the coordinatesyµ0 and has the projections of velocities yµ0 (µ = 1, 3N), the constraintsare imposed on the motion of system. According to the releasability principlethis leads to the occurrence of the force of reaction R. Therefore, beginningfrom time t = t0, Newton’s second low can be written as

MW = R .

This equation allows us to find uniquely the subsequent motion in the casewhen by the equations of constraints the vector R can be found as the func-tion of time, position, and velocities of system.

1. Some general remarks 79

In the case of simultaneous operation of active forces and the forces, gen-erated by constraints, Newton’s secondlaw takes the form

MW = Y + R . (1.5)

Obviously, in this case for the given initial data the subsequent motion canalso be found only when the force R is represented as a function of time,coordinates, and velocities of system. Clear up how this can be made.

Consider the vectors

εεεl+κ = al+κ

µjµ , l = 3N − k , κ = 1, k . (1.6)

For holonomic and nonholonomic constraints (1.1), (1.3), (1.4) vectors(1.6) can be represented, respectively, in the form

εεεl+κ =∂fκ

∂yµ

jµ = ∇∇∇fκ ,

εεεl+κ =∂ϕκ

∂yµ

jµ = ∇∇∇′ϕκ , (1.7)

εεεl+κ =∂ψκ

∂yµ

jµ = ∇∇∇′′ψκ , l = 3N − k , κ = 1, k , µ = 1, 3N .

The introduction of the vectors εεεl+κ permits us to represent equations ofconstraints (1.4) in the following way:

εεεl+κ · W = χκ(t, y, y) , χκ = −al+κ

0 , κ = 1, k . (1.8)

These relations imply that in the 3N -dimensional Euclidean space we use itis rational to consider a subspace, the basis of which are the vectors εεεl+κ,κ = 1, k. In this case the 3N -dimensional Euclidean space can be representedas a direct sum of this subspace and the orthogonal to it complement, thebasis of which are the vectors εεελ, λ = 1, l, satisfying the relations

εεεl+κ · εεελ = 0 , κ = 1, k , λ = 1, l .

Denote the introduced subspaces by K-space and L-space. These subspacespermit us to represent the acceleration of representation point as the followingsum

W = WL + WK , WL = Wλ εεελ , WK = Wl+κ εεεl+κ ,

WL · WK = 0 .(1.9)

It is to be noted that this representation of the vector W corresponds to thefixed values of the variables t, yµ, yµ (µ = 1, 3N).

In fact, the idea of introduction of K-space is based on the Chetaev postu-late on the ideality of nonlinear nonholonomic constraints [245]. This postu-late can be interpreted as the requirement of the orthogonality of the vectorsof virtual displacements of system to the vectors εεεl+κ , κ = 1, k.

80 III. Linear Transformation of Forces

Using relations (1.9), we can replace equation (1.5) by two followingequations

MWL = YL + RL , (1.10)

MWK = YK + RK . (1.11)

HereYL = Qλ εεελ , YK = Ql+κ εεεl+κ ,

RL = Rλ εεελ , RK = Λκ εεεl+κ .

The covariant components Rl+κ of the vector RK are denoted by Λκ sincein the sequel they turns out equal to the Lagrange multipliers.

Vector equation (1.11) is equivalent to the following k scalar equations

Λκ = MWl+κ − Ql+κ , κ = 1, k . (1.12)

We show that the vector WK , given by the components Wl+κ , is com-pletely determined by the equations of constraints. Really, relations (1.8) and(1.9) give

hκκ∗

Wl+κ∗ = χκ(t, y, y) , κ, κ∗ = 1, k , (1.13)

wherehκκ

∗

= εεεl+κ · εεεl+κ∗

, κ, κ∗ = 1, k .

If|hκκ

∗

| = 0 , (1.14)

then the solution of system of linear algebraic equations (1.13) can be repre-sented as

Wl+κ∗ = hκ

∗κ χκ(t, y, y) , κ, κ∗ = 1, k . (1.15)

Here hκ∗κ are the elements of a matrix inverse to the matrix of elements hκκ

∗

.Thus, the vector WK , entering into equation (1.11), is a vector, which

as the function of time, a position of system, and its velocities is uniquelydetermined by equations of constraints. The second vector YK in equation(1.11) is assumed to be the function of the same arguments. Hence by (1.11)the vector RK also can be found as a function of time, position, and velocitiesof system. The representation of this vector in the form

RK = Λκ εεεl+κ

implies that its determination is reduced to the numerical computation of thequantities Λκ , κ = 1, k, by formulas (1.12) and (1.15).

Thus, for the equations of constraints to be satisfied it is necessary to addthe force RK to the active force YK . We shall show that this condition isalso sufficient. Really, the influence of equations of constraints on the vectorW is given by formulas (1.8). Since εεεl+κ · WL = 0, κ = 1, k, formulas (1.8)can be represented as

εεεl+κ · WK = χκ(t, y, y) , κ = 1, k . (1.16)

1. Some general remarks 81

It follows that the constraints are satisfied for any vector WL and the form ofequations of constraints does not allow us to say anything about the vectorWL. Therefore without violating the condition that the equations of con-straints are satisfied we can assume in equation (1.10) that RL = 0.

Thus, we can solve the problem, formulated above: under which conditionsand in which form the vector of reaction R can be expressed via the quantitiesΛκ , the number of which is equal to the number of constraints, and how thisvector can be represented in terms of a function of time, a position of system,and its velocities. Firstly, this problem can be solved in the case when theequations of constraints are independent, i. e. condition (1.14) is satisfiedand, secondly, when the vector RL , which does not directly connected withthe equations of constraints, is equal to zero. The procedure of numericalcomputation of the vector R = RK is shown as part of the study.

The holonomic and nonholonomic constraints under the assumption RL =0, are called ideal. This implies that these constraints are completely defined

by their analytic representations.We pay attention to the following fact, resulting from the above. In the

case of free mechanical system the vector of acceleration W is defined as afunction of time, coordinates, and velocities by Newton’slaw of the form

W = W(t, y, y) = Y(t, y, y)/M .

In the case of the existence of constraints in K-space, the vector WK as afunction of time, coordinates, and velocities is uniquely defined by equationsof constraints (1.8). In other words, in this subspace the motionlaw is givenby equations of constraints (1.16). In K-space Newton’s secondlaw itself iswritten, if necessary, only for that the force of reaction RK can be found bymeans of thislaw.

The equations of constraints cannot affect the vector WL belonging toL-space and therefore in this subspace the vector WL under ideal constraintscan be found by Newton’s secondlaw, i. e. from the equation

WL = WL(t, y, y) = YL(t, y, y)/M .

Multiplying the last vector relation by the vectors εεελ, generating the basis ofL-space, we obtain

(MWL − YL

)· εεελ = 0 , λ = 1, l .

Since WK ·εεελ = 0 and YK ·εεελ = 0, the system of these scalar equations canbe represented as

(MW − Y

)· εεελ = 0 , λ = 1, l . (1.17)

Supplementing system of equations (1.17) with equations (1.8), we obtainthe closed system of equations since in this case the vector W as a functionof time, position, and velocities of system is given in the whole space. Note

82 III. Linear Transformation of Forces

that system of equations (1.17) does not contain the reactions of constraints.The concrete form of them depends on the form of a given system of thevectors εεελ, λ = 1, l, orthogonal to the vectors εεεl+κ , κ = 1, k. For holonomicsystems such equations were obtained by Lagrange (see Chapter I), for first-order nonholonomic systems they were suggested by P. Appell, L. Boltzmann,P. V. Voronets, G. Hamel, G. A. Maggi, L. M. Markhashov, V. S. Novoselov,V.V. Rumyantsev, Fam Guen, S. A. Chaplygin and others (see Chapter II).The equations with high-order nonholonomic constraints will be discussed inChapter V.

So, under ideal constraints Newton’s law can be written as

MW = Y + Λκ εεεl+κ , κ = 1, k . (1.18)

Therefore the motion of representation point under ideal holonomic or non-holonomic constraints is described by Lagrange’s equations of the first kind,which give with (1.7) for constraints (1.1), (1.3), and (1.4), respectively, thefollowing relations

Myµ = Yµ + Λκ

∂fκ

∂yµ

, µ = 1, 3N , κ = 1, k , (1.19)

Myµ = Yµ + Λκ

∂ϕκ

∂yµ

, µ = 1, 3N , κ = 1, k , (1.20)

Myµ = Yµ + Λκ

∂ψκ

∂yµ

, µ = 1, 3N , κ = 1, k . (1.21)

From relations (1.12), (1.15) it follows that the Lagrange multipliers (general-ized reactions) Λκ , entering into these equations, are uniquely determined bythe equations of constraints and the active forces in the form of the functionsof t, y, y. Hence, by equations (1.19)–(1.21) we can always find a motion,satisfying the equations of constraints.

Thus, we obtain the answer to the question how the number of unknownscan be diminished and the reactions Rµ can be expressed via the quantitiesΛκ , the number of which is equal to the number of constraints. To dependupon the form of ideal constraints, we obtain the following representationsfor reactions:

Rµ = Λκ

∂fκ

∂yµ

, µ = 1, 3N , κ = 1, k , (1.22)

Rµ = Λκ

∂ϕκ

∂yµ

, µ = 1, 3N , κ = 1, k , (1.23)

Rµ = Λκ

∂ψκ

∂yµ

, µ = 1, 3N , κ = 1, k . (1.24)

Further we shall show that the form itself of these linear relations betweenthe quantities Rµ and Λκ permits us to answer many questions and, in par-ticular, to establish why the quantities Λκ is called a generalized reaction.

2. Theorem on the forces of holonomic constraints 83

The structure of linear transformations (1.22)–(1.24), as will be shown fur-ther, allows us to separate the equations for determining the motion, fromthe equations for obtaining the generalized reactions.

We remark that by representations (1.22)–(1.24) we can construct a singlesolution only in the case when condition (1.14) is satisfied.

§ 2. Theorem on the forces providing

the satisfaction of holonomic constraints

If the representation point is free, then any of its positions can be givenby the Cartesian coordinates as well as by the curvilinear ones. In the generalcase these coordinates are related as

yµ = yµ(t, q) , qσ = qσ(t, y) , µ, σ = 1, s , s = 3N , (2.1)

and depend on time.The formulas, introduced in the present section, will be written, where

possible, in the form, which admits the development in the case of any me-chanical system. Therefore in the sequel the curvilinear coordinates of repre-sentation point will be called generalized Lagrangian coordinates or, simply,generalized coordinates and their number will be assumed equal to s.

Newton’s secondlaw, describing the motion of representation point, in thecase of free system takes the form

MW = Y . (2.2)

Multiplying this vector relation by the vectors of the fundamental basisof the introduced curvilinear system of coordinates eσ, σ = 1, 3N , we obtainLagrange’s equations of the second kind

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ , σ = 1, s , s = 3N , (2.3)

which can be represented in explicit form as

M(gστ qτ + Γσ,αβ qαqβ) = Qσ ,

σ, τ = 1, s , α, β = 0, s , s = 3N .(2.4)

Here

Γσ,αβ =1

2

(∂gσα

∂qβ+

∂gσβ

∂qα−

∂gαβ

∂qσ

),

eα =∂y

∂qα, gαβ = eα · eβ , q0 = t , q0 = 1 ,

σ = 1, s , α, β = 0, s , s = 3N .

84 III. Linear Transformation of Forces

Given constraints (1.1) it is rational to involve the functions, enteringinto the equations of constraints, in the system of functions, realizing thetransition to generalized coordinates, i. e. to put

qλ = fλ

∗(t, y) , λ = 1, l , l = 3N − k ,

ql+κ = f l+κ

∗(t, y) = fκ(t, y) , κ = 1, k .

(2.5)

The independent coordinates qλ, λ = 1, l, are chosen arbitrary but in sucha way that the coordinates yµ, µ = 1, 3N , can uniquely be expressed fromsystem (2.5) via the coordinates qσ, σ = 1, 3N .

Together with the introduction of generalized coordinates we considergeneralized forces, using the relation for virtual elementary work.

The invariant differential form, by which a virtual elementary work isgiven, can be represented as the chain of relations

δA = Yµ δyµ = Yµ

∂yµ

∂qσδqσ = Qσ δqσ = Qσ

∂qσ

∂yµ

δyµ .

Whence it follows that the forces Yµ and the generalized forces Qσ are relatedlinearly, namely

Yµ = Qσ

∂qσ

∂yµ

, Qσ = Yµ

∂yµ

∂qσ, µ, σ = 1, 3N .

Then, the reactions Rµ and the generalized reactions Rσ are also relatedlinearly as

Rµ = Rσ

∂qσ

∂yµ

, Rσ = Rµ

∂yµ

∂qσ, µ, σ = 1, 3N . (2.6)

When generated Lagrange’s equations of the first kind (1.19) we show thatthe imposition of ideal holonomic constraints (1.1) leads to the occurrence ofreactions, given by formulas (see formulas (1.22))

Rµ = Λκ

∂fκ

∂yµ

, κ = 1, k , µ = 1, 3N .

Taking into account relations (2.5) and (2.6), we obtain

Rλ = 0 , λ = 1, l ,

Rl+κ = Λκ , κ = 1, k .(2.7)

Thus, the generalized reactions, corresponding to the arbitrary generalizedcoordinates qλ, λ = 1, l, are equal to zero and the quantities Λκ , κ =1, k, are equal to the generalized reactions, corresponding to the equationsof constraints. In other words, from Lagrange’s equations of the first kind itfollows that the imposition of each ideal constraint results in the occurrence

of generalized reaction, providing the satisfaction of this constraint.

2. Theorem on the forces of holonomic constraints 85

This implies that under the constraints

ql+κ = fκ(t, y) = 0 , κ = 1, k ,

in the left-hand sides of equations (2.4) it is to be assumed ql+κ = ql+κ =ql+κ = 0, κ = 1, k. In the right-hand sides nothing is added to the generalizedforces Qλ, λ = 1, l, but the generalized reactions Λκ , κ = 1, k, are addedto the generalized forces Ql+κ . In this case it is rational to divide system ofequations (2.3) by two groups

d

dt

∂T

∂qλ−

∂T

∂qλ= Qλ , λ = 1, l , (2.8)

d

dt

∂T

∂ql+κ

−∂T

∂ql+κ

− Ql+κ = Λκ , κ = 1, k . (2.9)

Thus, Lagrange’s equations of the second kind (2.8) results, in fact, fromLagrange’s equations of the first kind (1.19).

As was already noted in Chapter I, for the given initial conditions fromequations (2.8) we can find the law of motion of system

qλ = qλ(t) , λ = 1, l ,

and determine then by formulas (2.9) the generalized reactions Λκ , κ = 1, k,in terms of the functions of time. For determining the generalized reactions,we make use of equations (2.9), in particular, in the case when in studying thedynamics of different mechanisms it is necessary to account for the dry frictionforces, which are distinctive for nonlinear nonideal holonomic constraints. Inthis case by means of a rational choice of the generalized coordinates ql+κ,κ = 1, k, the generalized reactions Λκ , κ = 1, k, can be related directly tothe forces of safe pressure of a system of elements on roughened surfaces. Thenthe dry friction forces, entering into the right-hand sides of equations (2.8),are connected directly with generalized reactions.

In this case it is necessary to consider systems of equations (2.8) and(2.9) together. Recall that to compute the left-hand sides of equations (2.9)it is necessary also to know the relation for kinetic energy in terms of thecoordinates ql+κ, κ = 1, k, and to assume that ql+κ = ql+κ = ql+κ = 0,κ = 1, k, but only after representing the left-hand sides in explicit form(2.4). In studying the motion under the releasing holonomic constraints wealso need to obtain generalized reactions from equations (2.9).

Lagrange’s equations of the second kind (2.8) are convenient for us sincethey can be constructed by means of the relations for kinetic energy andvirtual elementary work. Therefore, as is already remarked, these equationscan be applied to any mechanical system. Marvelling at the perfection ofLagrange’s equations, L. Pars [179, p. 90] writes: "The work of Lagrange [340]is a capital source of ideas of analytic mechanics and is legally considered asone of superlative spiritual achievements of humankind".

86 III. Linear Transformation of Forces

We pay attention to that the established above connection between La-grange’s equations of the first kind (1.19) and those of the second kind (2.8),(2.9) implies the following theorem on the forces providing the satisfaction ofholonomic constraints:

The motion, under which one of generalized coordinates is a given function

of time, can be obtained by the introduction of one additional generalized

forces, corresponding to this coordinate.

A direct corollary of this theorem is a more general assertion that themotion, given at the same time by several coordinates, can be provided bythe same number of the corresponding additional generalized forces. In otherwords, if in system of equations (2.8), (2.9) we assume ql+κ = F κ(t), κ =1, k, then from this system we can obtain the functions qλ(t),λ = 1, l, as wellas the functions Λκ(t) , κ = 1, k.

Really, assuming that the constraints are given by the following equations

fκ(t, y) = f l+κ

∗(t, y) − F κ(t) = 0 , κ = 1, k ,

we can conclude that the satisfaction of each of them can be provided by thecorresponding generalized reaction Λκ , κ = 1, k. The quantities Λκ , as isshown in § 1, are uniquely defined by the equations of constraints and by theactive forces. If the expression for reactions is obtained, then from equations(1.19) we can find a motion, i. e. the functions yµ(t), µ = 1, 3N , and thenwe obtain the functions qσ(t), σ = 1, 3N . Since system of equations (2.8),(2.9) is equivalent to equations (1.19) from this system we can determine thefunctions qλ(t), λ = 1, l, as well as the functions Λκ(t) , κ = 1, k.

The above theorem can easily be proved directly, without a reference tothe results, obtained earlier. Below we give this proof.

For definiteness, we assume that the only given function of time is thecoordinate qs. Adding in equations (2.4) the quantities Λs only to the gener-alized force Qs, we obtain

M(gρτ qτ + Γρ,αβ qαqβ+

+ 2Γρ,sτ qsqτ ) = Qρ − M(gρsqs+ Γρ,ss(q

s)2+ 2Γρ,0sqs),

ρ, τ = 1, s − 1 , α, β = 0, s − 1 , (2.10)

M(gsσ qσ + Γs,αβ qαqβ) = Qs + Λs , σ = 1, s , α, β = 0, s . (2.11)

The sum gρτ qρqτ (ρ, τ = 1, s − 1) is a positively defined quadratic form.Therefore the matrix with the elements gρτ (ρ, τ = 1, s − 1) is nonsingular.In this case system (2.10) is solvable for qτ (τ = 1, s − 1) and can be integrat-ed for the given initial data. Then we find the functions qτ (t) (τ = 1, s − 1).Substituting them and the function qs(t) into the left-hand side of equa-tion (2.11), we obtain the additional generalized force Λs(t) , providing thegiven motion over the coordinate qs.

Note that the assertion proved does not mean that the motion over anycoordinate does not effect the motion over the rest of coordinates. It means

2. Theorem on the forces of holonomic constraints 87

only that it is unnecessary to apply another additional forces except for theforce, corresponding to the chosen coordinate. Really, the function qs(t) entersinto the right-hand side of system (2.10) and therefore effects the functionsqτ (t) (τ = 1, s − 1).

Lagrange’s equations of the first and second kinds result from the abovetheorem. Really, from this theorem, generalized to the case of few coordinates,it follows that under the holonomic constraints

ql+κ = fκ(t, y) = 0 , κ = 1, k ,

in the linear transformations of forces (2.6) the generalized reactions mustbe given in the form (2.7). In this case linear transformations (2.6) can berepresented in the following way:

Rµ

∂yµ

∂qλ= Rλ = 0 , λ = 1, l , µ = 1, 3N , (2.12)

Rµ

∂yµ

∂ql+κ

= Rl+κ = Λκ , κ = 1, k , µ = 1, 3N , (2.13)

Rµ = Λκ

∂fκ

∂yµ

, κ = 1, k , µ = 1, 3N . (2.14)

Taking into account the relations

Myµ

∂yµ

∂qσ=

d

dt

∂T

∂qσ−

∂T

∂qσ, µ, σ = 1, 3N ,

which have been proved by Lagrange, and also the fact that

Rµ = Myµ − Yµ , Qσ = Yµ

∂yµ

∂qσ,

we can conclude that from relations (2.12) it follows Lagrange’s equations ofthe second kind (2.8), from relations (2.13) the equations for determining ofgeneralized reactions (2.9), and from relations (2.14) Lagrange’s equations ofthe first kind (1.19).

Thus, relations (2.12)–(2.14), which are, in essence, a short analytic formof the theorem, involve all the types of equations of holonomic mechanics.They also permit us to show the covariance property of Lagrange’s equa-tions of the second kind, i. e. the fact that these equations in terms of theindependent coordinates qλ and qλ

∗

∗, λ, λ∗ = 1, l, are related as

Rλ

∂qλ

∂qλ∗

∗

= R∗

λ∗ = 0 , λ, λ∗ = 1, l ,

where

Rλ =d

dt

∂T

∂qλ−

∂T

∂qλ− Qλ , R∗

λ∗ =d

dt

∂T

∂qλ∗

∗

−∂T

∂qλ∗

∗

− Q∗

λ∗ .

88 III. Linear Transformation of Forces

Really, relations (2.12), written in terms of the variables qλ∗

∗, λ∗ = 1, l, yield

that

Rµ

∂yµ

∂qλ∗

∗

= Rµ

∂yµ

∂qλ

∂qλ

∂qλ∗

∗

= Rλ

∂qλ

∂qλ∗

∗

= R∗

λ∗ = 0 , λ, λ∗ = 1, l ,

which was to be proved.We now turn to Lagrange’s equations of the second kind (2.8) and remark

that this system of equations is equivalent to one vector relation

MWL = YL , (2.15)

represented in the vector space, the basis of which is the vectors eλ, λ = 1, l.The index L of the vectors in equation (2.15) points out a dimension ofspace, in which this equation is represented. Deleting the index L, we arriveat equation (2.2). Note that in this case case in equations (1.16) we haveεεεl+κ = el+κ = ∇∇∇fκ , κ = 1, k, and in equations (1.17) εεελ = eλ, λ = 1, l.

The vector form of a system of Lagrange’s equations of the second kindwas used here for the simplest holonomic mechanical system with a finitenumber of mass points. In the case of arbitrary mechanical system, consist-ing of as rigid bodies as solids, the system of Lagrange’s equations of thesecond kind is also equivalent to one vector relation, represented in the spacetangential to the manifold of all possible positions of mechanical system (inmore detail, see Chapter IV).

§ 3. An example of the application of theorem on the forces

providing the satisfaction of

holonomic constraints

In the present section and then in § 5 we study some problems on a guid-ance of mass point on target by the methods of analytic mechanics.

The pursuitlaw to depend upon its form can be regarded as the ideal holo-nomic or nonholonomic constraint, respectively. In this case the constraint isa sought control force. Among the works, devoted the study of controllablemotion with applying the theory of constrained motion it should be notedthe work of V. I. Kirgetov [105].

We remark that not only the problem on the approach to target is highlyactual but the opposite problem when one studies the optimal deviations oftarget from the object, which points to the target by various ways. (see, forexample, [244]).

Example III. 1 . The target guidance by the method of parallel approach.

In the case when the aircraft of the mass m moves in horizontal plane Oxy

we account for the tractive force P, directed along the velocity v, and theforce of air drag Ra, operating in opposite direction. It is required to find the

3. An example of the application of theorem 89

control force R, under which the described below method provides a guidanceon target moving by the knownlaw

ξ = ξ(t) , η = η(t) . (3.1)

We consider a target guidance by the scheme of parallel approach [129]. Asis known, in this case in moving the aircraft the line of sighting a target mustmove in parallel itself, what provides the continuous guidance of aircraft atthe point of instantaneous collision. If for t = 0 the aircraft (rocket) is at theorigin of coordinates M0(0, 0) and the target have the following coordinates

ξ(0) = ξ0 , η(0) = η0 ,

then the line of sight is directed along the straight line

y =η0

ξ0(x − ξ) + η . (3.2)

It means that if the target has the position (ξ(t), η(t)), then the coordinatesx, y of rocket satisfy equation (3.2), which can be rewritten as

(x − ξ) sin ϕ0 + (η − y) cos ϕ0 = 0 , tg ϕ0 = η0/ξ0 .

In other words, on the coordinates of aircraft it is imposed the nonstationaryholonomic constraint

f(t, x, y) ≡ (x − ξ) sin ϕ0 + (η − y) cos ϕ0 = 0 . (3.3)

Consider a new system of coordinates Oq1q2, which is rotated clockwiserelative to the original system by the angle π/2 − ϕ0. In these two systemsthe coordinates of aircraft are related by the transition formulas:

q1 = x sin ϕ0 − y cos ϕ0 , q2 = x cos ϕ0 + y sin ϕ0 , (3.4)

x = q1 sin ϕ0 + q2 cos ϕ0 , y = −q1 cos ϕ0 + q2 sin ϕ0 . (3.5)

We make use of the following linear transformations of forces:

Rx = R1∂q1

∂x+ R2

∂q2

∂x, Ry = R1

∂q1

∂y+ R2

∂q2

∂y,

R1 = Rx

∂x

∂q1+ Ry

∂y

∂q1, R2 = Rx

∂x

∂q2+ Ry

∂y

∂q2.

(3.6)

Substitute the expressions of x and y from (3.5) into equation of con-straints (3.3). Then we obtain

q1 = ξ sin ϕ0 − η cos ϕ0 . (3.7)

We pay attention to the fact that according to the first formulas of (3.4)in the right-hand side of this relation there is the projection q1

targ of the

90 III. Linear Transformation of Forces

radius-vector of target on the axis Oq1 and therefore relation (3.7) can berewritten as

q1 = q1targ(t) , q1

targ(t) = ξ(t) sin ϕ0 − η(t) cos ϕ0 . (3.8)

Thus, we can formulate the following problem: it is required to find theadditional force, under which for the given active forces a motion of themechanical system in the system of coordinates Oq1q2 is provided such thatthe coordinate q1 is varied by given low (3.8). By the theorem, proved in theprevious section, such a motion can be obtained by the introduction of oneadditional force Λ, corresponding to the coordinate q1. Thus, forlaw (3.8) tobe satisfied it is sufficient to put

R1 = Λ , R2 = 0 .

In this case linear transformations of forces (3.6) take the form

Rx = Λ sin ϕ0 , Ry = −Λ cos ϕ0 , (3.9)

Λ = Rx sin ϕ0 − Ry cos ϕ0 , Rx cos ϕ0 + Ry sin ϕ0 = 0 . (3.10)

Construct the projections of given active forces, acting on the aircraft:

X = (P − Ra) x/v , Y = (P − Ra) y/v ,

Q1 = X∂x

∂q1+ Y

∂y

∂q1= (P − Ra) q1/v ,

Q2 = X∂x

∂q2+ Y

∂y

∂q2= (P − Ra) q2/v ,

v2 = x2 + y2 = (q1)2 + (q2)2 .

(3.11)

The projection Rx, Ry for aircraft is as follows

Rx = mx − X , Ry = my − Y . (3.12)

Therefore to linear transformation (3.9) correspond Lagrange’s equations ofthe first kind

mx = X + Λ sin ϕ0 , my = Y − Λ cos ϕ0 , (3.13)

and to linear transformation of forces (3.10) Lagrange’s equations of thesecond kind

mq1 − Q1 = Λ , (3.14)

mq2 = Q2 . (3.15)

It is easily seen that, taking into account the simplest algebraic transforma-tions, we can obtain equations (3.14), (3.15) from equations (3.13) and viceversa. This example demonstrates the reciprocity of two kinds of Lagrange’sequations.

3. An example of the application of theorem 91

Given the initial data and function (3.8) and using relations of forces(3.11) we can numerically integrate equation (3.15) and obtain the law ofvariation of the coordinate q2:

q2 = q2(t) . (3.16)

Using functions (3.8) and (3.16), we can obtain then from formula (3.14) thecontrol force

Λ = Λ(t) , (3.17)

which provides thelaw of a guidance of aircraft on a target moving bylaw(3.1) (or (3.8)). After determining function (3.17) by formulas (3.9) we canfind the components of control force in the system Oxy.

As an example of concrete numerical computation we consider a motionof target by the followinglaws (t is given in seconds, ξ, η in meters):

(I) ξ(t) = v0t + ξ0 , η(t) = η0 , (3.18)

(II) ξ(t) = v0t cos ϕ0 + ξ0 , η(t) = −9.812 t2/2 + v0t sin ϕ0 + η0 . (3.19)

In Fig. III. 1 curves 11, 12 show the trajectories of the target, moving bylaws(3.18) and (3.19), and curves 21, 22 the corresponding motions of aircraft,respectively,. In this case for a hypothetical aircraft (rocket) we put

m = 200 kg , P = 2500 N , Ra = 0.01v2 N ,

ξ0 = η0 = 5000 m , ϕ0 = π/4 , v0 = 194.44 m/s ,

x(0) = y(0) = 0 , x(0) = v0 cos ϕ0 , y(0) = v0 sin ϕ0 .

(3.20)

Fig. III. 1

We consider now a technical realization of control forces. For the aircraftpursues a target by laws (3.18) or (3.19), it is necessary together with amotor, which generates the tractive force P, to use an additional motor,which generates the required by value and direction control force R. One canutilize only one motor if it can vary a tractive force by value and direction,i. e. can generate the tractive force P∗ such that

P∗ = P + R . (3.21)

92 III. Linear Transformation of Forces

Fig. III. 2

In Fig. III. 2 are represented the hodographs of the vectors PI∗

and PII∗,

providing the motion of aircraft, which pursues by the method of parallelapproach the target moving bylaw (I) (3.18) or bylaw (II) (3.19). The arrowsin the figure point out the increasing of time. The corresponding curves aredenoted by digits 41 and 42.

In Chapter VI according to Lagrange’s multipliers a special form of equa-tions of the dynamics of rigid bodies system is obtained. This form of theequations of dynamics is used for a motion control of the platform of roboticstand by means of the bars of variable length. The orientation of platform isgiven by the positions of the vector of a center of mass and those of the unitvectors of principal central axes of inertia. The vector equations with respectto these four vectors are constructed. The stresses in bars, regarded as con-trol parameters, enter into the equations linearly. If the position of platformis given by six generalized coordinates, which are the lengths of bars, thenthe theorem of holonomic mechanics we use in the present section becomesdescriptive. Recall that according to this theorem the motion, under whichone of generalized coordinates is given by a function of time, can be obtainedby means of the introduction of one additional force, corresponding to thiscoordinate.

§ 4. Chetaev’s postulates and the theorem on the forces

providing the satisfaction of

nonholonomic constraints

Introduce the generalized forces, corresponding to the equations of con-straints. Then from the D’Alembert–Lagrange principle, extended to the caseof Chetaev’s type constraints, the base complex of equations of nonholonomicmechanics in compact form can be obtained. Let us formulate the theorem,involving this complex of equations.

For the sake of generality, we consider an arbitrary mechanical system,the position of which is defined by the generalized coordinates qσ, σ = 1, s.Suppose that this system is under the nonlinear nonholonomic constraints

ϕκ(t, q, q) = 0 , κ = 1, k , k < s . (4.1)

4. Chetaev’s postulates and the theorem 93

Recall that the nonholonomicity of these constraints consists in that inspite of their occurrence the transition of the system from any its positionwith the coordinates qσ

0 , σ = 1, s, to other position with the coordinates qσ1 ,

σ = 1, s, is kinematically possible.N. G. Chetaev showed that for the real motion of the considered system,

the D’Alembert–Lagrange principle(

d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 (4.2)

is satisfied. The kinetic energy T is given in the form

T =M

2gαβ(t, q)qαqβ , α, β = 0, s , q0 = t , q0 = 1 ,

where M is a mass of complete system and the generalized forces Qσ areassumed to be the function of time t, the coordinates qσ, and the generalizedvelocities qσ (σ = 1, s).

N. G. Chetaev assume that the quantities δqσ, entering into theD’Alembert–Lagrange principle, satisfy the following conditions (see formulas(6.3) of Chapter II)

∂ϕκ

∂qσδqσ = 0 , κ = 1, k . (4.3)

Equations (4.2) and (4.3) we shall call further the Chetaev postulates.

Now we consider the constraint of the Chetaev postulates with the genera-tion of Maggi’s equations, which are based on the introduction of generalizedforces, corresponding to quasivelocities. Using the principle of releasabilityof constraints (4.1), we consider their generalized reactions Rσ and representLagrange’s equations in the form

d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ = Rσ , σ = 1, s . (4.4)

From Chetaev’s postulates (4.2), (4.3) it follows that the quantities Rσ canbe represented as

Rσ = Λκ

∂ϕκ

∂qσ, σ = 1, s , (4.5)

where Λκ are Lagrange’s multipliers.Thus, the Chetaev postulates (4.2), (4.3) are equivalent to one postulate,

which is given by formulas (4.5).Constraints (4.1) are nonholonomic and therefore their left-hand sides can

be regarded only as the certain quasivelocity vl+κ

∗, l = s − k, i. e.

vl+κ

∗= ϕκ(t, q, q) , κ = 1, k . (4.6)

We supplement this system of quasivelocities by the following quasivelocities

vλ

∗= vλ

∗(t, q, q) , λ = 1, l ,

94 III. Linear Transformation of Forces

in such a way that the transition from the generalized velocities qσ to thequasivelocities v

ρ

∗ (ρ, σ = 1, s) is one-to-one, i. e. there exists the followinginverse transformation

qσ = qσ(t, q, v∗) , σ = 1, s . (4.7)

Note that the quantities vλ∗, λ = 1, l, not necessarily must be quasivelocities,

they can be generalized velocities.A distinctive example of the usage of quasivelocities is dynamic Euler’s

equations

Adωx

dt+ (C − B)ωyωz = Lx ,

Bdωy

dt+ (A − C)ωzωx = Ly ,

Cdωz

dt+ (B − A)ωxωy = Lz .

(4.8)

Here A, B, C are the moments of inertia of rigid body about the principal ax-es of inertia x, y, z, rigidly bound with body, and ωx, ωy, ωz and Lx, Ly, Lz

are the projections of the vector of the instantaneous angle velocity ωωω and ofthe principal moment of exterior forces L about the center of mass on theseexes, respectively.

The quantities ωx, ωy, ωz are, as is well known, the quasivelocities sincethey cannot be regarded as time derivatives with respect to the certain threenew angles, uniquely related to the Euler angles.

We regard the moment Lx as the generalized force, corresponding to thequasivelocity ωx . We can similarly regard the moments Ly and Lz . Notethat for the forces, having potential, the generalized forces, correspondingto quasivelocities, were introduced by Poincare [373]. N. G.Chetaev [248] ex-tends then this notion on the forces of any nature. In this case the relationbetween the generalized velocities and the quasivelocities is assumed to belinear. Consider now the case when this relation are nonlinear.

Above we assumed that to the generalized coordinates qσ correspond thegeneralized forces Qσ and the transition from the quasivelocity v

ρ

∗ to the gen-eralized velocities qσ (ρ, σ = 1, s) is given by formulas (4.7). Then the gen-

eralized force Qρ , corresponding to quasivelocities vρ

∗ , are given by formulas

Qρ = Qσ

∂qσ

∂vρ

∗

, ρ = 1, s . (4.9)

Consider also the inverse linear transformation of forces, which is given bythe relation

Qσ = Qρ

∂vρ

∗

∂qσ, σ = 1, s . (4.10)

Replacing in formulas (4.9) and (4.10) the forces Qσ and Qρ by the reac-

tions Rσ and R∗

ρ, respectively, we obtain

R∗

ρ= Rσ

∂qσ

∂vρ

∗

, Rσ = R∗

ρ

∂vρ

∗

∂qσ, ρ, σ = 1, s . (4.11)

4. Chetaev’s postulates and the theorem 95

Here R∗

ρis a generalized reaction, corresponding to the quasivelocity v

ρ

∗ (ρ =

1, s).Consider now postulate (4.5) from the point of view of general formu-

las (4.11). Relations (4.5) and (4.6) implies that postulate (4.5) means thatin formulas (4.11) it should be assumed the following

R∗

λ= 0 , λ = 1, l , R∗

l+κ= Λκ , κ = 1, k . (4.12)

The postulate (4.5), as is remarked above, is equivalent to Chetaev’s postu-lates (4.2) and (4.3). Therefore relations (4.11) and (4.12) can be regarded asone of forms of notation of Chetaev’s postulates. Relations (4.12) imply thatit is rational to divide the transition formulas from the quantities Rσ to thequantities R∗

ρinto two groups:

Rσ

∂qσ

∂vλ∗

= 0 , λ = 1, l , Rσ

∂qσ

∂vl+κ

∗

= Λκ , κ = 1, k ,

which can be represented in accordance with relations (4.4) as

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vλ∗

= 0 , λ = 1, l , (4.13)

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vl+κ

∗

= Λκ , κ = 1, k . (4.14)

Equations (4.13), obtained from the Chetaev postulates, are Maggi’s equa-tions. From these equations the many famous forms of equations of motionfor nonholonomic systems can be obtained (see § 3 Chapter II).

We now consider one of theorems of nonholonomic mechanics. As is known,the introduction of generalized coordinates leads to the introduction of gener-alized forces, corresponding to these coordinates. In § 2 of the present Chap-ters we prove the theorem expressing the following property of generalizedforces: the motion such that one of generalized coordinates is a given timefunction, can be obtained by the introduction of one additional generalizedforce corresponding to this coordinate.

Above, the introduction of generalized forces, corresponding to quasive-locities, was demonstrated on an example of dynamic Euler’s equations (4.8).Considering these equations once more, we can conclude that if, for exam-ple, the quasivelocity ωx is the given function of time, then we can obtainsuch motion by means of introduction of one additional moment R∗

x, cor-

responding to the quasivelocity ωx. Extending this example to an arbitrarymechanical system, we arrive at the following theorem on the forces, underwhich the nonholonomic constraints are satisfied:

Let the quasivelocities be the given time functions vl+κ

∗= χκ

∗(t), and be

related with generalized velocities as

vl+κ

∗= ϕκ

∗(t, q, q) , κ = 1, k . (4.15)

96 III. Linear Transformation of Forces

On the rest of quasivelocities vλ∗, λ = 1, l, no restrictions are imposed. Then

for the mentioned above motion to be realized, it is sufficiently to apply the

additional generalized forces R∗

l+κ= Λκ , corresponding to the quasivelocities

vl+κ

∗, κ = 1, k.

Relations (4.15) can be represented as

ϕκ(t, q, q) = ϕκ

∗(t, q, q) − χκ

∗(t) = 0 , κ = 1, k . (4.16)

We observe that they coincide, by form, with the equations of constraints(4.1) and therefore we can use the theory of constrained motion. In § 1 ofthis Chapter it was shown that for first-order nonholonomic constraints tobe satisfied, it is sufficient to add the forces Rµ, given in the form of (1.23),to the active forces Yµ:

Rµ = Λκ

∂ϕκ

∂yµ

, µ = 1, 3N , κ = 1, k .

When passed from the Cartesian coordinates to the generalized ones, theabove relations take the form

Rσ = Λκ

∂ϕκ

∂qσ, σ = 1, 3N , κ = 1, k .

Relations (4.11), (4.12) imply that the quantities Λκ, κ = 1, k, are additionalgeneralized forces, which provide the satisfaction of constraints (4.16). Thismeans that in order that the quasivelocity vl+κ

∗, κ = 1, k, are the given

functions of time, it is sufficient to add the corresponding to them additionalgeneralized forces R∗

l+κ= Λκ , κ = 1, k.

Thus, to each equation of nonholonomic constraint can be assigned a gen-

eralized force, controlling this constraint.

Equations (4.15) can also be regarded as the equations of noncompleteprogram, of motion, given in quasivelocities. Thus, the present theorem canalso be used to study a controllable motion.

System of differential Maggi’s equations (4.13), supplemented by equa-tions (4.16) differentiated in time, can be reduced to the following system

qσ = F σ(t, q, q) , σ = 1, s . (4.17)

Substituting these relations into the left-hand sides of equations (4.14), weobtain the additional generalized forces Λκ in the form

Λκ = Λκ(t, q, q) , κ = 1, k .

Note that the quantities Rσ as the functions of the same variables can beobtained if relations (4.17) are substituted into formulas (4.4).

By the theorem we have the following three base groups of relations:

Rσ

∂qσ

∂vλ∗

= 0 , λ = 1, l ,

5. An example of the application of theorem 97

Rσ

∂qσ

∂vl+κ

∗

= Λκ , κ = 1, k , (4.18)

Rσ = Λκ

∂ϕκ

∂qσ, σ = 1, s .

Using relations (4.4), they make transition to Maggi’s equations (4.13), (4.14),and Lagrange’s equations of the first kind in curvilinear coordinates for non-holonomic systems, respectively:

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ + Λκ

∂ϕκ

∂qσ, σ = 1, s .

So, we can say that the present theorem involves the main complex ofequations of nonholonomic mechanics.

The first two groups of relations (4.18) follows from the third relation,which, as already was remarked, is equivalent to Chetaev’s postulates (4.2)and (4.3). Thus, using Chetaev’s postulates, we show that the fundamen-tal equations of nonholonomic mechanics can be constructed by using thetheorem, according to which the generalized forces corresponding to quasive-locities permit us to control the varying of these quasivelocities.

§ 5. An example of the application of theorem on forces

providing the satisfaction of

nonholonomic constraints

Example III .2 . A target guidance by pursuit method. Consider the prob-lem, formulated in Example III.1 in the case when the aircraft pursues atarget by the pursuit method. As is known [129], in the method of targetguidance, the vector of velocity of aircraft is continuously directed to themoving target. For such motion the following relation

x

x − ξ=

y

y − η(5.1)

must be satisfied. Here the coordinates of target ξ, η are assumed to be givenfunctions of time (3.1). We shall regard formula (5.1) as a nonstationarynonholonomic constraint, imposed on the motion of the point M(x, y) beingan aircraft (rocket):

ϕ(t, x, y, x, y) ≡ (y − η)x − (x − ξ)y = 0 . (5.2)

The equations of motion are generated by the theorem of nonholonomic me-chanics, proved in § 4.

We introduce the new quasivelocity v1∗, v2

∗, related with the initial veloc-

ities x, y asv1∗

= x , v2∗

= (y − η)x − (x − ξ)y . (5.3)

98 III. Linear Transformation of Forces

Then it is easy to find the inverse transformation

x = v1∗, y = (y − η)v1

∗/(x − ξ) − v2

∗/(x − ξ) . (5.4)

Introduce a linear transformation of forces (4.11) using transition formulas(5.3) and (5.4). We pay attention that in the example under considerationthe usual Cartesian system x, y is regarded as the original system of thecoordinates q1, q2. Therefore in formulas (4.11) in place of R1,R2 we takeRx, Ry. Thus, we have

Rx = R∗

1

∂v1∗

∂x+ R∗

2

∂v2∗

∂x, Ry = R∗

1

∂v1∗

∂y+ R∗

2

∂v2∗

∂y, (5.5)

R∗

1 = Rx

∂x

∂v1∗

+ Ry

∂y

∂v1∗

, R∗

2 = Rx

∂x

∂v2∗

+ Ry

∂y

∂v2∗

. (5.6)

For the condition of guidance (5.2) to be satisfied it is necessary for theintroduced quasivelocity v2

∗to be equal to zero:

v2∗

= 0 . (5.7)

By the theorem of nonholonomic mechanics, proved in § 4, the given changeof the quasivelocity (5.7) can be provided by the one additional force Λ,corresponding to this quasivelocity v2

∗, i. e. in order to satisfylaw (5.7) it is

sufficient to assume that

R∗

1 = 0 , R∗

2 = Λ .

In accordance with formulas (5.3), (5.4) the linear transformations of forces(5.5), (5.6) are the following:

Rx = Λ(y − η) , Ry = Λ(ξ − x) , (5.8)

0 = Rx + Ry

y − η

x − ξ, Λ = Ry

1

ξ − x. (5.9)

In these transformations the reactions Rx, Ry take the form (3.12). Thentransformations (5.8) lead to Lagrange’s equations of the first kind

mx − X = Λ(y − η) , my − Y = Λ(ξ − x) , (5.10)

and transformations (5.9) to Maggi’s equations

mx − X + (my − Y )y − η

x + ξ= 0 , (5.11)

Λ =my − Y

ξ − x. (5.12)

It is easily seen that eliminating Λ from equations (5.10) according to (5.12),we obtain the equation of motion coincident with the Maggi’s equations (5.11).

5. An example of the application of theorem 99

Equation of motion (5.11) and equation of constraint (5.2) give a closedsystem. For representing it in normal form we differentiate equation of con-straint (5.2) with respect to time. Finally, we obtain

x =(ηx − ξy)(y − η)

(x − ξ)2 + (y − η)2+

(P − Ra)x

m√

x2 + y2,

y =(ξy − ηx)(x − ξ)

(x − ξ)2 + (y − η)2+

(P − Ra)x

m√

x2 + y2.

(5.13)

In Fig. III. 3 curves 31, 32 show the trajectories of aircraft, obtained as aresult of the integration of system (5.13) in the cases when the target movesby laws (I) (3.18) and (II) (3.19), respectively. The characteristics of aircraftare given by formulas (3.20).

Let R = Rxi + Ryj be the sought control force. Taking into account thatthe motion of aircraft must satisfy equations (5.13) and the equations

mx = X + Rx , my = Y + Ry ,

we obtain that

Rx =m(ηx − ξy)(y − η)

(x − ξ)2 + (y − η)2, Ry =

m(ξy − ηx)(x − ξ)

(x − ξ)2 + (y − η)2,

R =√

R2x

+ R2y

=m|ξy − ηx|√

(x − ξ)2 + (y − η)2.

The force Q = P+Ra is directed along the tangent line to the trajectoryof aircraft. The force R, as follows from formulas (5.1) and (5.8), is perpen-dicular the vector of aircraft velocity. Thus, the force Q is proportional tothe tangential acceleration and the force R to the normal one.

In the case when the control is realized by one motor, which can generatethe required traction P∗ of the given value and direction, arguing similarto that at the end of Example III.1, we can determine the vector P∗ byformula (3.21). In Fig. III. 4 are represented the hodographs of the vectors

Fig. III. 3

100 III. Linear Transformation of Forces

Fig. III. 4

PI∗

and PII∗, providing the motion of aircraft, which pursues by the pursuit

method a target, moving by law (I) (3.18) or by law (II) (3.19), respectively.In the figure arrows corresponds to increasing the time. The correspondingcurves are denoted by the symbols 51 and 52.

§ 6. Linear transformation of forces

and Gaussian principle

The motion of free mechanical system of the general type in generalizedcoordinates is described by Lagrange’s equations of the second kind (2.3). Wechange from the variables qσ to the new coordinates q

ρ

∗ by formulas

qρ

∗= qρ

∗(t, q) , qσ = qσ(t, q∗) , ρ, σ = 1, s ,

|∂qρ

∗/∂qσ| = 0 .

(6.1)

Let the given functions of time be the variables ql+κ

∗= ql+κ

∗(t, q), l =

s − k, κ = 1, k (noncomplete program of motion) only. The generalizedforces Rσ, which must be added to the forces Qσ in equations (2.3) for thementioned above program to be satisfied, we seek in the form

Rσ = R∗

ρ

∂qρ

∗

∂qσ, R∗

ρ= Rσ

∂qσ

∂qρ

∗

, ρ, σ = 1, s . (6.2)

The variables qλ∗, λ = 1, l, are free coordinates. Therefore the corresponding

to them additional forces R∗

λcan be assumed equal to zero. In this case

relations (6.2) take the form

Rσ = R∗

l+κ

∂ql+κ

∗

∂qσ, Rσ

∂qσ

∂qλ∗

= 0 , Rσ

∂qσ

∂ql+κ

∗

= R∗

l+κ,

κ = 1, k , λ = 1, l .

6. Linear transformation of forces and Gaussian principle 101

Discriminate from these relations the following equations

Rσ

∂qσ

∂qλ∗

= 0 , λ = 1, l .

The latter is, in fact, a short form of notation for Lagrange’s equations of thesecond kind in the new free variables qλ

∗. This results from the proved above

covariance property of Lagrange’s equations of the second kind.In a similar way, if we have the following transformation of velocities

vρ

∗= vρ

∗(t, q, q) , qσ = qσ(t, q, v∗) , ρ, σ = 1, s ,

|∂vρ

∗/∂qσ| = 0 ,

(6.3)

then the corresponding to it transformation of forces is as follows

Rσ = R∗

ρ

∂vρ

∗

∂qσ, R∗

ρ= Rσ

∂qσ

∂vρ

∗

, ρ, σ = 1, s . (6.4)

For noncomplete program in the case when the given functions of time arethe variables vl+κ

∗, l = s − k, κ = 1, k, only, in relations (6.4) we can put

R∗

λ= 0, λ = 1, l.Analogously, for the linear transformation of accelerations

wρ

∗= aρ

σ(t, q, q) qσ + a

ρ

0(t, q, q) ,

qσ = bσ

ρ(t, q, q)wρ

∗+ bσ

0 (t, q, q) ,

|aρ

σ| = 0 , ρ, σ = 1, s ,

(6.5)

it is rational to introduce a transformation of forces by formulas

Rσ =˜R

∗

ρ

∂wρ

∗

∂qσ=

˜R

∗

ρaρ

σ,

˜R

∗

ρ= Rσ

∂qσ

∂wρ

∗

= Rσ bσ

ρ, ρ, σ = 1, s , (6.6)

and for noncomplete program in the case when there are given only the

variables wl+κ

∗, l = s − k, κ = 1, k, we can put

˜R

∗

λ= 0, λ = 1, l.

In the case R∗

ρ=

˜R

∗

λ= 0, λ = 1, l, relations (6.4), (6.6) yield, in particu-

lar, the relations

Rσ

∂qσ

∂vλ∗

= 0 , Rσ

∂qσ

∂wλ∗

= 0 , λ = 1, l ,

which are Maggi’s equations and Appell’s equations, respectively.Differentiating twice the transformation of coordinates (6.1) with respect

to time and assuming qρ

∗ = wρ

∗, we arrive at linear relations (6.5). Similarly,differentiating the transformation of velocities (6.3) with respect to time anddenoting v

ρ

∗ = wρ

∗, we obtain the relations, which can be represented inthe form (6.5). However linear transformations (6.5) can also be introducedin the case when for certain ρ there do not exist the coordinates q

ρ

∗ and

102 III. Linear Transformation of Forces

quasivelocities vρ

∗ such that we can put qρ

∗ = wρ

∗ and vρ

∗ = wρ

∗. In this senseformulas (6.6) are of a more general form of the transformation of forces.

We shall show that the equations

Rσ

∂qσ

∂wλ∗

= 0 , λ = 1, l , (6.7)

give Gaussian principle in the free variables wλ∗.

A concrete form of equations (6.7) depends on a choice of variables wλ∗.

We reduce a family of equations (6.7) to the form invariant under a choice ofthe free variables wλ

∗. For this purpose we write equations (6.5) in differential

form:δ′′wρ

∗= aρ

σδ′′qσ , δ′′qσ = bσ

ρδ′′wρ

∗, ρ, σ = 1, s .

The primes of the derivative δ underline that we compute the partial deriva-tives for the fixed t, qσ, qσ. The assumptions that δ′′wl+κ

∗= 0, κ = 1, k, and

the derivatives δ′′wλ∗, λ = 1, l, are regarded as arbitrary and independent

variables means that the variables wl+κ

∗are to be given and the variables wλ

∗

under the condition |aρσ| = 0 are chosen arbitrary. The linear independence

of the quantities δ′′wλ∗

permits us to represent system of equations (6.7) as aunique equation

Rσ

∂qσ

∂wλ∗

δ′′wλ

∗= Rσ δ′′qσ = 0 . (6.8)

In the next Chapter we shall show that the family of Lagrange’s equationsof the second kind

MWσ ≡ M(gστ qτ + Γσ,αβ qαqβ

)= Qσ + Rσ ,

Γσ,αβ =1

2

(∂gσβ

∂qα+

∂gσα

∂qβ−

∂gαβ

∂qσ

),

τ, σ = 1, s , α, β = 0, s , q0 = t , q0 = 1 ,

describing the motion of mechanical system of any structure, can be repre-sented as a one vector relation written in tangential space

MW = Y + R , W = Wσeσ , Y = Qσeσ , R = Rσeσ .

Let the vector W be given in the contravariant form:

W = Wσeσ , Wσ = qσ + Γσ

αβqαqβ .

Since the Christoffel symbols Γσ

αβis independent of accelerations, we have

δ′′W σ = δ′′qσ ,

and therefore the sum Rσ δ′′qσ can be represented as the scalar productR · δ′′W. Then equation (6.8) has the form

R · δ′′W = 0 . (6.9)

6. Linear transformation of forces and Gaussian principle 103

Taking into account that the force Y is independent of the accelerations qσ,we obtain

δ′′R = δ′′(MW − Y) = Mδ′′W .

Therefore in place of equation (6.9) we can write

δ′′R2 = 2R · δ′′R = 0 . (6.10)

It means that the vector R is chosen from the condition of a minimality ofits modulus.

Relations (6.10), expressing Gaussian principle, can be represented inusual form:

δ′′Z = 0 , Z = M(W − Y/M)2 . (6.11)

Formulas (6.10) and (6.11) give Gaussian principle in invariant form andequation (6.7) gives it in terms of the variables wλ

∗. We remark that this

principle is here the principle of optimal choice of control forces, providing thegiven conditions of motion. The additional discussion of Gaussian principlewill be given in the next Chapter.

C h a p t e r IV

APPLICATION OF A TANGENT SPACE

TO THE STUDY OF CONSTRAINED MOTION

By means of a tangent space we introduce, a system of Lagrange’s equations

of the second kind is represented in the vector form. It is shown that the tan-

gential space is partitioned by equations of constraints into the direct sum

of two subspaces. In one of them the component of an acceleration vector

of system is uniquely determined by the equations of constraints. The no-

tion of ideality of holonomic constraints and nonholonomic constraints of the

first and second orders is analyzed. This notion is extended to high-order con-

straints. The relationship and equivalence of differential variational principles

of mechanics are considered. A geometric interpretation of the ideality of con-

straints is given. Generalized Gaussian principle is formulated. By means of

this principle for nonholonomic systems with third-order constraints the equa-

tions in Maggi’s form and in Appell’s form are obtained.

§ 1. The partition of tangent space

into two subspaces by equations of constraints.

Ideality of constraints

We assume that the motion of free mechanical system in the generalizedcoordinates qσ, σ = 1, s, is described by Lagrange’s equations of the secondkind [189]

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ , T =

M

2gαβ qαqβ ,

σ = 1, s , α, β = 0, s , q0 = t , q0 = 1 ,

(1.1)

where Qσ is a generalized force corresponding to the coordinate qσ and M isa mass of the whole system.

Consider a manifold of all positions of the considered mechanical systemat time t. Fix a certain point of this manifold, given by the coordinates qσ,σ = 1, s. Suppose, the old and new coordinates of this point are related byformulas

qσ = qσ(t, q∗) , qρ

∗= qρ

∗(t, q) , ρ, σ = 1, s ,

or in differential form

δqσ =∂qσ

∂qρ

∗

δqρ

∗, δqρ

∗=

∂qρ

∗

∂qσδqσ , ρ, σ = 1, s .

The quantities δqσ and δqρ

∗ , related by the these expressions, are called con-travariant components of the tangent vector δy and the whole set of the

105

106 IV. Application of a Tangent Space

vectors δy is called a tangent space to the above-introduced manifold at thegiven point [63]. It is useful to represent the vector δy as

δy = δqσ eσ , σ = 1, s ,

and to regard a family of the vectors eσ as a fundamental basis of tangentspace in the system of coordinates qσ.

We introduce an Euclidean structure in a tangent space, making use ofthe invariance of the positively defined quadratic form

(δy) 2 = gστ δqσ δqτ = g∗σ∗τ∗ δqσ

∗

∗δqτ

∗

∗, σ, τ, σ∗, τ∗ = 1, s .

Here gστ and g∗σ∗τ∗ are coefficients, entering into the relation of kinetic energy

in terms of the coordinates qσ and qρ

∗ (ρ, σ = 1, s), respectively. They, thus,prescribe the metric tensor, by which the scalar product of the vectors a =aσeσ and b = bτeτ can be represented as

a · b = gστ aσbτ , gστ = eσ · eτ , σ, τ = 1, s .

The components δqσ, σ = 1, s, of the tangent vector δy are also called thevariations of the coordinates qσ or the admissible (virtual) displacements. Bydefinition, the generalized forces Qσ, entering into system of equations (1.1),are the coefficients of the variations of coordinates δqσ in the expression forthe virtual elementary work δA. Using through numbering µ = 1, 2, 3, . . . fordenoting as the Cartesian coordinates of the points of forces application asthe projections of these forces, we can write

δA = Xµ δxµ .

Taking into account that

δxµ =∂xµ

∂qσδqσ =

∂xµ

∂qρ

∗

δqρ

∗,

we obtainδA = Qσ δqσ = Q∗

ρδqρ

∗, (1.2)

where

Qσ = Xµ

∂xµ

∂qσ, Q∗

ρ= Xµ

∂xµ

∂qρ

∗

= Qσ

∂qσ

∂qρ

∗

.

Relation (1.2) is a linear invariant differential form of the vector δy. Thecoefficients Qσ and Q∗

ρof this form, in terms of the coordinates qσ and q

ρ

∗ ,respectively, become the components of the covariant vector Y [63]. Takinginto account the Euclidean structure of tangent space, we can represent thequantity δA as the scalar product

δA = Y · δy , Y = Qσeσ , σ = 1, s .

1. The partition of tangent space 107

Here eσ, σ = 1, s, are the vectors of reciprocal basis, which are given by thefollowing relations

eσ · eτ = δσ

τ=

0, σ = τ ,

1, σ = τ .

Then from the relations gστ = eσ · eτ we have

eτ = gστeσ , eσ = gστeτ .

The coefficients gστ are the elements of the matrix inverse to the matrix withthe elements gστ .

The introduction of the covariant vector Y, using the relation for thevirtual elementary work δA, permits us to regard system of equations (1.1)as a one vector relation

MW = Y . (1.3)

Here

W = Wσeσ =1

M

(d

dt

∂T

∂qσ−

∂T

∂qσ

)eσ =

=(gστ qτ + Γσ,αβ qαqβ

)eσ = W σeσ =

(qσ + Γσ

αβqαqβ

)eσ ,

Γσ

αβ= gστΓτ,αβ =

1

2gστ

(∂gτβ

∂qα+

∂gτα

∂qβ−

∂gαβ

∂qτ

),

τ, σ = 1, s , α, β = 0, s .

(1.4)

Thus, by formulas (1.3) and (1.4) we can introduce the acceleration vectorW for arbitrary mechanical system with s degrees of freedom.

Consider now a constrained motion. By the releasability principle the im-position of constraints leads to occurrence the reaction force R and thereforeNewton’s second law can be written in the following way:

MW = Y + R .

The reaction force arises out of the acceleration, generated by constraints.Therefore it is necessary to clear up the influence of constraints on generatingthe vector W.

Consider first nonlinear first-order nonholonomic constraints, given by therelations

fκ

1 (t, q, q) = 0 , κ = 1, k .

Differentiating these constraints in time, we obtain

fκ

2 (t, q, q, q) ≡ al+κ

2σ(t, q, q) qσ + al+κ

20 (t, q, q) = 0 ,

κ = 1, k , l = s − k .(1.5)

Note that the linear second-order nonholonomic constraints can be given inthe same form. The holonomic constraints give relations (1.5) after a doubledifferentiation in time.

108 IV. Application of a Tangent Space

The introduction of a tangent space and the vector W, given by formu-las (1.4), allows us to represent system of equations (1.5) in the vector form:

εεεl+κ · W = χκ

2 (t, q, q) ,

εεεl+κ = al+κ

2σeσ , χκ

2 = −al+κ

20 + al+κ

2σΓσ

αβqαqβ ,

κ = 1, k , α, β = 0, s .

(1.6)

The vectors εεεl+κ, κ = 1, k, corresponding to constraints (1.5), are assumedto be linearly independent. Therefore in an s-dimensional tangential space wecan consider a subspace with the basis consisting of these vectors (K-space).Then the whole space can be represented as the direct sum of this subspaceand its orthogonal complement with the basis εεελ, λ = 1, l (L-space), in whichcase we have

εεελ · εεεl+κ = 0 , λ = 1, l , κ = 1, k .

We remark that this partition of a tangent space by the equations of con-straints corresponds to the fixed values of the variables t, qσ, qσ (σ = 1, s).

Substituting the acceleration W given by

W = WL + WK ,

WL = Wλ εεελ , WK = Wl+κ εεεl+κ WL · WK = 0 ,(1.7)

in equations (1.6), we obtain

Wl+κ∗ = hκ

∗κ χκ

2 (t, q, q) , κ, κ∗ = 1, k . (1.8)

Here hκ∗κ are elements of the matrix inverse to the matrix with the ele-

ments hκκ∗

, given by the following relations

hκκ∗

= εεεl+κ · εεεl+κ∗

, κ, κ∗ = 1, k . (1.9)

The vectors εεεl+κ, κ = 1, k, are linearly independent and therefore we have

|hκκ∗

| = 0 . (1.10)

Taking into account relations (1.7), we represent Newton’s second law astwo equations

MWL = YL + RL ,

YL = Qλεεελ , RL = Rλεεελ , λ = 1, l ,

MWK = YK + RK ,

YK = Ql+κ εεεl+κ, RK = Λκ εεεl+κ, κ = 1, k .

(1.11)

Here R = RL + RK is a constraint reaction, in which case the componentsRl+κ of the vector RK are denoted by Λκ since they are just the Lagrangemultipliers. If condition (1.10) is satisfied, then relations (1.7)–(1.10) imply

2. The connection of differential variational principles of mechanics 109

that the vector WK is uniquely defined by the equations of constraints interms of a function of the variables t, qσ, qσ. Thus, in K-space the law ofmotion is given by equations of constraints and takes the form (1.6). Thecomponent of constraint RK , which occurs in this case, is computed by meansof the second equation of system (1.11).

The equations of constraints cannot influence the vector WL since it canbe eliminated from equations (1.6). Therefore we have only indirect influenceof constraints on the component of acceleration WL via the vector RL. Inparticular, the equations of constraints can also be satisfied for RL = 0. Suchconstraints are called ideal. Thus, the influence of ideal constraints on the

acceleration W is completely defined by their analytic representations.Pay attention, that it was necessary to represent all kinds of constraints

in a single differential form (1.5) to find out how the constraints influence thegeneration of the constraint reaction. It is this form of writing the constraintequations that makes it possible to show that the whole space is partitioned bythe constraint equations into two orthogonal subspaces. As this takes place,the analytical expressions for constraint reactions were found. We note, thatfor the first time these results were given in the monograph [189. 1985]. Aswas shown in "Survey of the main stages of development of nonholonomicmechanics"the same results were obtained with the help of matrix calculusby F. E. Udwadia and R. E. Kalaba [394] in 1992.

§ 2. The connection of differential

variational principles of mechanics

Taking the partial derivative δ′′ of both sides of relations (1.6) for fixedt, qσ, qσ, we obtain

εεεl+κ · δ′′W = εεεl+κ · δ′′WL = 0 , κ = 1, k . (2.1)

Then from relation RK = Λκ εεεl+κ it follows that RK · δ′′W = 0. For idealconstraints we find

RK = R = MW − Y ,

and therefore(MW − Y) · δ′′W = 0 . (2.2)

Henceδ′′(W − Y/M)2 = 0 . (2.3)

The relation represents Gaussian principle. This principle is obtained herefrom the fact that constraints are ideal but it can also be regarded as adefinition of ideal constraints.

Consider now how the condition of ideality of constraints can be describedwith the usage of definition of the vector of reaction R in terms of a covariantvector, which is given by the relation for the invariant differential form

δAR = R · δy .

110 IV. Application of a Tangent Space

Relations (2.1) and (2.2) yield that the quantity δAR, which is given on a setof virtual displacements δy satisfying the system of equations

εεεl+κ · δy = 0 , κ = 1, k , (2.4)

is equal to zero for R = RK , i. e. for RL = 0.Thus, the condition of the ideality of constraints (1.5) is as follows

(MW − Y) · δy = 0 , (2.5)

which is the generalization of the notation of the D’Alembert–Lagrange prin-ciple. Taking into account that

εεεl+κ =∂fκ

1

∂qσeσ = ∇∇∇′fκ

1 ,

relations (2.4) and (2.5) become

∂fκ

1

∂qσδqσ = 0 , κ = 1, k , (2.6)

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 . (2.7)

The connection of Gaussian principle (2.2), (2.3) with the generalizedD’Alembert–Lagrange principle (2.5) — (2.7) was considered by N.G. Chetaev[245, p. 68], using another approach. As is already mentioned in § 6 of Chap-ter II, he imposed conditions (2.6) on virtual displacements and attemptedto ". . . introduce a notion of virtual displacement for nonlinear constraintsso that the D’Alembert principle and Gaussian principle . . . to be saved to-gether". Note that in the same way this question is considered in the paperof G. Hamel, published in 1938 [314]. As is already mentioned in Chapter II,the Chetaev–Hamel conditions (2.4), (2.6) have played the great importantrole in the development of the nonholonomic mechanics. From these condi-tions, in particular, it follows that for the fixed t, qσ, qσ the tangential spaceis partitioned by the equations of constraints into the subspaces K and L.

In the works [31, 280] the partition into two orthogonal subspaces is givenin the matrix form. The authors make use of this partition in order to elim-inate the Lagrange multipliers from equations of motion. In these works forthe study of the dynamics of system of rigid bodies the computer algorithmsare constructed. The partition into two orthogonal subspaces is also used inthe works [50, 275, 305, 392].

We now turn to equation (2.5). This equation together with equations(2.4) shows that for ideal constraints the component of reaction vector islacking in the subspace such that for fixed values of the variables t, qσ, qσ,σ = 1, s, any acceleration WL is kinematically admissible. Therefore theD’Alembert–Lagrange principle (2.5) is, in fact, a principle of virtual acceler-ations. It was called first a principle of virtual velocities [4] and then a general

2. The connection of differential variational principles of mechanics 111

(fundamental) equation of dynamics. It was applied to linear nonholonomicconstraints, in particular, in the works [379, 398]. A detail survey of the workson the nonholonomic mechanics can be found in the work [370. 1998, 2002].

In the case of holonomic constraints, given by the equations

fκ

0 (t, q) = 0 , κ = 1, k , (2.8)

the restrictions on the vector W can also be represented in the form of (1.6).In this case the vectors εεεl+κ, entering into equations (1.6) and (2.4), are thefollowing

εεεl+κ =∂fκ

0

∂qσeσ = ∇∇∇fκ

0 , κ = 1, k ,

i. e. they coincide with the usual gradients to surfaces, given by the equa-tions of constraints. In the particular case of the equilibrium of mechanicalsystem the D’Alembert–Lagrange principle passes to the principle of virtualdisplacements, namely

Qσ δqσ = Xµ δxµ = 0 , (2.9)

in which case by virtue of the constraints given by equations (2.8), missingwith time t, the quantities δqσ satisfy the relations

∂fκ

0

∂qσδqσ = 0 , κ = 1, k .

The conditions, under which the D’Alembert–Lagrange principle and, in par-ticular, the principle of virtual displacements (2.9) are satisfied, are consid-ered in detail in the treatises of G. K. Suslov [220] and A. M. Lyapunov [138].In this connection G. K. Suslov writes [220, p. 380]: "Many attempts weremade to find a strong proof of the principle of virtual displacements . . . ".Considering in detail two such attempts realized by Lagrange and Ampere, hearrived at the following general conclusion "any proof of the considered princi-ples . . . , strictly speaking, cannot be called a proof, i. e. a reduction to the ac-knowledged verities only". A similar conclusion was made by A. M. Lyapunov.In the present section we show how principles (2.7), (2.9) are connected withthe restrictions, imposed on the acceleration vector of system W by meansof the equations of constraints.

Consider equation (2.2). It implies that by ideal constraints the invariantdifferential form δAR = R·δy goes identically to zero on a set of the tangentialvectors δy such that, as is shown by V. V. Rumyantsev [199], they are thefollowing

δy =τ2

2δ′′W =

τ2

2δ′′WL . (2.10)

Here τ is an infinitely small time interval, introduced by Gauss.

112 IV. Application of a Tangent Space

By formulas (1.5) and (1.6) and taking the partial differential δ′ of theequations of nonholonomic constraints for the fixed values of the variables t

and qσ, σ = 1, s, we obtain

δ′fκ

1 =∂fκ

1

∂qσδ′qσ = ∇∇∇′fκ

1 · δ′V = εεεl+κ · δ′V = 0 , κ = 1, k ,

whereδ′V = δ′qσ eσ .

Then equations (2.4) yields that, following the work [199], we can representthe tangential vector δy, written in the form (2.10), as as

δy = τ δ′V . (2.11)

This relation permits us to identify the notion of virtual velocity and virtualdisplacement (see the notion: "virtual velocities (displacements) in the work[202, p. 5]). By the change (2.11) the generalized D’Alembert–Lagrange (2.5)principle pass to the Suslov–Jourdain principle

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δ′qσ = 0 .

A detailed analysis of the differential and integral variational principlesof mechanics and a detailed survey of the works, devoted to them, can befound in the treatise of V. N. Shchelkachev [254].

The representation of the vector δy, entering into the generalizedD’Alembert–Lagrange (2.5) principle in the form of (2.10), shows clearlythe substance of these principles. It established that in the case of idealconstraints in L-space, which the vectors δ′′WL belong to, the reaction islacking since the equations of constraints cannot effect the acceleration WL

and generate it. They generate the acceleration WK only, what is providedby the reaction RK together with the active force YK . In other words, inL-space the reaction is lacking since in it the equations of constraints "do notconstrain"the mechanical system to have the acceleration WL. The expres-sion: "do not constrain is taken from the formulation of Newton’s first law inthe translation of A. N. Krylov.

Formulas (2.10) and (2.11), as was emphasized by V. V. Rumyantsev [199],confirm the conclusion of N. G. Chetaev on the compatibility of differentialvariational principles of Gauss, D’Alembert–Lagrange, and Suslov–Jourdain.

Generalizing the approach to the notion of ideality of constraints, we ar-rive at the following definition: the nonholonomic constraints of any order,

which are linear in high derivatives, are ideal if for the fixed values of the

variables t, qσ, qσ, σ = 1, s, the equations of constraints discriminate in tan-

gential space the L-space such that the mechanical system is not constrained

by equations of constraints to have the acceleration WL different from the

acceleration given by Newton’s law

MWL = YL .

3. Generalized Gaussian principle 113

In § 3 of Chapter V the problem is considered: what constraints are idealfor n 3 ?

§ 3. Geometric interpretation of linear

and nonlinear nonholonomic constraints.

Generalized Gaussian principle

Consider the velocity vector V of mechanical system, using the vector ofgeneralized impulse

p =∂T

∂qσeσ ,

and putting, by definition, that

V = p/M = Vσeσ = (gστ qτ + gσ0)eσ =

= (gστ qτ + gσ0)gσρeρ = V ρ eρ = (qρ + gσ0g

σρ)eρ ,

ρ, σ, τ = 1, s .

(3.1)

Denotinge0 = gσ0 eσ , σ = 1, s ,

we havegαβ = eα · eβ , V = qα eα , α, β = 0, s .

The introduction of the velocity vector V makes it possible to represent akinetic energy of an arbitrary mechanical system and a vector of its accelera-tion in just the same form as in studying the motion of one mass point. Thispermits us to use in the theory of constrained motion of arbitrary mechanicalsystem the apparatus of analytic and differential geometry, extended to thecase of s-dimensional Euclidean space.

Then we have

T =MV2

2=

MVσV σ

2,

W = V = Vσ eσ + Vσ eσ = V σ eσ + V σ eσ , σ = 1, s .

(3.2)

Taking into account that eσ · eτ = −eσ · eτ , we obtain

Wτ = W · eτ = Vτ − V · eτ , τ = 1, s .

In this case relations (1.1), (1.4), (3.1), (3.2) yields the relations

eσ =∂V

∂qσ=

∂W

∂qσ=

∂(n)

V

∂(n+1)

qσ

, σ = 1, s , (3.3)

114 IV. Application of a Tangent Space

eσ = Γτ

σαqα eτ =

∂V

∂qσ=

1

2

∂W

∂qσ=

1

(n + 1)

∂(n)

V

∂(n)

qσ

,

α = 0, s σ = 1, s .

(3.4)

The notation(n)

qσ and(n)

V , correspond to the n-th derivatives in time of thefunction qσ and the vector V, respectively.

By formulas (3.3) and (3.4) for computing the n-th derivative in time ofthe vector MV we can use the generalized operator of Appell

M(n)

V =∂Tn

∂(n+1)

qσ

eσ , Tn =M

(n)

V2

2, n 0 , T0 = T ,

and the generalized operator of Lagrange

M(n)

V =

(d

dt

∂Tn−1

∂(n)

qσ

−1

n

∂Tn−1

∂(n−1)

qσ

)eσ ,

(for detail, see [252]).From formulas (3.2) and (3.4) we obtain that the contravariant compo-

nents of the vectors(n)

V and(n−1)

V are related as

(n)

V · eσ =d

dt

((n−1)

V · eσ

)+

((n−1)

V · eτ

)Γσ

ταqα ,

σ, τ = 1, s , α = 0, s .

(3.5)

This relation between the derivatives(n)

qσ and the covariant and contravari-

ant components of the vector(n−1)

V for the fixed values of the variables

t, qσ, qσ, . . . ,(n−1)

qσ , σ = 1, s, gives geometric interpretation as linear, as non-linear nonholonomic constraints of any order. Consider first the constraintsof the first order.

Relations (3.1) implies that the equations of linear nonholonomic con-straints

fκ

1 (t, q, q) ≡ al+κ

1σ(t, q) qσ + al+κ

10 (t, q) = 0 , κ = 1, k , l = s − k ,

impose the following restrictions on the velocity vector V:

εεεl+κ · V = χκ

1 (t, q) , χκ

1 = −al+κ

10 + al+κ

1σgστgτ0 ,

κ = 1, k , σ, τ = 1, s .(3.6)

In the present case the vectors

εεεl+κ = ∇∇∇′fκ

1 =∂fκ

1

∂qσeσ = al+κ

1σ(t, q) eσ , κ = 1, k ,

3. Generalized Gaussian principle 115

which partition, under condition (1.10), the tangential space into the sub-spaces K and L, are independent of the generalized velocities qσ, σ = 1, s.

System of equations (3.6) is similar to system (1.6). Therefore we canconclude that the component VK of the vector V as the function of thevariables t, qσ, σ = 1, s, is uniquely defined by equations of constraints. Thespace, which the vectors V belong to, has Euclidean structure. In this casefrom system of equations (3.6) it follows that the set of velocities V, admittedby linear first-order nonholonomic constraints, is an l-dimension plane. Fors = 2, l = k = 1, we have a straight line (Fig. IV. 1).

Fixed the values of the variables t and qσ, σ = 1, s, by means of for-mulas (3.1) we can relate the generalized velocities qσ, σ = 1, s, and thecomponents of the vector V. This implies that for the fixed values of thequantities t, qσ, σ = 1, s, the system of equations of nonlinear first-ordernonholonomic constraints can be regarded as a system of equations, whichgives an l-dimensional surface in the space of the velocities V. For s = 2,l = κ = 1, this is a curve on a plane. In particular, it is the closed curveshown in Fig. IV. 2.

Fig. IV. 1 Fig. IV. 2

In the case of linear nonholonomic constraints for the fixed values of t

and qσ, σ = 1, s, and for any velocity V, admitted by constraints, the me-chanical system has the same component VK of the vector V (Fig. IV. 1). Inparticular, for the uniform equations of constraints this component is equalto zero. The technical realization of such constraint can be easily made: forexample, it is the component of velocity perpendicular to a knife-edge andequal to zero. In the case of nonlinear constraints the directions εεελ, λ = 1, l,along which the equations of constraints admit the infinitely small changesof the vector V for the fixed values of the variables t and qσ, σ = 1, s, and,respectively, the directions εεεl+κ , κ = 1, k, along which such changes are im-possible, depend on the vector V (Fig. IV. 2). This substantially complicatesthe technical realization of nonlinear nonholonomic constraints as the con-straints between the mass bodies. Another point of view, connected with theexistence of nonlinear nonholonomic constraints, is discussed in the treatises[72, 166].

116 IV. Application of a Tangent Space

Consider the nonlinear second-order nonholonomic constraints

fκ

2 (t, q, q, q) = 0 , κ = 1, k . (3.7)

At present, we cannot cite an example such that even a linear second-order nonholonomic constraint is a result of a certain mechanical interactionof mass bodies. (An exception is the work [335]). Therefore we shall regardconstraints (3.7) as equations of a certain program of motion (as program

constraints). We can obtain a descriptive representation on the restriction,which one constraint (3.7) imposes on the vector W for s = 2, if in Fig. IV. 2the vector V is changed to the vector W.

Note that if in Figs. IV. 1 and IV. 2 we change the vector V and its com-

ponents VK and VL to the vectors(n−1)

V ,(n−1)

VK ,(n−1)

VL , respectively, thenFig. IV. 1 corresponds to linear n-order constraint and Fig. IV. 2 to nonlinearthat.

Consider now second-order constraints. For the linear in accelerationsconstraints the assumption that they are ideal, as shown above, results inthe minimality of Gaussian function

Z = (W − Y/M)2 , (3.8)

given on a set of the accelerations W, admitted by constraints, for the fixedt, qσ, qσ, σ = 1, s. We extend this property of ideality on nonlinear con-straints (3.7).

Necessary conditions of minimality of the function Z for constraints (3.7)are the following [210]

δ′′fκ

2 = ∇∇∇′′fκ

2 · δ′′W = 0 , κ = 1, k ,

2M δ′′Z = (MW − Y) · δ′′W = 0 .(3.9)

We can represent the derivatives δ′′fκ

2 , κ = 1, k, in above form since byrelations (1.4) the vector δ′′W can be represented as

δ′′W = δ′′qσeσ .

Introduce the Lagrange multipliers, using the classical approach [210]. Thenfrom system of equations (3.9) we obtain the relation

MW = Y + Λκ ∇∇∇′′fκ

2 , (3.10)

which in passing to the scalar form becomes the system of Lagrange’s equa-tions of the first kind. Note that for the nonlinear in accelerations constraintsthis system is used in the work [60].

Consider a geometric interpretation of equation (3.10). We consider firstthe case of linear constraints, given by equations (1.5). For the first-ordernonholonomic constraints and the holonomic constraints, we obtain these

3. Generalized Gaussian principle 117

equations by differentiating in time one and two times, respectively, the equa-tion of these constraints. Then for n = 2, n = 1, and n = 0 the vectors∇∇∇′′fκ

2 , κ = 1, k, entering into equation (3.10), can be represented as

∇∇∇′′fκ

2 = al+κ

2σeσ = εεεl+κ , κ = 1, k .

For n = 1 and n = 0 we have

εεεl+κ = ∇∇∇′fκ

1 , εεεl+κ = ∇∇∇fκ

0 , κ = 1, k ,

respectively. The l-dimensional plane in a space of the accelerations W isgiven by system of equations (1.5), represented in vector form (1.6). This ge-ometric approach is illustrated in Fig. IV. 3, which corresponds to the motionof one mass point with one constraint.

From all acceleration vectors, admitted by constraints, we discriminatethe vector WK that is minimal in absolute magnitude. To construct thisvector, we drop a k-dimensional perpendicular from the origin of coordinateson the considered plane. The crosspoint of this perpendicular and the planecorresponds to the end of the vector WK , beginning from the origin of coor-dinates. This implies that the vector WK is uniquely defined by equations ofconstraints and can be represented as WK = Wl+κ εεεl+κ . Adding to the vec-tor WK the arbitrary vector WL, which is in the plane, we obtain the vectorW, satisfying the equations of constraints. Using vector Lagrange’s equationof the first kind (3.10), we choose then from all these vectors a unique accel-eration W such that a distance between the point, given by the vector Y/M ,and the l-dimensional plane considered is minimal. This method to choice thevector W is geometric interpretation of the notion of ideality of constraints.

To this simple and natural choice of the vector W from all accelerations,admitted by constraints, corresponds Gaussian principle of virtual accelera-tions, which, as shown in § 2, can, in turn, be represented in the form of theD’Alembert–Lagrange principle.

We proceed now to the case when equations of constraints (3.7) dependnonlinearly on the generalized accelerations qσ, σ = 1, s. Using formulas

Fig. IV. 3

118 IV. Application of a Tangent Space

(1.4), we obtain the relation between the quantities qσ and the componentsof the vector W. This allows us to regard system of equations (3.7) for thefixed values of the quantities t, qσ, qσ, σ = 1, s, as a system of equations,which gives an l-dimensional surface in the space of the accelerations W. Theacceleration W, satisfying equation (3.10) and equations of constraints (3.7),corresponds to the point, of this surface, which is at the minimal distancefrom the end of the vector Y/M . The difficulties occur in the case when thispoint is not single. The difficulties also occur in the case when the vectors∇∇∇′′fκ

2 , κ = 1, k, depend nonlinearly on the generalized accelerations qσ,σ = 1, s. In the case when this dependence is linear the solution of problem,as shown in a concrete example in Chapter V, can be sufficiently simple.

Now we turn to equations (3.9). Following the work of V. V. Rumyantsev[199], we consider the vector

δy =τ2

2δ′′W .

Then system of equations (3.9), represented in scalar form, is as follows

∂fκ

2

∂qσδqσ = 0 , κ = 1, k , (3.11)

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 . (3.12)

It follows that for the nonlinear in velocities nonholonomic constraints, theapproach of N. G. Chetaev [245] to the virtual displacements δqσ, σ = 1, s,which enter into the D’Alembert–Lagrange principle (3.12), can also be ex-tended to the nonlinear accelerations of constraints (3.7) by means of formulas(3.11). For the first time this was made by G. Hamel in 1938 [314].

For the nonlinear in accelerations constraints, to each point of l-dimensionalsurface, introduced above, corresponds its own tangential plane, while for theconstraints, depending linearly on the generalized accelerations qσ, σ = 1, s,we have a unique plane, given by equations (1.6). The point, of this surface,for which the tangential plane, containing the vectors δy = (τ2/2) δ′′W, isintroduced, is at the shortest distance from the end of the vector Y/M . Thenfor the nonlinear in accelerations constraints, in equation (3.12), representedin vector form

(MW − Y) · δy = 0 , (3.13)

the directions of the vector of reactions R = MW − Y and the vectors δy

depend on the vector Y. Unlike this case for the linear in accelerations con-straints (1.5), the directions of these vectors are independent of the vector Y.

Generalized Gaussian principle. Consider now linear third-order con-straints. In this case an l-dimensional plane in the space W is given by equa-tions of these constraints. Replacing in Fig. IV. 3 the quantities xk by

...x k,

k = 1, 2, 3, and the vectors W, WK , WL, Y by the vectors W, WK , WL,

4. Gaussian principle in Maggi’s form 119

Y, respectively, we obtain the picture, illustrating a geometric interpretationof generalized Gaussian principle. According to this principle the function

Z1 = (W − Y/M)2

is minimal on a set of the vectors W, admitted by the constraints for thefixed values of the variables t, qσ, qσ, qσ, σ = 1, s [59. 1976, 177. 1997, 188.1983, 252. 1974]. This assertion is equivalent to that the following variation

δ′′′(W − Y/M)2 = 0 (3.14)

is equal to zero. Here three accents after the differentiation symbol δ meanthat the third derivatives in time of generalized coordinates is varied only.

The above reasonings can be extended to the case of constraints of anyorder. Thus, if the linear nonholonomic constraints of any order (programconstraints) are given in the form of linear differential equations of order(n + 2), then it is rational to construct a system of differential equations,which completes the given system, on the principle that

δ(n+2)((n)

W −(n)

Y/M)2 = 0 , n 1 . (3.15)

Here the index (n) points out an order of derivative in time with resoect tothe vector and the index (n + 2) means that a partial derivative is computed

for the fixed t, qσ, qσ, . . . ,(n+1)

qσ .

Formula (3.15) is a generalization of Gaussian principle to the case ofhigh-order nonholonomic constraints. Note that, in using principle (3.15), atthe initial moment of time we assume that all of the coordinates qσ and alltheir derivatives up to be order (n + 1) and therefore the vector R and itsderivatives up to be order (n − 1) are given.

We make use of generalized Gaussian principle (3.14) for the certain typesof equations of motion of nonholonomic systems for the linear third-orderconstraints to be found. Besides, generalized Gaussian principle will be usedin § 4 of Chapter V when studying the motion of a spacecraft with constantin modulo acceleration and in § 9 of Chapter VI when solving the problem ofdamping the vibration of elastic systems.

§ 4. The representation of equations of motion

following from generalized Gaussian principle

in Maggi’s form

Generalized Gaussian principle for constraints of any of orders was consid-ered first by M. A. Chuev in 1974 [252]. He also suggests the different forms,of notation for equations of motion, following from this principle [59, 252].In the present and next sections we consider Maggi’s and Appell’s forms forlinear third-order constraints only [76].

120 IV. Application of a Tangent Space

So, we assume that the linear third-order differential constraints

αl+κ

∗= cl+κ

σ(t, q, q, q)

...q σ + cl+κ

0 (t, q, q, q) = 0 ,

κ = 1, k , l = s − k ,(4.1)

are imposed on the motion of a system, the position of which is given by thegeneralized coordinates qσ, σ = 1, s. As shown in § 1, the system of Lagrange’sequations of the second kind can be reduced to one vector relation equivalentto Newton’s second law:

MW = Y + R ,

where the vectors belonging to the space tangential to the manifold of virtualpositions of system can be represented as

MW =

(d

dt

∂T

∂qσ−

∂T

∂qσ

)eσ = MWσeσ , Y = Qσeσ , R = Rσeσ .

Given constraints (4.1), for a generalized Gaussian principle to be used, thisexpression for Newton’s second law is differentiated in time. Then we obtain

MU = P + G , U = W , P = Y , G = R . (4.2)

According to formulas (4.2), generalized Gaussian principle (3.14) takes theform

δ′′′(MU − P)2 = 0 . (4.3)

In this formula three accents after the differentiation symbol δ mean that thepartial derivative is computed for the fixed t, qσ, qσ, qσ, σ = 1, s. By (4.2)principle (4.3) is as follows

δ′′′G2 = 0 .

Thus, given constraints (4.1), in accordance with generalized Gaussian prin-ciple the vector G is chosen to be minimal in absolute magnitude.

We rewrite principle (4.3) in the form

(MUσ − Pσ) δ′′′...q σ = 0 . (4.4)

Using formula (A.52) from Appendix A, we have

MUσ − Pσ = MWσ − Qσ − Γρ

στ(MWρ − Qρ) qτ . (4.5)

Then, we supplement system (4.1) with the equations

αλ

∗= cλ

σ(t, q, q, q)

...q σ + cλ

0 (t, q, q, q) , λ = 1, l , l = s − k . (4.6)

The family of equations (4.1) and (4.6) can be regarded as the transitionformulas from

...q σ to α

ρ

∗, σ, ρ = 1, s. If det [cρσ] is not equal to zero, then we

can write the inverse transformation

...q σ = hσ

ρ(t, q, q, q)αρ

∗+ hσ

0 (t, q, q, q) , ρ, σ = 1, s . (4.7)

5. Gaussian principle in Appell’s form 121

It follows thatδ′′′

...q σ = hσ

ρδ′′′αρ

∗, ρ, σ = 1, s .

However, since the constraints (4.1) are valid, we obtain δ′′′αl+κ

∗= 0, κ =

1, k, and therefore

δ′′′...q σ = hσ

λδ′′′αλ

∗, λ = 1, l , ρ, σ = 1, s . (4.8)

Substituting formulas (4.8) into principle (4.4), we have

(MUσ − Pσ)hσ

λδ′′′αλ

∗= 0 , λ = 1, l .

Since the variations δ′′′αλ∗, λ = 1, l, are independent, we obtain the following

equations of motion of system

(MUσ − Pσ)hσ

λ= 0 , λ = 1, l , (4.9)

which by formulas (4.5) can be represented finally in the form

(MWσ − Qσ − Γρ

στ(MWρ − Qρ) qτ

)hσ

λ= 0 , λ = 1, l . (4.10)

Equations (4.9) have the same structure as Maggi’s equations and there-fore they can be called the equations, represented in Maggi’s form, for third-

order constraints (4.1).The law of motion can be found after the solution of system of equa-

tions (4.10) and (4.1), in which case for their integration the initial values ofgeneralized coordinates, velocities, and accelerations have to be given.

§ 5. The representation of equations of motion

following from generalized Gaussian principle

in Appell’s form

Since the vectors eσ, σ = 1, s, can be represented by formulas

eσ =∂W

∂qσ=

∂W

∂...q σ , W = U ,

the quantities MUσ, σ = 1, s, introduced in the previous section, take theform

MUσ = MU · eσ = MU ·∂U

∂...q σ =

∂S1(t, q, q, q,...q )

∂...q σ , (5.1)

where S1 = MU2/2. According to relations (4.7) and (5.1) equations (4.9)are the following

∂S1

∂...q σ

∂...q σ

∂αλ∗

= Pσ

∂...q σ

∂αλ∗

, λ = 1, l , σ = 1, s . (5.2)

122 IV. Application of a Tangent Space

Now we represent equations (5.2) as

∂S1

∂αλ∗

= P ∗

λ, λ = 1, l , (5.3)

where

P ∗

λ= Pσ

∂...q σ

∂αλ∗

.

Taking into account that

P ∗

λ= Pσ

∂...q σ

∂αλ∗

= P ·∂U

∂αλ∗

=∂(P · U)

∂αλ∗

,

equations (5.3) have the form

∂(S1 − P · U)

∂αλ∗

= 0 , λ = 1, l . (5.4)

We introduce in place of the function S1 − P · U the function

Z1 = S1 − P · U +1

2MP2 =

M

2

(U −

P

M

)2

> 0 ,

for which the following relation

∂Z1

∂αλ∗

=∂(S1 − P · U)

∂αλ∗

,

is satisfied since ∂P2/∂αλ∗

= 0. In this notation equations of motion (5.4) arethe following

∂Z1

∂αλ∗

=∂Z1

∂...q σ

∂...q σ

∂αλ∗

= 0 , λ = 1, l .

Represent these equations in the form of the scalar products

∇∇∇′′′Z1 · εεελ = 0 , λ = 1, l , (5.5)

where

∇∇∇′′′ =∂

∂...q σ eσ , εεελ =

∂...q σ

∂αλ∗

eσ = hσ

λeσ .

Comparing equations (5.5) and equations (4.9), we obtain

MU − P = ∇∇∇′′′Z1 . (5.6)

From equations (5.5) it follows that for the value U, corresponding to areal motion, the function Z1(U) has the value, which is minimal in comparisonwith the value Z1(U1) for any other U1, which is kinematically admissiblefor the same t, qσ, qσ, qσ. This shall be shown below.

5. Gaussian principle in Appell’s form 123

The vector U can be represented as

U = UL + UK , UL · UK = 0 .

Here

UL = Uλεεελ , UK = Ul+κ∇∇∇′′′fκ

3 , λ = 1, l , κ = 1, k ,

in which case for the fixed t, qσ, qσ, qσ the vector UK is completely determinedby the equations of constraints and the vector UL remains arbitrary. In otherwords, any vector UL is kinematically admissible. This implies that

U1 − U = U1L + UK − UL − UK = ∆Uλεεελ , λ = 1, l ,

where ∆Uλ are arbitrary. Using this relation for U1 in the function Z1 andtaking into account relations (5.5) and (5.6), we have

Z1(U1) =M

2

(U −

P

M+ ∆Uλεεελ

)2

=

= Z1(U) +∇∇∇′′′Z1 · ∆Uλεεελ +M

2

(∆Uλεεελ

)2=

= Z1(U) +M

2

(∆Uλεεελ

)2> Z1(U) , U1 = U .

Condition Z1(U1) > Z1(U) is obtained here from equations of constrainedmotion (5.5). However equations (5.2)–(5.5) can also be regarded as necessaryconditions of minimality for the function Z1 under constraints (4.1).

Equations (5.2)–(5.5) have a structure, suggested by Appell for mechan-ical systems with the constraints up to be the second order. Therefore theycan be called equations of Appell’s form with third-order constraints.

In many works (see, for example, [193]) the equations of motion for non-holonomic system with the constraints of the form (4.1) are represented (usingthe notation of the work [193]) as

∂S

∂αλ∗

= Q∗

λ, λ = 1, l , (5.7)

where

S =dS

dt= MW · W , Q∗

λ= Qσhσ

λ.

Equations (5.7) are obtained from Appell’s equations

∂S

∂qσ= Qσ + Rσ , σ = 1, s ,

in a formal way and therefore cannot be regarded as a minimum condition forthe function Z1. This is a foundational distinction between equations (4.9),(5.2)–(5.5) and equations (5.7).

124 IV. Application of a Tangent Space

Finally, it should be specially accented that in a number of problems a for-mal application of generalized Gaussian principle to the analysis of motion ofmechanical system can give rise to unexpected results [79]. Therefore the gen-eral theory of motion of nonholonomic systems with high-order constraints,established in Chapter V, is noteworthy.

C h a p t e r V

THE MIXED PROBLEM OF DYNAMICS.

NEW CLASS OF CONTROL PROBLEMS

The law, of a motion of mechanical system, represented in the vector form,

is applied to the solution of the mixed problem of dynamics. The essence of

the problem is to find additional generalized forces such that the program

constraints, given in the form of system of differential equations of order

n 3, are satisfied. The notion of generalized control force is introduced.

The fact is proved that if the number of program constraints is equal to the

number of generalized control forces, then the latter can be found as the time

functions from the system of differential equations in generalized coordinates

and these forces. The conditions, under which this system of equations has

a unique solution, are determined. The conditions are also obtained under

which for the constraints of any order the motion control is realized according

to Gauss’ principle. Thus, the theory is constructed with the help of which a

new class of control problems can be solved. This theory is used to consider

two problems connected with the dynamics of spacecraft motion. In the first

problem a radial control force, providing the motion of spacecraft with modulo

constant acceleration, is determined as a time function. In the second problem

we seek the law, of varying in time the radial and tangential control forces,

by which a smooth passage of spacecraft from one circular orbit to another

occur.

§ 1. The generalized problem of P. L. Chebyshev.

A new class of control problems

As is well known, P. L. Chebyshev is a founder of the theory of synthesisof mechanisms. He has posed and solved the problem of constructing themachines, the concrete points of which are subject to a given motion. Amongsuch devices we can recall, for example, the mechanisms with the stoppingsof definite elements in the given positions. We generalize this Chebyshev’sproblem to the case that the motion of certain points of mechanism is asolution of the given differential equations of order n 3.

The following is noteworthy. As is noted by L. A. Pars [179] andV. V. Rumyantsev [199], the forces cannot depend on accelerations. Howeverwe can always choose the system of forces, realizing the required motion ofmechanical system qσ = qσ(t), σ = 1, s, in which case any law of varyingany of derivatives with respect to generalized coordinates can be satisfied.We can, by that, provide the vanishing of any combination with respect toderivatives of generalized coordinates of system. Therefore it is obvious that

125

126 V. The Mixed Problem of Dynamics. New Class of Control Problems

we can require such a motion of mechanical system that the system of dif-ferential equations of any order is satisfied. Taking into account the abovenotes, we consider the following problem.

Suppose, in the generalized coordinates qσ the motion of mechanical sys-tem acted by the given generalized forces Qσ is described by Lagrange’sequations of the second kind

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ , T =

M

2gαβ qα qβ ,

σ = 1, s , α, β = 0, s , q0 = t , q0 = 1 ,

(1.1)

where M is a mass of a whole system.It is necessary to find as functions of time the forces Rσ, which have to

be added to the forces Qσ in order that the motion satisfies the followingsystem of differential equations

fκ

n≡ aκ

nσ(t, q, q, ... ,

(n−1)q )

(n)

qσ + aκ

n0(t, q, q, ... ,(n−1)

q ) = 0 ,

σ = 1, s , κ = 1, k , k s .

(1.2)

For n 3 we call this problem the generalized problem of P. L. Chebyshev.However, before we attack the problem, recall briefly some results, obtainedfor the constraints given in the form (1.2) for n = 1, 2. In this case the soughtadditional forces, which represented as

Rσ = Λκ

∂fκ

n

∂(n)

qσ

,

where the Lagrange multipliers Λκ, κ = 1, k, are uniquely determined asfunctions of variables t, qσ, qσ, σ = 1, s, if

det

[∂fκ

n

∂(n)

qσ

∂fµn

∂(n)

qτ

gστ

]= 0 ,

σ, τ = 1, s , κ, µ = 1, k .

Here gστ are elements of the matrix inverse to the matrix with the elementsgστ .

Note that for n = 1 there exists a sufficiently large class of problems, inwhich the forces Rσ, σ = 1, s, result from interactions of mass bodies, whatleads to the occurrence of the constraints of the form (1.2). A distinctivefeature here is that the generalized reactions Λκ , κ = 1, k, i. e. the forces,arising out of these interactions, are necessary and sufficient for realizing themotion that satisfy the equations of constraints. In this case the methods ofnonholonomic mechanics make it possible to find this motion without deter-mining the generalized reactions. For n = 2 we can also obtain this motioneven if the functions Λκ(t, q, q), κ = 1, k, are unknown. However they cannot

1. The generalized problem of P. L. Chebyshev. 127

be called generalized reactions since at present one cannot find the interactionbetween mass bodies, which leads to the occurrence of reactions, providingthe validity of the absolutely nonintegrable equations (1.2) for n = 2 (exceptfor the example, considered in the work [335]). Then the values Λκ , κ = 1, k,can be regarded only as the forces, which are given by control system. Itis clear that for n 3 the validity of equations (1.2) is also provided by acontrol system only. Therefore we introduce a notion of generalized controlforce.

So, we assume that a control system generates a certain force such thatits virtual elementary work is as follows

δA = Λ bσ(t, q, q)δqσ , σ = 1, s .

We shall say that the value Λ, entering into this relation, is a generalized

control force.Suppose, the k control forces Λκ , κ = 1, k, can be generated by the control

system. Then we have

δA = Λκbκ

σ(t, q, q)δqσ , κ = 1, k , σ = 1, s . (1.3)

Note that a mechanism, by which the control forces arise, is, as a rule, suchthat in relation (1.3) the coefficients bκ

σare either constant or the functions

of generalized coordinates only.From formulas (1.3) it follows that the additional generalized forces Rσ, σ =

1, s, corresponding to the generalized control forces Λκ , κ = 1, k, take theform

Rσ = Λκbκ

σ.

Below we show that if in the time interval t0 t t∗ we have

det[bκ

σaµ

nτgστ ] = 0 ,

σ, τ = 1, s , κ, µ = 1, k ,(1.4)

then for n = 1, 2, the generalized control forces Λκ, κ = 1, k, can uniquelybe found as the functions of the variables t, qσ, qσ, If n 3, then, as will beshown below, the generalized control forces Λκ, κ = 1, k, can be found as thefunctions of time only. In this case the differential equation with respect toeach of the functions Λκ is of order (n − 2). Thus, for n 3 the generalizedcontrol forces Λκ, κ = 1, k, and the generalized coordinates qσ, σ = 1, s, areregarded as the sought functions of time, satisfying the following initial data

Λκ(t0) = Λ0κ

, Λκ(t0) = Λ0κ

, ... ,(n−3)

Λκ (t0) =(n−3)

Λκ

0 ,

qσ(t0) = qσ

0 , qσ(t0) = qσ

0 , κ = 1, k , σ = 1, s .

(1.5)

In the next section we show that if in the time interval t0 t t∗ condition(1.4) is satisfied, then in this time interval for the given initial data (1.5) the

128 V. The Mixed Problem of Dynamics. New Class of Control Problems

motion and the generalized control forces can uniquely be found such thatrelations (1.2) are satisfied.

Thus, the generalized problem of P. L. Chebyshev can uniquely be solved.The statement of this problem has the features of as the direct, as inverseproblem of dynamics. Really, on the one hand, for the given forces Qσ, σ =1, s, we seek the motion of system and, on the other hand, we seek as thefunctions of time the additional forces Rσ, σ = 1, s, providing a motionsuch that relations (1.2) are satisfied for n 3. Therefore the academicianS. S. Grigoryan calls the generalized problem of P. L. Chebyshev the mixed

problem of dynamics. The solution of the problem, constructed below, per-mits us to find as functions of time the generalized control forces, the existenceof which is a necessary and sufficient condition in order that the motion sat-isfies a system of equations of any order. This construction is actual since itpermits us to solve a new sufficiently large class of control problems.

§ 2. A generation of a closed system of differential

equations in generalized coordinates

and the generalized control forces

For the solution of the mixed problem of dynamics it is rational to usethe notion of a tangent space, introduced in § 1 of Chapter IV. In this case ifthe forces Rσ are added to the forces Qσ, system of equations (1.1) can bewritten as one vector equation

MW = Y + Λκbκ , κ = 1, k , (2.1)

whereY = Qσeσ , bκ = bκ

σeσ ,

W = (gστ qτ + Γσ,αβ qαqβ)eσ = (qσ + Γσ

αβqαqβ)eσ ,

Γσ

αβ= gστΓτ,αβ =

1

2gστ

(∂gτβ

∂qα+

∂gτα

∂qβ−

∂gαβ

∂qτ

),

σ, τ = 1, s , α, β = 0, s , κ = 1, k .

(2.2)

Here eσ and eσ are the vectors of the fundamental and reciprocal bases of atangent space, respectively.

From relations (2.1) and (2.2) we have

qσ = F σ

2 (t, q, q, Λ) , Fσ

2 = −Γσ

αβqαqβ + (Qτ + Λκbκ

τ)gστ/M ,

σ, τ = 1, s , α, β = 0, s , κ = 1, k .(2.3)

Consider first the case n = 3. Using formulas (2.2) and taking into accountthat

eτ = Γσ

ταqαeσ , σ, τ = 1, s , α = 0, s ,

2. A generation of a closed system of differential equations 129

we represent system of equations (1.2) in vector form

aκ

3 · W = χκ

3 (t, q, q, q) , aκ

3 = aκ

3σeσ ,

χκ

3 = −aκ

30 + aκ

3σ

(d

dt(Γσ

αβqαqβ) + (qτ + Γτ

αβqαqβ)Γσ

ταqα

),

σ, τ = 1, s , α, β = 0, s , κ = 1, k .

(2.4)

Differentiating in time equation (2.1), we obtain

MW = Y + Λκbκ + Λκbκ , κ = 1, k , (2.5)

where

Y = (Qτ − QσΓσ

ταqα)eτ , bκ = (bκ

τ− bκ

σΓσ

ταqα)eτ ,

σ, τ = 1, s , α, β = 0, s , κ = 1, k .

Multiplying equation (2.5) scalarly by the vectors aµ

3 and taking into accountrelations (2.4), we get

Λκhκµ

3 = Bµ

3 (t, q, q, q, Λ) , Bµ

3 = Mχµ

3 − Y · aµ

3 − Λκbκ · aµ

3 ,

hκµ

3 = bκ · aµ

3 = bκ

σa

µ

3τgστ , σ, τ = 1, s , κ, µ = 1, k .

Condition (1.4) is satisfied by assumption and therefore we have

Λκ = h3κµ

(t, q, q, q)Bµ

3 (t, q, q, q, Λ) , κ, µ = 1, k . (2.6)

Here h3κµ

are the elements of the matrix inverse to the matrix with theelements h

κµ

3 . By formulas (2.3) we can eliminate the derivatives qσ from thefunctions h3

κµ, B

µ

3 and represent the right-hand sides of equations (2.6) as

Λκ = C3κ(t, q, q, Λ) , κ = 1, k . (2.7)

For arbitrary n the functions hnκµ

, Bµn

occur such that we need to eliminate

the derivatives qσ, ... ,(n−1)

qσ , from them. By (2.3) we have

...q σ =

∂F σ2

∂t+

∂F σ2

∂qτqτ +

∂F σ2

∂qτqτ +

∂F σ2

∂Λκ

Λκ , σ, τ = 1, s , κ = 1, k . (2.8)

Using formulas (2.3), we can eliminate the derivatives qσ from relations (2.8)and represent them in the form

...q σ = F σ

3 (t, q, q, Λ, Λ) , σ = 1, s .

Reasoning as above, we obtain

(n−1)

qσ = F σ

n−1(t, q, q, Λ, Λ, ... ,(n−3)

Λ ) , σ = 1, s .

130 V. The Mixed Problem of Dynamics. New Class of Control Problems

Thus, in the general case we have

(n−2)

Λκ = Cn

κ(t, q, q, Λ, Λ, ... ,

(n−3)

Λ ) , κ = 1, k , n 3 . (2.9)

The particular case of these equations is system (2.7).Equations (2.3) and (2.9) make up the closed system of equations with

respect to the functions qσ(t) and Λκ(t). By initial data (1.5) it has a uniquesolution, which was to be proved.

We remark that if the differential, describing the motion, depends nonlin-

early on the higher derivatives(n−1)

qσ , then, differentiating in time this equa-

tion, we obtain the equation, which depends linearly on the derivatives(n)

qσ .Therefore the theory suggested can also be applied to high-order nonlinearequations.

Now we turn to the second-order equations. Representing them in vectorform, we obtain

aκ

2 · W = χκ

2 (t, q, q) , aκ

2 = aκ

σeσ ,

χκ

2 = −aκ

20 + aκ

2σΓσ

αβqαqβ ,

σ = 1, s , α, β = 0, s , κ = 1, k .

(2.10)

Multiplying equation (2.1) scalarly on the vectors aµ

2 , µ = 1, k, we find

Λκhκµ

2 = Bµ

2 (t, q, q) , Bµ

2 = Mχµ

2 − Y · aµ

2 ,

hκµ

2 = bκ · aµ

2 = bκ

σa

µ

2τgστ , σ, τ = 1, s , κ, µ = 1, k .

By assumption, condition (1.4) is satisfied. Then

Λκ = h2κµ

(t, q, q)Bµ

2 (t, q, q) , κ, µ = 1, k .

Thus, for n = 2 the generalized control forces can uniquely be found as thefunctions of time, generalized coordinates, and generalized velocities. This isalso valid in the case when in equations (1.2) n = 1. In fact, differentiatingthem in time, we obtain equations such that

aκ

2σ= aκ

1σ, aκ

20 = aκ

1σqσ + aκ

10 , σ = 1, s , κ = 1, k .

The generation of control forces as the functions of the variables t, qσ(t),qσ(t), σ = 1, s, is more complicated problem than its generation as the func-tions of time, which can be obtained by means of integrating a system ofdifferential equations. Taking into account these reasonings it is rational todifferentiate in time the equations of program motion for n = 1, 2, twice andonce, respectively, and to reduce them, thus, to third-order equations.

3. The mixed problem of dynamics and Gaussian principle 131

§ 3. The mixed problem of dynamics

and Gaussian principle

The obtained solution of the mixed problem of dynamics depends sub-stantially on as the form of equations (1.2), as the system of the vectorsbκ , in which the sought force R(t) is expanded. Consider the particular casewhen the coefficients aκ

nσ, entering into equations (1.2), are functions of the

variables t, qσ(t), qσ(t), σ = 1, s, only. Using these coefficients, we representa tangential space as a direct sum of K and L–spaces. For n = 2 this wasmade in § 1 of Chapter IV. In this Chapter a system of linearly independentvectors

εεεl+κ = ∇∇∇′′fκ

2 , l = −k , κ = 1, k ,

is a basis of K–space. These vectors have no the index n since for n = 0, 1,they have respestively the form

εεεl+κ = ∇∇∇fκ

0 , εεεl+κ = ∇∇∇′fκ

1 , κ = 1, k .

For n 3 the vectors

aκ

n= ∇∇∇(n)fκ

n= aκ

nσ(t, q, q)eσ , σ = 1, s , κ = 1, k ,

which are also assumed to be independent, must have the index n since forthe given n, by assumption, equations (1.2) cannot, generally speaking, beobtained by means of differentiation in time of lower-order equations.

The vectors aκ

n, κ = 1, k, are assumed to be a basis of K–space and the

vectors aλn such that

aλn · aκ

n= 0 , λ = 1, l , l = s − k , κ = 1, k ,

are assumed to be a basis of L–space. In this case the vectors W and Y likethose, considered in § 1 of Chapter IV, can be represented as

W = WL + WK , Y = YL + YK , WL · WK = 0 , YL · YK = 0 .

We shall show that if equations (1.2) take the form

fκ

n≡ aκ

nσ(t, q, q)

(n)

qσ + aκ

n0(t, q, q, ... ,(n−1)

q ) = 0 ,

σ = 1, s , κ = 1, k , n 3 ,

(3.1)

then for the given values of the variables t, qσ, qσ, σ = 1, s, these equationsdo not constrain the mechanical system to have the acceleration WL differentfrom the acceleration, given by Newton’s law

MWL = YL . (3.2)

We begin the proof from the case n = 3. Representing equations (2.4) as

d

dt(aκ

3 · W) = χκ

3 + aκ

3 · W , κ = 1, k ,

132 V. The Mixed Problem of Dynamics. New Class of Control Problems

we obtain

aκ

3 · W = aκ

3 · W|t=t0+

t∫

t0

(χκ

3 + aκ

3 · W)dt , κ = 1, k . (3.3)

The right-hand sides of these relations cannot be found as the functionsof the variables t, qσ, qσ, σ = 1, s, since, generally speaking, the order ofequations (2.4) cannot be lowered. We shall regard these right-hand sides asthe functions of time, which are equal to Ψκ

3 (t). Note that in order to findthem we need to know the motion of system, satisfying Newton’s law (2.1).This law involves the unknown control forces Λκ(t), κ = 1, k, which, as isshown, are to be determined from system of equations (2.3), (2.7). Thus, thefunctions Ψκ

3 (t), κ = 1, k, are certain unknown functions. From equations(3.3), represented in the form

aκ

3 (t, q, q) · W = Ψκ

3 (t) , κ = 1, k ,

it follows that for the given values of variables t, qσ, qσ, σ = 1, s, equations(2.4) are satisfied for any vector WL , and, in particular, for that, given byNewton’s law (3.2).

For n-order constraints, equations (3.1) yields that the (n − 2)-nd timederivatives of the scalar products aκ

n· W, κ = 1, k, are known functions of

the variables t, qσ, qσ, . . . ,(n−1)

qσ , σ = 1, s. Hence these products themselvescan be represented in terms of definite integrals. Therefore for any n and thefixed values of the variables t, qσ, qσ, σ = 1, s, the mechanical system cannotbe constrained by equations (3.1) to have in an L–space the accelerationdifferent from that, given by the Newton equations (3.2).

Recall that at the end of § 2 of Chapter IV the linear nonholonomic con-straints of any order, by which equation (3.2) is satisfied, were called ideal.As is shown, this is the constraints, given in the form (3.1). The definition ofideality of these constraints has a geometric interpretation. For constraints,given by equations (3.1), the scalar products aκ

n·W, κ = 1, k, are expressed

via a definite integral. They cannot be found as the functions of the variablest, qσ, qσ and therefore it is necessary to regard them as the unknown functionsΨκ

n(t), κ = 1, k.For the fixed values of the variables t, qσ, qσ, σ = 1, s, in the space of

accelerations W the system of equations

aκ

n· W = Ψκ

n(t) , κ = 1, k , (3.4)

gives an l-dimensional plane. Its position relative to the origin is determinedby a system of the independent sought functions Ψκ

n(t), κ = 1, k. This plane

is similar to that, represented in Fig. IV. 3 of Chapter IV.Substituting into equations (3.3) the acceleration W, represented in the

formW = WL + WK ,

3. The mixed problem of dynamics and Gaussian principle 133

we obtainWK = hκµΨµ

n(t)aκ

n, κ, µ = 1, k .

Here hκµ are elements of the matrix inverse to the matrix with the ele-ments hκµ, which are given by the relations

hκµ = aκ

n· aµ

n, κ, µ = 1, k .

For n = 1, 2, the vector WK , which is equal to k-dimensional perpendicu-lar, dropped from the origin of coordinates to the introduced l-dimensionalplane, is uniquely defined by equations (3.1) as a function of the variablest, qσ, qσ, σ = 1, s. For n 3 this vector is defined by a value assignment ofthe unknown functions Ψκ

n(t), κ = 1, k. This is a vital difference of equations

(3.1) for n 3 and the same equations for n = 1, 2. Equation (2.1) yieldsthat the position of the considered l-dimensional plane relative to the end ofthe vector Y/M is given by the vector Λκ(t)bκ/M . Thus, the position of thisplane is defined by a value assignment of as the functions Ψκ

n(t), κ = 1, k,

as the functions Λκ(t), κ = 1, k. This implies that if it is impossible, inprinciple, to lower the order of equations (3.1) up to n = 2 and to find thevalues Ψκ

n(t), κ = 1, k, as the functions of the variables t, qσ, qσ, σ = 1, s,

then it is also impossible, in principle, to find the generalized control forcesΛκ , κ = 1, k, as the functions of the same variables. Therefore in the previoussection we sought them as the functions of time.

In accordance with Gaussian principle the constraint measure, given bythe following relation

Zg =τ4

4(W − Y/M)2 ,

must be minimal. In the enunciation of this principle, which is due to Gaussitself, it is not said that the value Zg is to be regarded as the function, whichis given on a set of the accelerations W admitted by constraints, and thatits minimum is sought on this set. The formulation of Gauss is more general.We give it, using the treatise of P. Appell [4, Vol. II, p. 421]: "The motion of asystem of mass points, which are arbitrary one another and are liable to anyinfluence, is at every moment in as perfect as possible agreement with themotion of these points under the condition that they became free, i. e. themotion is under the minimally possible constraint if we regard as a measureof constraint, applied in an infinitely small time, a sum of the products of amass of each point by the square value of its deviation from the position ofthis point under the condition that it is free".

Now we apply the formulation of Gaussian principle to the case whenthe constraints are given by equations (3.1). Under the condition that thesystem considered is free, it might have the acceleration Y/M . However,really, on account of the existence of constraints its acceleration W is anelement of the set that is an l-dimensional plane. The position of this planerelative to the origin of coordinates is given by a system of the independentfunctions Ψκ

n(t), κ = 1, k. For all that, on the above-mentioned plane we can

134 V. The Mixed Problem of Dynamics. New Class of Control Problems

find the point, for which the value Zg is minimal in accordance with Gaussianprinciple.

This point can be obtained if the generation of the generalized controlforces is in agreement with equations (3.1) so that bκ

σ= aκ

nσ, σ = 1, s, κ =

1, k. In this case equation (2.1) takes the form

MW = Y + Λκaκ

n, κ = 1, k . (3.5)

This implies that a k-dimensional perpendicular, dropped from the pointY/M to the considered plane, is given by the vector Λκaκ

nsuch that the

value Zg is minimal.Thus, for bκ = aκ

n, κ = 1, k, the motion control is realized in accordance

with Gaussian principle. We call such control ideal. The condition that forideal control the vector of control force R = MW − Y is orthogonal to theintroduced l-dimensional plane can be represented as

δA = (MW − Y) · δy = 0 . (3.6)

Here δy is an arbitrary tangential vector, satisfying the following system ofequations

aκ

n· δy = 0 , κ = 1, k . (3.7)

Representing relations (3.6), (3.7) in scalar form and taking into account that

aκ

n=

∂fκ

n

∂(n)

qσ

eσ , σ = 1, s , κ = 1, k ,

we obtain ( d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 , (3.8)

∂fκ

n

∂(n)

qσ

δqσ = 0 , σ = 1, s , κ = 1, k . (3.9)

As is remarked in § 6 of Chapter II and in § 2 of Chapter IV, N. G. Chetaevmakes use of the imposing of the conditions

∂fκ

1

∂qσδqσ = 0 , σ = 1, s , κ = 1, k ,

on the virtual displacement δqσ, σ = 1, s, for "... the D’Alembert and Gaus-sian principles to be saved together ..."[245, p. 68].

The equations (3.8), (3.9) are obtained under the assumption that themotion control is realized in accordance with Gaussian principle. This impliesthat the imposing of conditions (3.9), which is similar by its structure tothe conditions of N. G. Chetaev, on the virtual displacements, entering intothe D’Alembertian–Lagrange principle (3.8), makes it possible to "... save theD’Alembert and Gaussian principles together ..."

3. The mixed problem of dynamics and Gaussian principle 135

We shall show that an ideality condition of control, written in the formof equations (3.6), (3.7), can also be represented in the form of the Mangeron–Deleanu principle. In fact, computing the partial differential δ(n) of theequations of program constraints (3.1) for the fixed values of the variables

t, qσ, qσ, . . . ,(n−1)

qσ , σ = 1, s, we obtain

δ(n)fκ

n=

∂fκ

n

∂(n)

qσ

δ(n)(n)

qσ = aκ

n· δ(n−1)

(n−1)

V = 0 ,

δ(n−1)(n−1)

V = δ(n)(n)

qσeσ , κ = 1, k .

This system is identical to system (3.7) under the assumption

δy =τn

n!δ(n−1)

(n−1)

V .

Thus, equation (3.6) can be represented in the form of the Mangeron–Deleanu

principle [59. 1976]

(MW − Y) · δ(n−1)(n−1)

V = 0 , (3.10)

which was to be proved. Note that the principle (3.10) can be used for pro-gram constraints (3.1) only if the vectors aκ

ndepend on the variables t, qσ, qσ,

σ = 1, s.From the representation of the control force R in the form

R = Λκ aκ

n= Λκ(t)∇∇∇(n)fκ

n, κ = 1, k ,

it follows that for ideal control, to each program constraint corresponds itsgeneralized control force Λκ(t), κ = 1, k. Note that the idea that the forcesare generated by constraints is due to G. Hertz.

In the simplest case of one holonomic constraint such that the mass pointis on the given surface, the ideality criterion is that this constraint can beprovided by imposing on the mass point the normal reaction only. In thiscase this reaction is a generalized reaction, corresponding to this constraint.If it is nonlinear, i. e. the surface is rough, the normal reaction is insufficientand it is necessary still to overcome the frictional force. Thus, the constraintis ideal in the case when it is provided by its generalized reaction, which, as isnoted by A. M. Lyapunov [138], is necessary and sufficient for the constraintto be ideal. This definition of ideality, as is shown above, can be extended tohigh-order program constraints, given in the form (3.1).

In the case when the order of constraints is less than three, the forcesΛκ , κ = 1, k, are known functions of the variables t, qσ, qσ, σ = 1, s. There-fore at initial time their values are defined by the initial values of coordinatesand velocities. For program constraints (3.1) the generalized control forces

136 V. The Mixed Problem of Dynamics. New Class of Control Problems

Λκ , κ = 1, k, are unknown functions of time, satisfying the system of (n−2)-order differential equations. Hence at initial time the values Λκ, κ = 1, k,must be given together with their time derivatives up to the (n−3)-rd order.Thus, for n 3 the problem must be solved with initial data (1.5).

Consider the question of initialization for our problem from another stand-points.

Equation (3.5), corresponding to ideal control, can be obtained from theMangeron–Deleanu principle. This was made in scalar (but not vector) formby M. A. Chuev [252. 1975]. We remark that the generalized Gaussian prin-ciple was proposed first in this work and in the paper [252. 1974] and thenindependently in the works [188, 189].

Multiplying equation (3.5) scalarly by the vectors aλn, λ = 1, l, l = s−k,such that

aλn · aκ

n= 0 , κ = 1, k ,

we obtain the equations

(MW − Y) · aλn = 0 , λ = 1, l . (3.11)

If the order of constraints is less than three, then, joining equations (3.11)and (2.10), we obtain the closed system of differential equations, describingthe motion with the given initial values of coordinates and velocities. Express-ing the Newton law in L–space by equations (3.11) and adding to them, inparticular, for n = 3 equation (2.4), for program constraints (3.1) we obtaina system of differential equations of general order (2s + k). This system doesnot involve the sought functions Λκ(t), κ = 1, k, which are, in principle,intrinsic to this problem. Therefore their elimination leads to the problemof initialization for the variables qσ, σ = 1, s. This problem is discussed, inparticular, in the work of M. A. Chuev. He remarks [252, p. 69] that ". . . theMangeron–Deleanu principle makes it possible to obtain the equations whichare not contradict to the principle of superposition for the constraints ofthe form (3.1) only and under very strong restriction on the initial data".M. A. Chuev writes that the principle of superposition is violated in the casewhen ". . . the forces depend on the derivatives of coordinates, the order ofwhich is greater than unity"[252, p. 69]. In this case he refers to the treatiseof L. Pars [179], § 1.4, where it is shown that the force cannot be a functionof acceleration. The validity of this assertion results from the following rea-sonings of V. I. Arnold. In his book [6, p. 8] he remarks: "The initial state ofmechanical system (a set of all positions and velocities of points of systemat any time) uniquely defines its whole motion". V. I. Arnold calls this law ofnature Newton’s determinacy principle.

According to this principle the position of mechanical system and itsvelocity V at time t defines a derivative of any order of the vector V at asthis moment as all subsequent moments of time. V. I. Arnold [6, p. 12] writes"In particular, the position and velocity define acceleration. In other words,there exists the function F . . ."of the variables t, qσ, qσ, σ = 1, s, such that

W = F(t, q, q) .

4. The motion of spacecraft with modulo constant acceleration 137

By (2.1) for the given active force Y(t, q, q) the existence of the function F

results from the fact that the vector of the control force R = Λκ bκ , κ = 1, k,providing the fulfilment of program constraints (1.2) for n = 1, 2, is uniquelydefined as the function of variables t, qσ, qσ, σ = 1, s. For n 3, the causalityprinciple is also preserved but by the fact that the generalized control forcesare sought as time functions. In this case if bκ = aκ

n, κ = 1, k, then the

control forces as functions of time are generated by Gaussian principle.

§ 4. The motion of spacecraft

with modulo constant acceleration

in Earth’s gravitational field

Let a spacecraft move in Earth’s gravitational field along elliptical or-bit. Suppose, beginning from a certain instant of time, the spacecraft has aconstant acceleration. This condition we regard as a second-order nonlinearnonholonomic program constraint. The imposing of constraint can occur atany point of orbit, the additional force at this moment of time is lacking.

The motion of spacecraft along elliptical orbit is described by the equation

d2ρρρ

dt2= −

µρρρ

ρ3, µ = γM , ρ = |ρρρ| . (4.1)

Here ρρρ is a radius–vector, connecting the geocenter and a spacecraft, γ is agravity constant, M is Earth’s mass. The constant µ can be represented as[189]

µ =4π2a3

T 2,

where a is a major semiaxis of the spacecraft elliptical orbit, T is a time ofcomplete revolution.

In the dimensionless variables

r = xi + yj = ρρρ/a , τ = 2πt/T ,

equation (4.1) takes the form

r = −r/r3 , r = |r| . (4.2)

A point denotes the derivative with respect to the dimensionless time τ . Theintegral of energy and the area integral in equation (4.2) are thefollowing [189]

v2 = 2/r − 1 , v = |r| , r2ϕ =√

1 − e2 , (4.3)

respectively. Here e is an eccentricity of elliptical orbit. Suppose, at initialtime, beginning from which a spacecraft must move with a constant acceler-ation, it is on the axis x. Without loss of generality, we can assume that inthis case the initial data are the following:

x(0) = x0 , x(0) = x0 =√

2x0 − x20 − 1 + e2/x0 ,

y(0) = y0 = 0 , y(0) = y0 =√

1 − e2/x0 , 1 − e x0 1 + e .

(4.4)

138 V. The Mixed Problem of Dynamics. New Class of Control Problems

The equation of constraint in the accepted notations can be represented as

r2 − 1/x40 = 0 . (4.5)

This equation is satisfied, in particular, when the vector r collinear to thevector r is constant in value. In this case the time derivative of the vector r

is orthogonal to the vector r, i. e. we have

er ·...r = 0 , er = r/r . (4.6)

This equation is a linear third-order nonholonomic program constraint. Thus,in this problem system of equations (2.4) is reduced to one equation (4.6).

We assume that a spacecraft is equipped by the generalized control forceΛ such that the vector of control force is as follows

R = Λer .

From equation (4.6) we conclude that for this force R the control is ideal,i. e. it is in accordance with Gaussian principle.

Beginning from the instant of imposing constraint (4.6), the motion ofspacecraft is described by the following equation

r = −r

r3+ Λ

r

r. (4.7)

At tine of imposing a constraint the control force is lacking, i. e. we have

Λ(0) = 0 . (4.8)

Differentiating relation (4.7) with respect to τ , we obtain

...r = −

r

r3+

3r

r4r + Λ

r

r+ Λ

r

r− Λ

rr

r2.

Multiplying this equation scalarly by r and taking into account equation ofconstraint (4.6) and the relation

r2 = r2 , r · r = rr ,

we have

Λ = −2r

r3. (4.9)

In this problem, system of equations (2.6) is reduced thus to one equation(4.9). Assuming in it

Λ = −dΛ

drr ,

we finddΛ

dr= −

2

r3.

4. The motion of spacecraft with modulo constant acceleration 139

Integrating this equation and taking into account that by (4.4) and (4.8)Λ = 0 for r = x0, we get

Λ =1

r2−

1

x20

.

Substituting this relation into equation (4.7), we obtain

r = −r/(rx20) . (4.10)

Now we shall show that this equation follows directly from Gaussian prin-ciple. In fact, determining a minimum of the function

Z = |r + r/r3|2

on a set of the values r, admitted by equations (4.5), we obtain

r + r/r3 + Λ∗r = 0 . (4.11)

Here Λ∗ is a sought Lagrange multiplier.This implies that

(1 + Λ∗)2r2 = 1/r4 .

Taking into account equation of constraint (4.5), we have

Λ∗ = x20/r2 − 1 .

Substituting this value of the Lagrange multiplier into equation (4.11), wearrive at equation (4.10), which was to be proved.

Using equation (4.10), we find a motion satisfying equation (4.5), in thecase when the control force R = Λr/r, due to which this motion occurs, isunknown. However for this motion to be found really, it is necessary to knowthe force as a function of time. Therefore we do not eliminate a control forcefrom equation (4.7) and consider it together with equation (4.9). Projectingthe vector equation (4.7) on the unit vectors of the polar system of coordinateser = r/r and eϕ, we find

r − rϕ2 +1

r2= Λ ,

rϕ + 2rϕ = 0 .

(4.12)

Adding these two equations to equation (4.9), we obtain the closed system ofequations, from which the motion and the control force can be found.

The numerical integration of system equations (4.9), (4.12) was performedwith the following initial data

r(0) = x0 = 1 − e , r(0) = 0 , ϕ(0) = 0 ,

ϕ(0) =√

1 − e2/x20 , Λ(0) = 0 .

The computation demonstrates that for any value of the eccentricity e differ-ent from zero and unity, the trajectory of spacecraft motion is a curve, which

140 V. The Mixed Problem of Dynamics. New Class of Control Problems

lies between two concentric circles. To find their radii and their dependenceon the values x0 and e, we need to consider equation (4.10). The integral ofenergy with arbitrary initial data (4.4) is as follows

v2

2+

r

x20

=x2

0 + y20

2+

1

x0=

4 − x0

2x0.

Then using the Binet formula [189]

v2 = c2[( du

dϕ

)2

+ u2], u =

1

r, c2 = r4ϕ2 = 1 − e2 ,

we obtain ( dr

dϕ

)2

=2r4

c2

(4 − x0

2x0−

r

x20

−1 − e2

2r2

).

The trajectory of spacecraft makes contact with the circle of radius r at

the point such thatdr

dϕ= 0. Therefore the sought radii r1 and r2 are positive

roots of the equation

2r3 − (4 − x0)x0r2 + x2

0(1 − e2) = 0 .

Note that the motion between the circles of these radii is not periodic in thesense that the point never returns at initial position in integer revolutions.

As an example, in Figs. V. 1, V. 2, and V. 3 are shown the results ofcomputation in the time interval 0 t T/2 (0 τ π) for e = 0.4.In Fig. V. 1 the original elliptical orbit and the concentric circles of radiir1 = 0.6 and r2 = 0.754, respectively, which the solution of equation (4.10)lies between, are shown by thin lines. The solution is shown by a bold line.

The hodograph of the control force R = Λ(τ)r/r, providing this solution,is shown in Fig. V. 2 by a thick line. The results shown are under the assump-tion that Λ 0. The graph of the function Λ(τ) is shown in Fig. V. 3. Notethat the value Λ, as follows from equations (4.1) and (4.7), is measured interms of the fractions of the gravitational force F , where

F =µm

a2.

Here m is a spacecraft mass.

Consider now the solution of this problem, resulting from generalizedGaussian principles. Differentiating equation of constraints (4.5) in time, weobtain

r ·...r = 0 . (4.13)

Determining a minimum of the function

Z =1

2

∣∣∣...r +r

r3−

3rr

r4

∣∣∣2

4. The motion of spacecraft with modulo constant acceleration 141

Fig. V. 1

on a set of the values...r , admitted by equations (4.13), we arrive at the

following equation...r = −

r

r3+

3r

r4r + Λ∗ r , (4.14)

where Λ∗ is a sought Lagrange multiplier. Relations (4.5), (4.3) yield therelation

Λ∗ =x4

0

r3r · r −

3x40 r

r4r · r .

The Cartesian coordinates were used for numerical integration of equation(4.14) after the substitution the value Λ∗. Initial data (4.4) were completedby the following initial data for accelerations:

x(0) = x0 = −1/x20 , y(0) = y0 = 0 .

The computation demonstrates that even for very small eccentricity (inde-pendently of x0) the trajectory goes at infinity. It tends asymptotically toa straight-line motion with constant acceleration. In Fig. V. 4 is shown thatthe tendency to a straight-line motion increases with value e. All the curvescorrespond to the case x0 = 1 − e.

Interesting feature of this solution is that the straight-line motion withconstant acceleration is obtained after approximately three revolutions for thevalue e ≈ 4 · 10−6 when the motion, before imposing a constraint, satisfiedthis constraint with great accuracy. Consider the cause of this phenomenon.

In the absence of active forces and constraints Gaussian principle leads tothe motion with the zero acceleration W, what is in agreement with Newton’s

142 V. The Mixed Problem of Dynamics. New Class of Control Problems

Fig. V. 2

first law. Note that the equations of dynamics can be obtained from thisprinciple.

In the case when the active forces and constraints are lacking the usage ofgeneralized Gaussian principle does not lead to the motion with the zero ac-celeration W but leads to that with the zero time derivative of order k of thevector W, where k is an order of principle. Hence for k = 1 the application of

Fig. V. 3

4. The motion of spacecraft with modulo constant acceleration 143

Fig. V. 4

generalized Gaussian principle in the absence of active forces and constraintsleads to a uniformly accelerated straight-line motion. A satellite (which be-comes a spacecraft) tends to such a "natural"in the frame of this principlemotion even for e ≈ 4 · 10−6. It is clear that the considered problem on themotion of satellite (spacecraft) with modulo constant acceleration can havea solution such that the satellite tends asymptotically to the straight-linemotion with this constant acceleration. Thus, such a solution is a result ofapplication of generalized Gaussian first-order principle to this problem.

However it should be kept in mind that a generalized Gaussian princi-ples, unlike the usual one, does not result from the equation of dynamicsand therefore its application to another problems can lead to unpredictableresults. Consider an example.

Suppose, on the motion of mass point on plane there is imposed the idealholonomic constraint: x2 + y2 = l2 or in vector form r2 = l2. In the absenceof external forces the point moves in a circle with the constant velocity v0.

Differentiating the equation of constraints in time thrice, we obtain

r · w = −3v · w .

The application of the generalized principle gives

mw = Λr .

Introducing the polar coordinates r and ϕ, we have

mw · eϕ =d

dt

∂T1

∂ϕ−

1

2

∂T1

∂ϕ= 0 ,

T1 =mw2

2=

m[(r − rϕ2)2 + (rϕ + 2rϕ)2]

2.

144 V. The Mixed Problem of Dynamics. New Class of Control Problems

Hence ...ϕ = ϕ3 .

Assuming that for t = 0 ϕ = 0, ϕ = ω0 = v0/l, ϕ = 0, we obtain

t =1

ω0F

(arccos

ω0

ω,

1√

2

), F (θ, k) =

θ∫

0

dα√1 − k2 sin2 α

.

This solution implies that the angular velocity ω = ϕ becomes infinite in atime

t∗ = F (π/2, 1/√

2)/ω0 = 1,854/ω0 .

The example considered shows that a generalized Gaussian principle shouldbe applied very carefully. However, as is shown in studying the motion of aspacecraft with modulo constant acceleration, exactly the generalized Gaus-sian principle permits for one of possible motions, which cannot be obtainedby usual Gaussian principle, to be found.

§ 5. The satellite maneuver alternative

to the Homann elliptic motion

In the previous section we study the spacecraft motion with modulo con-stant acceleration. Now we consider a more complicated problem of the pas-sage of a spacecraft from one elliptical orbit to another close to circular. Thispassage, as is known, can be realized for the Homann elliptic motion by meansof an instantaneous application of impulses at the beginning and at the endof orbital passage [189]. Theory, given in § § 2 and 3 of this Chapter makesit possible to realize this orbital passage in the case of a soft application ofcontrol forces.

For the solution of this problem we make use of the dimensionless variablesand equations, introduced in the previous section.

As is known, the radial component wr of the vector of acceleration r hasthe form [189]

wr = r − rϕ2 . (5.1)

At the initial point of elliptical orbit this component is equal to (−1/r20). For

a smooth passage on a circular orbit of radius r1 it is necessary for the valuewr, which continuously increases (or decreases, respectively) beginning fromthe value (−1/r2

0), to be tended asymptotically to the value (−1/r21). For the

description of this passage we make use of the function, which occurs whensolved the problem on a longitudinal collision of bars with circled ends.

English scholar Sears showed that in this problem a smooth increasing ofthe dimensionless force of collusion Q(t) from zero to unity is described bythe following equation [77]

dQ

dt= Q1/3(1 − Q) .

5. The satellite maneuver alternative to the Homann elliptic motion 145

The Sears equation is integrated in closed form but the integration leads toa complicated dependence of t on Q:

t =1

2ln

1 + Q1/3 + Q2/3

(1 − Q1/3)2−√

3 arctan2Q1/3 + 1

√3

+π√

3

6.

Therefore it is convenient to regard at once the Sears function as a solutionof the Sears differential equation.

Taking into account the properties of the Sears function, we assume thatthe varying of the value wr of satellite for τ 0 is described by the generalizedSears equation

Q = pQq(1 − Q) . (5.2)

HereQ = (wr + 1/r2

0)/(1/r20 − 1/r2

1) . (5.3)

The definition of the generalized Sears function Q(τ) involves two parameterp and q. The parameter p controls a time, beginning from which Q 1 −ε. The second parameter q defines a behavior of the function Q(τ) in theneighborhood of zero. In Fig. V. 5 are shown the graphs of the function Q(τ),satisfying equation (5.2). To the value p = 2 corresponds the curve with longdashes, to p = 1 the solid curve, to p = 0.5 the curve with short dashes. Forall these curves we assume that q = 1/3.

By (5.1) and (5.3) equation (5.2) becomes

f3 ≡...r − rϕ2 − 2rϕϕ + (1/r2

1 − 1/r20)pQq(1 − Q) = 0 . (5.4)

This equation is a linear third-order nonholonomic program constraint.We introduce, as in the previous section, the generalized control force Λ,

which the vectorR = Λer = Λr/r

corresponds to. In this case the program constraint (5.4) control is ideal sincethe vector ∇∇∇′′′f3 = er coincides with that, entering into the relation for the

Fig. V. 5

146 V. The Mixed Problem of Dynamics. New Class of Control Problems

control force R. Then the equation of motion over the coordinate r takes theform

r − rϕ2 +1

r2= Λ . (5.5)

From relations (5.4) and (5.5) it follows that the sought generalized radialcontrol force Λ(τ) must satisfy the equation

Λ +2r

r3= (1/r2

0 − 1/r21)pQq(1 − Q) = 0 . (5.6)

In accordance with the technique of generating the differential equationwith respect to the generalized control force Λ the value Q in equation (5.6)in accordane with (5.1), (5.3), (5.5) takes the form

Q = (Λ − 1/r2 + 1/r20)/(1/r2

0 − 1/r21) .

We shall regard this relation as the change of the variables Λ to the newvariables Q. Then equation (5.6) is reduced to equation (5.2) and equation(5.5) becomes

r − rϕ2 = −1/r20 + (1/r2

0 − 1/r21)Q . (5.7)

Consider now the equation with respect to the coordinate ϕ:

rϕ + 2rϕ ≡1

r

d

dτ(r2ϕ) = P (τ) . (5.8)

Here P (τ) is a sought tangential control force.It is necessary to perform a passage of spacecraft on circular orbit of radius

a1 = ar1. Find the value r2ϕ on this orbit. For this purpose we introducenew dimensionless variables, assuming that

r∗ =ρρρ

a1, τ1 =

2πt

T1.

Using formula (4.3), we have

r2∗

dϕ

dτ1= 1 .

Returning in this relation to the variables

r =ρρρ

a, τ =

2πt

T,

and taking into account that [189]

a3

a31

=T 2

T 21

, r1 =a1

a,

we obtainr2ϕ =

√r1 .

5. The satellite maneuver alternative to the Homann elliptic motion 147

Fig. V. 6

The function wr(τ) = r − rϕ2 and the Sears function Q are related by(5.7). Suppose, for τ 0 the function c(τ) = r2ϕ is varied similarly, i. e. weput

c(τ) =√

1 − e2 + (√

r1 −√

1 − e2)Q(τ) .

Then equation (5.8) can be represented as

rϕ + 2rϕ = (√

r1 −√

1 − e2) · Q/r . (5.9)

The problem is reduced, thus, to the solution of equations (5.2), (5.7), and(5.9).

Equation (5.2) has as the zero solution Q(τ) ≡ 0, as a nonzero solution.When numerical computing we assumed that

Q(0) = 0.0001 .

For simplicity, the initial data for the variables r and ϕ were given as

r(0) = r0 = 1 − e , ϕ(0) = 0 , r(0) = 0 , ϕ(0) =√

1 − e2/r20 .

Fig. V. 7

148 V. The Mixed Problem of Dynamics. New Class of Control Problems

Fig. V. 8

The results of computation for e = 0.01 , r0 = 0.99 , r1 = 3 , p = 0.25 , q =1/3 in the time interval 0 t 5T (0 τ 10π) are shown in Figs. V. 6,V. 7, V. 8.

We remark that the control forces Λ and P obtained are measured interms of the fractions of the gravitational force F = µm/a2, where m is asatellite mass.

Ch a p t e r VI

APPLICATION OF THE LAGRANGE MULTIPLIERS

TO THE CONSTRUCTION OF THREE NEW METHODS

FOR THE STUDY OF MECHANICAL SYSTEMS

The Lagrange multipliers are used to construct three new methods for the

study of mechanical systems. The first of them corresponds to the problem

of determining the normal frequencies and normal forms of oscillations of

elastic system, consisting of the elements with the known normal frequencies

and normal forms. In this method the conditions of connection of elastic

bodies to one another are regarded as holonomic constraints. Their reactions

equal to the Lagrange multipliers are the forces of interaction between the

bodies of system. Using the equations of constraints, the system of linear

uniform equations with respect to the amplitudes of the Lagrange multipliers

for normal oscillations is obtained. By the solution of this system the normal

frequencies and normal forms of complete system are expressed in terms of

the normal frequencies and normal forms of its elements. An approximate

algorithm for determining the normal frequencies and normal forms, based

on a quasistatic account of higher forms of its elements, is proposed.

The second method suggested is connected with the study of the dynamics

of system of rigid bodies. In this case the Lagrange multipliers are introduced

for the abstract constraints taking into account that the number of introduced

coordinates, by which the kinetic energy of rigid body has a simple form, is

excessive. In this case the elimination of the Lagrange multipliers leads to a

new special form of equations of motion of rigid body. This form is utilized

to describe a motion of a dynamic stand, which lets us to imitate the state of

a pilot in the cabin in extremal situations.

The third method is used in the problem of vibration suppression (damp-

ing) of mechanical systems. It is shown that formulation of such problems

is equivalent to imposing a high-order constraint on the motion of system.

This makes it necessary to solve a mixed problem of dynamics. It turns out,

that the Pontryagin maximum principle chooses from the possible class of

mixed problems that one in which a control force is given by a series in nat-

ural frequencies of system. In the suggested method of vibration suppression

(damping) the generalized Gauss principle is used, which makes it possible

to find the control force as a polynomial in time. The computational results

obtained by the Pontryagin maximum principle and the generalized Gauss

principle are compared.

149

150 VI. Application Lagrange Multipliers

§ 1. Some remarks on the Lagrange multipliers

The quantities Λκ, entering into equations (2.22) of Chapter I, are usu-ally called the Lagrange multipliers or generalized reactions. Note that it isrational to call them generalized reactions and to compute them, for neces-sity, only if equations (2.17) of Chapter I describe the constraints betweenthe elements of system or the constraints between the elements of the systemand the bodies, no entering into the system. In the first case a distinctiveexample is a mechanical system consisting of two mass points, connected bya weightless nonstretchable bar, and in the second case that is a one masspoint on the given ideally ground surface. In the first case the generalizedreaction is a stretching (pressing) force of bar and in the second one a force,holding the point on a surface and directed at right angles to it.

It is also possible more general and abstract situation when equations(2.17) of Chapter I do not describe the imposing of materially realizable con-straints on the system but express the fact that equations (2.17) relate thegeneralized coordinates, which are convenient to compute the kinetic energy,and a possible elementary work.

The reasonings, given below, are used in § 7 of this Chapter.Suppose, for example, the mechanical system consists of a one element,

which is a free rigid body. We shall show that its kinetic energy can berepresented as

T =Mρρρ2

2+

Ix

2i2 +

Iy

2j2 +

Iz

2k2 . (1.1)

Here M is a mass of body, ρρρ is a radius-vector of its center of mass, i, j,k are the unit vectors of the central axes of inertia of the bodies x, y, z,respectively, and

Ix =

∫

m

x2 dm , Iy =

∫

m

y2 dm , Iz =

∫

m

z2 dm .

By definition of kinetic energy we have

T =1

2

∫

m

v2 dm =1

2

∫

m

(ρρρ + xi + yj + zk

)2dm .

Since the axis x, y, z are the principal axes of inertia of body, we obtain∫

m

x dm =

∫

m

y dm =

∫

m

z dm =

∫

m

xy dm =

∫

m

yz dm =

∫

m

zx dm = 0

and therefore the quantity T can really be represented in the form (1.1).The unit vectors i, j, k, as is known, satisfy the relations

f1 ≡ i2 − 1 = 0 , f2 ≡ j2 − 1 = 0 , f3 ≡ k2 − 1 = 0 ,

f4 ≡ i · j = 0 , f5 ≡ j · k = 0 , f6 ≡ k · i = 0 .(1.2)

1. Some remarks on the Lagrange multipliers 151

It is clear that these equations are of purely mathematical nature and thefact of their existence do not mean that on an rigid body some materiallyrealizable constraints are imposed. In this case the quantities Λκ , enteringinto equations (2.22) of Chapter I, are auxiliary and their computation doesnot make a sense. Therefore it is better to call them the Lagrange multipliersbut not generalized reactions.

In the case of such abstract constraints the application of equations (2.22)of Chapter I is not obvious. This was shown above in a purely mathematicalway. Now we shall consider this case on the example of one mass point.Represent the radius-vector r of this point as

r = rn , r = |r| .

In this case the position of point is given by four parameters: the quantity r

and three components of the unit vector n. The equation of constraint, thekinetic energy, and the virtual elementary work are the following

f ≡ n2 − 1 = 0 ,

T =m

2(r2 + r2 n2) ,

δA = F · δr = Qrδr + Qn · δn ,

whereQr = F · n , Qn = r F .

This implies that it is useful to represent Lagrange’s equations of the firstkind (2.22) of Chapter I, corresponding to three components of the vector n,in vector form

d

dt

∂T

∂n−

∂T

∂n= Qn + Λ

∂f

∂n. (1.3)

Here we make use of the following notations

∂

∂n=

∂

∂nx

i +∂

∂ny

j +∂

∂nz

k .

The vector form of (1.3) is very convenient since the vector n, enteringinto the equation of constraint, the kinetic energy, and the virtual elementarywork, can be regarded formally as a usual variable. In accordance with thissimple rule, the system of equations (2.22) of Chapter I is as follows

mr = mrn2 + F · n ,

m(r2n + 2rrn) = r F + 2Λn .(1.4)

Since by the equation of constraint we have

n · n = 0 , n · n = −n2 ,

152 VI. Application Lagrange Multipliers

system of equations (1.4) gives

2Λ = −rF · n − mr2n2 = −mrr .

Substituting this relation into the second equation of system (1.4), we obtain

mr(rn + 2rn + rn) = mrr = rF ,

what is equivalent to Newton’s second law, represented in usual form.Thus, for one mass point the second Newton’s law in the Cartesian co-

ordinates can be written in usual simple form, in curvilinear coordinates(cylindrical, spherical, and so on) in the form of Lagrange’s equations of thesecond kind, and in the excessive coordinates it can be represented in theform of Lagrange’s equations of the first kind.

The considered example of applying the excessive coordinates to the de-scription of motion of one free mass point demonstrates that for abstractconstraints the reactions as the real forces are lacking. Then we cannot saythat the sum of their works on virtual displacements of system is equal tozero and therefore on the base of this definition of the ideality of holonomicconstraints (2.17) of Chapter I, we cannot generate equations (2.22) of thisChapter for abstract constraints.

Note that the fact that equations (2.22) of Chapter I can be used for suchabstract constraints is a bright demonstration of the perfect mathematicalapparatus introduced in the mechanics by Lagrange.

§ 2. Generalized Lagrangian coordinates

of elastic body

In the next sections Lagrange’s equations with multipliers are applied tomechanical system consisting of elastic bodies. In this case for each elasticbody a certain system of generalized Lagrangian coordinates is introduced.A rational method for their introduction is based on the notion of normalforms of oscillations of elastic body. Consider this approach, assuming, forthe sake of generality, that the body can move freely.

We introduce the Cartesian system of coordinates Cxyz, connected rigidlywith a body before its deformation. Suppose, the axes of this system are theprincipal central axes of inertia of this body.

Before deformation, an arbitrary point of body with the coordinates x, y, z

can be displaced, firstly, by virtue of the motion of this body as the rigid oneand, secondly, by the deformation of body. Both displacements are measuredfrom the position of system Cxyz at time t = 0. We assume that thesedisplacements are so small that the vector of displacement u(x, y, z, t) of thepoint with the coordinates x, y, z at time t can be represented as

u(x, y, z, t) = (ξ(t) + ϕy(t)z − ϕz(t)y)i + (η(t) + ϕz(t)x − ϕx(t)z)j+

+(ζ(t) + ϕx(t)y − ϕy(t)x)k +∞∑

σ=1

qσ(t)uσ(x, y, z) .(2.1)

2. Generalized Lagrangian coordinates of elastic body 153

Here i, j,k are the unit vectors of the axes x, y, z, respectively, the quantitiesξ, η, ζ give the displacement of the center of mass of body, and the quantitiesϕx, ϕy, ϕz are the angles of rotation of body about the axes x, y, z, respec-tively. By assumption, these angles are so much small that the differencebetween the projections of the vector u on the original and rotated axes canbe neglected.

The functions uσ(x, y, z), entering into relation (2.1) are normal forms ofoscillations. This means that if at time t = 0 all the points of body have azero velocity and the vector of displacement is equal to Cσuσ(x, y, z), thenfor t = 0 this vector takes the form

u(x, y, z, t) = Cσcosωσtuσ(x, y, z) . (2.2)

Here ωσ is a normal frequency, corresponding to the normal form uσ(x, y, z),and Cσ is a small constant. The system of functions uσ is complete andtherefore any displacement of a point of body by virtue of its deforma-tion can be expanded in these functions. This implies that the quantitiesξ, η, ζ, ϕx, ϕy, ϕz, q1, q2, . . . , given at time t, define uniquely the position ofall points of body at this instant of time. Hence these quantities are general-ized Lagrangian coordinates.

Substituting the vector of displacement given into the form (2.1) in rela-tion for kinetic energy of elastic body, we obtain

T =1

2

∫ ∫

V

∫ρ(x, y, z)

(∂u

∂t

)2

dxdydz =

=M

2(ξ2 + η2 + ζ2) +

Aϕ2x

+ Bϕ2y

+ Cϕ2z

2+

∞∑

σ=1

Mσ q2σ

2,

Mσ =

∫ ∫

V

∫ρ(x, y, z)u2

σ(x, y, z)dxdydz .

(2.3)

Here ρ(x, y, z) is a density of body material, M is a mass of body, A,B,C

are the principal central moments of inertia of body. The kinetic energyobtained depends on the squared generalized velocities only. Such a simpleform results from that the vector of displacement u(x, y, z, t) is representedas a series in normal forms of oscillations.

Find now a potential energy of deformation of elastic body. Using relation(2.1), we obtain a deformation tensor and, assuming that generalized Hooke’slaw is satisfied, we compute then a stress tensor. Substituting these tensorsinto the relation for potential energy of deformation of elastic body, we find

Π =

∞∑

σ=1

Mσω2σ

2q2σ

. (2.4)

Such form of the potential energy of deformation follows from that the in-troduced generalized Lagrangian coordinates qσ are the principal coordinates.By formulas (2.1) and (2.2) these coordinates must satisfy the equations

154 VI. Application Lagrange Multipliers

qσ + ω2σqσ = 0 , σ = 1, 2, ... .

We can obtain them from Lagrange’s equations of the second kind

d

dt

∂T

∂qσ

−∂T

∂qσ

= −∂Π

∂qσ

, σ = 1, 2, ... ,

if the kinetic and potential energies of elastic body are represented in theform (2.3) and (2.4), respectively.

§ 3. The application of Lagrange’s equations of the first kind

to the study of normal oscillations

of mechanical systems

with distributed parameters

Developing the ideas of the works of S. A. Gershgorin and P. F. Papkovich(see, for example, the book: Collection of works on a ship vibration, L.: Sud-promgiz. 1960), we apply Lagrange’s equations of the first kind to the analysisof normal oscillations of elastic system. Let its elements be bars, rings, plates,shells, and rigid bodies, connected to each other. For any of these elements,distinguished mentally from such elastic system, their normal frequencies andnormal forms of oscillations are known. In other words, for any of elements itis known the principal or normal coordinates in which its kinetic and poten-tial energies have the simplest form. These elements make up united systemsince their coordinates are related by expression (2.17) of Chapter I. Thus,using these relations we can construct in analytic form the considered elasticsystem by its elements. Therefore they can also be used to obtain the rela-tion between the normal frequencies and forms of this elastic system and thenormal frequencies and forms of its elements. This is a base of the methodto be suggested for determining the normal frequencies and normal forms ofoscillations of elastic systems. For its realization it is used the approximateapproach, based on the dynamic account of the first forms of oscillations ofa system of elements and on a quasistatic account of the rest of them. Theefficiency of a quasistatic account of higher forms of normal oscillations inthe dynamical problems of the elasticity theory is shown in the works [36,75, 77].

The method considered can be applied to any system of the connectedto each other elastic bodies with distributed parameters. The essence of thismethod is demonstrated on the example of lateral oscillations of the bar, ofthe length l, possessing the weight, and such that in the sections xk, k = 1, n,the disks with the mass mk and the moments of inertia Jk are fitted on it.For the solution of this problem some methods were developed [12, 71, 107].In particular, for determining the critical velocities of rotating loaded shafts,possessing the weight, the method of equivalent disk [258] is proposed. How-ever all these approaches do not use the fact that for the bar without disks, its

3. The application of Lagrange’s equations of mechanical systems 155

normal forms of the oscillations Xσ(x), σ = 1, 2, . . . , which represent a com-plete system of functions, as a rule, are known. In this case it is useful recallthat the lateral oscillations of the bar with disks can also be represented as

y(x, t) =

∞∑

σ=1

qσ(t)Xσ(x) .

Suppose, uk is a displacement of the center of mass of the k-th disk in theaxis y and ϕk is its angle of rotation. The quantities qσ (σ = 1, 2, . . . ), uk,ϕk (k = 1, n) we regard as generalized Lagrangian coordinates under theholonomic constraints

fk ≡

∞∑

σ=1

qσXσ(xk) − uk = 0 ,

fn+k ≡

∞∑

σ=1

qσX ′

σ(xk) − ϕk = 0 , k = 1, n .

(3.1)

Formulas (2.3) and (2.4) yield that the kinetic and potential energies of sys-tem take the form

T =

∞∑

σ=1

Mσ q2σ

2+

n∑

k=1

(mku2

k

2+

Jkϕ2k

2

),

Π =

∞∑

σ=1

Mσω2σq2σ

2, Mσ =

l∫

0

ρSX2σ

dx .

Here ωσ are the normal frequencies of a shaft without disks, ρ is a density,and S is a cross-section area of shaft.

Using Lagrange’s equations of the first kind (2.22) of Chapter I we obtain

Mσ(qσ + ω2σqσ) =

n∑

k=1

[ΛkXσ(xk) + Λn+kX ′

σ(xk)

],

mkuk = −Λk , Jkϕk = −Λn+k , σ = 1, 2, ... , k = 1, n .

(3.2)

From these equations it follows that the quantity Λk, k = 1, n, is equal to theforce, acting on the bar due to the disk k and the quantity Λn+k, k = 1, n, isequal to the moment, applied to the bar by virtue of the k-th disk. Note thatif in the equation of constraint the quantity f(t, q) is equal to displacement

of the point A relative to the point B along the line−−→AB, then the Lagrange

multiplier Λ, corresponding to this constrain, is equal to the force, which is

applied to the point A and acts along the line−−→AB. A similar rule is valid for

the moment of the force of reaction if the equation of constraint shows thatthe angles of rotation are equal.

156 VI. Application Lagrange Multipliers

In the case of the oscillations of a system with the sought normal frequencyp, the quantities qσ, uk, ϕk, Λk, Λn+k are varied by the harmonic laws

qσ = qσeipt , uk = ukeipt , ϕk = ϕkeipt ,

Λk = Λkeipt , Λn+k = Λn+keipt ,

σ = 1, 2, ... , k = 1, n ,

(3.3)

where i is the imaginary unity. From equations (3.2) we have

qσ =

n∑

k=1

ΛkXσ(xk) + Λn+kX ′

σ(xk)

Mσ(ω2σ− p2)

,

uk =Λk

mkp2, ϕk =

Λn+k

Jkp2, σ = 1, 2, ... , k = 1, n .

(3.4)

Substituting relations (3.3) and formulas (3.4) into the equations of con-straints (3.1), we obtain

2n∑

j=1

αij(p2)Λj = 0 , i = 1, 2n , αij = αji . (3.5)

Here the index i corresponds to the number of constraints and

αij = βij + γij ,

βkl(p2) =

0 , k = l ,

−1

mkp2, k = l ,

βn+k,n+l(p2) =

0 , k = l ,

−1

Jkp2, k = l ,

γkl(p2) =

∞∑

σ=1

Xσ(xk)Xσ(xl)

Mσ(ω2σ− p2)

, γn+k,n+l(p2) =

∞∑

σ=1

X ′

σ(xk)X ′

σ(xl)

Mσ(ω2σ− p2)

,

γk,n+l(p2) =

∞∑

σ=1

Xσ(xk)X ′

σ(xl)

Mσ(ω2σ− p2)

, k, l = 1, n .

The quantities γij(p2) are called dynamic compliances [14, 36]. Note that the

coefficients γkl(p2), k, l = 1, n, are introduced first by I. M. Babakov and are

called the harmonic coefficients of influence of the frequency p [12]. For p = 0they become usual coefficients of influence.

If the determinant of system (3.5) is not equal to zero, then Λj = 0, j =

1, 2n. However this is impossible. The quantity Λk for certain k n is equalto zero if the point xk is a node of the sought normal form of a shaft withdisks. The angle of inclination of tangent line at the node is not equal tozero and therefore Λn+k = 0. On the contrary, if at the point xk the angle of

inclination is equal to zero, then Λn+k = 0, and Λk = 0. Thus, the shaft with

3. The application of Lagrange’s equations of mechanical systems 157

disks has no normal forms such that Λj = 0, j = 1, 2n. Then the equation offrequencies is as follows

det[αij(p2)] = 0 . (3.6)

For the shaft, having at the points xk the lumped masses mk in place ofdisks, all quantities Λk, k = 1, n, can be vanished. This occurs in the casewhen the points xk, k = 1, n, are the nodes of the certain original formsXσ(x) of a shaft without masses. For example, for the one lumped mass m1

and

Xσ = sinσπx

l, σ = 1, 2, ... , x1 =

l

2,

the normal forms X2ν and the normal frequencies ω2ν are saved. In this casefrom the equation

α11(p2) = 0

we can find the frequencies, for which the normal forms are symmetric aboutthe midpoint of shaft.

This example demonstrates that the method suggested does not allowus to find the normal forms, of oscillations of elastic system, such that theconstraints between the elements of system are satisfied if the reactions ofconstraint are lacking. However these forms have the frequencies from thespectrum of frequencies of its elements and, as a rule, these forms can easilybe found.

We now return to equations (3.5) and (3.6).

Suppose, pρ, ρ = 1, 2, ... , are roots of equation (3.6) and Λρj , j = 1, 2n,are the corresponding to them solutions of system (3.5). Given in formulas

(3.4) p = pρ and Λj = Λρj , we obtain the eigenfunctions of this problem

X∗ρ(x) =

∞∑

σ=1

n∑

k=1

ΛρkXσ(xk) + Λρ,n+kX ′

σ(xk)

Mσ(ω2σ− p2

ρ)

Xσ(x) ,

ρ = 1, 2, ... .

(3.7)

Thus, we obtain the normal forms of oscillations of a shaft with disks as seriesin the normal forms of oscillations of a shaft without disks.

In the works [36, 75, 77] it is shown the efficiency of the approximateapproach consisting in that in the relation for dynamic compliance the first N

forms of normal oscillations are accounted dynamically and the rest of themquasistatically. Using this approach, we shall say that the problem consideredis solved in the N -th approximation if, by assumption, the coefficients γij(p

2)in system (3.5) and in equation (3.6) have the form

γN

kl(p2) =

N∑

σ=1

Xσ(xk)Xσ(xl)

Mσ(ω2σ− p2)

+ γkl(0) −

N∑

σ=1

Xσ(xk)Xσ(xl)

Mσω2σ

,

158 VI. Application Lagrange Multipliers

γN

n+k,n+l(p2) =

N∑

σ=1

X ′

σ(xk)X ′

σ(xl)

Mσ(ω2σ− p2)

+ γn+k,n+l(0)−

N∑

σ=1

X ′

σ(xk)X ′

σ(xl)

Mσω2σ

,

γN

k,n+l(p2) =

N∑

σ=1

Xσ(xk)X ′

σ(xl)

Mσ(ω2σ− p2)

+ γk,n+l(0) −

N∑

σ=1

Xσ(xk)X ′

σ(xl)

Mσω2σ

.

In what follows all the quantities of the N -th approximation are marked bythe upper index N . Recall that the new addends γij(0) in these formulas areusual coefficients of influence and can be found in closed form. For N = 0equation (3.6) reduced to the equation of frequency for a system with n diskson a weightless shaft [12].

It is useful to consider the curves of static bend Xstρ

(x) of shaft under the

generalized reactions Λρj . They can be found in closed form by the methodsof the resistance of material and be represented in the form of infinite series:

Xstρ

(x) =∞∑

σ=1

n∑

k=1

ΛρkXσ(xk) + Λρ,n+kX ′

σ(xk)

Mσω2σ

Xσ(x) .

Then relations (3.7) imply that the normal forms of oscillations of a shaftwith disks can be rewritten as

X∗ρ(x) =∞∑

σ=1

n∑

k=1

ΛρkXσ(xk) + Λρ,n+kX ′

σ(xk)

Mσω2σ(ω2

σ− p2

ρ)

p2ρXσ(x) + Xst

ρ(x) ,

ρ = 1, 2, ... .

(3.8)

We remark that infinite series, entering into this relation, converges muchfaster than series (3.7). In fact, the frequencies ω2

σgrow as σ4 and the quan-

tities X ′

σ(xk) as σ. Hence the series, entering into representation (3.8), con-

verges as 1/σ7. Such a rapid convergence is provided by discrimination of acurve of static bend.

By formulas (3.8) the normal forms in the N -th approximation are thefollowing

XN

∗ρ(x) =

N∑

σ=1

n∑

k=1

ΛN

ρkXσ(xk) + ΛN

ρ,n+kX ′

σ(xk)

Mσω2σ

[ω2

σ− (pN

ρ)2

](pN

ρ

)2Xσ(x) + Xst,N

ρ(x) .

Here Xst,Nρ

(x) is a curve of static bend under the generalized reactions ΛN

ρj.

The zero approximation corresponds to a weightless shaft. Already the firstapproximation permits us to find with high accuracy the first normal fre-quency and form.

E x a m p l e VI .1 . In the case of one lumped mass m1, fixed at themidpoint of the pin-ended beam of the mass M and the length l, the equationof frequencies, as is remarked above, takes the form

α11(p2) ≡ −

1

m1p2+

∞∑

ν=1

2

M(ω22ν−1 − p2)

= 0 ,

ω22ν−1 =

EJ

ρSl4π4(2ν − 1)4 ,

(3.9)

3. The application of Lagrange’s equations of mechanical systems 159

where EJ is a flexural rigidity of shaft. Note that, using the representations oftrigonometric and hyperbolic tangents in the form of infinite sums of simplefractions, we can show that equation (3.9) is equivalent to equation [51]

2M

m1= ξ(tg ξ − th ξ) , ξ2 =

l2p

4

√ρS

EJ. (3.10)

For the dynamic account of the first form and the quasistatic account ofthe rest of forms, equations (3.9) take the form

−1

m1p2+

2

M(ω21 − p2)

+l3

48EJ−

2

Mω21

= 0 . (3.11)

In the whole size of changing the quotient m1/M the frequency p1, obtainedfrom this square equation, differs from the first frequency, determined fromthe exact equation (3.10), no more than 0.1 %. Neglecting the last two ad-dends in relation (3.11), we arrive at the following simple approximate for-mula for the first normal frequencies:

p21 =

Mω21

2m1 + M. (3.12)

The error of this formula grows with the quotient m1/M but does not attain1 %.

In this case from relations (3.8) and (3.9) it follows that the first form ofoscillations has the form

X∗1(x) =2 Λ11 p2

1 l3

π4EJ(1 − p21)

(sin

πx

l+

1 − p21

34(34 − p21)

sin3πx

l+ . . .

)+

+Λ11x(3l2 − 4x2)

48EJ,

p21 =

p21

ω21

, 0 x l

2.

Being restricted to the first term of this rapidly convergent series and takinginto account that the frequency p1 can be represented most exactly in theform (3.12), we obtain for the first eigenfunction the following approximaterelation

X∗1(x) = sinπx

l+

π4m1

48M

(3x

l−

4x3

l3

), 0 x

l

2.

Since the quantity π4/96 slightly differs from unity, we can assume ap-proximately that

X∗1(x) = sinπx

l+

2m1

M

(3x

l−

4x3

l3

), 0 x

l

2.

160 VI. Application Lagrange Multipliers

This implies that the form, corresponding to a shaft without the mass m1, istaken with the weight Mequiv = M/2 equal to the equivalent mass of shaft,which is computed at the point of holding the weight, and the form for aweightless shaft is taken with the weight m1. This approximate method toconstruct the first form can also be applied to another similar problems, forexample, to the problem on the lateral oscillations of beam with the weightat the end.

E x a m p l e VI .2 . In the case of the disk, holden at the distance of x1

from the left support of pin-ended beam, the exact equation of frequencies,which is in terms of the Krylov functions, is rather lengthy [14]. The compu-tation demonstrates that the first normal frequency, which is found from thecubic equation obtained from determinant (3.6) for N = 1, differs from theexact value no more than 0.1 % if the parameters of system are varied withinlimits

0.25M m1 2M , 0.05Ml2 J1 0.5Ml2 , 0 < x1 0.5 l ,

where J1 is an equatorial moment of inertia of disk.

§ 4. Lateral vibration of a beam with immovable supports

At the beginning of § 3 we remark that the method suggested can beused to consider the oscillations of different elastic systems. As is known, instudying the oscillations of bars and plates with fixed supports, it is necessaryto consider the nonlinear equations [1, 250]. We shall show, for example,how the similar equation can be obtained for free lateral oscillations of pin-ended beam of length l and rigidity EJ in the case whith no displacement ofsupports.

The lateral and longitudinal oscillations of bar are given by the functionsy(x, t) and u(x, t), respectively. The lateral oscillations of the pin-ended beamand the longitudinal oscillations of bar with the fixed left end point and thefree right end point can be represented as [227]

y(x, t) =

∞∑

k=1

qk sinkπx

l, ω2

k=

k4π4EJ

ρSl4,

u(x, t) =

∞∑

j=1

u2j−1 sin(2j − 1)πx

2l, p2

2j−1 =(2j − 1)2π2E

(2l)2ρ.

Note that here the longitudinal oscillations are considered for the bar, whichis assumed to be fixed at the left end point only and at the right end pointthe bar is assumed first to be free though at the both end points there isno displacement. Further we introduce a holonomic constraint, by which thedisplacement at the right end point is absent.

4. Lateral vibration of a beam with immovable supports 161

Now we write the relations for the kinetic and potential energies of bar inthe case of its lateral oscillations:

T1 =ρS

2

l∫

0

(∂y

∂t

)2

dx =1

2

∞∑

k=1

mk q2k, mk =

ρSl

2=

M

2,

Π1 =EJ

2

l∫

0

(∂2y

∂x2

)2

dx =1

2

∞∑

k=1

ω2kmkq2

k.

(4.1)

In the case of longitudinal oscillations we have

T2 =ρS

2

l∫

0

(∂u

∂t

)2

dx =1

2

∞∑

j=1

m2j−1u22j−1 , m2j−1 =

M

2,

Π2 =ES

2

l∫

0

(∂u

∂x

)2

dx =1

2

∞∑

j=1

p22j−1m2j−1u

22j−1 .

(4.2)

Compute the stretch of bar in the case of lateral oscillations:

∆ =

l∫

0

√1 + (y′

x)2 dx − l =

1

2

l∫

0

(y′

x)2dx =

π2

4l

∞∑

k=1

k2q2k, (4.3)

and in the case of longitudinal oscillations:

u|x=l =

∞∑

j=1

(−1)j+1u2j−1 . (4.4)

In the case of the lateral oscillations of bar when the displacement of supportsis absent relations (4.3) and (4.4) must coincide, what can be regarded as anonlinear holonomic constraint

f ≡∞∑

j=1

(−1)j+1u2j−1 −π2

4l

∞∑

k=1

k2q2k

= 0 , (4.5)

imposed on the generalized coordinates u1, u3, ... and q1, q2, ... .

Using Lagrange’s equations of the first kind (2.22) of Chapter I and alsorelations (4.1), (4.2), and (4.5), we obtain

qk + ω2kqk = −Λ

π2k2

Mlqk , k = 1, 2, ... , (4.6)

u2j−1 + p22j−1u2j−1 = (−1)j+1 2Λ

M, j = 1, 2, ... . (4.7)

For the approximate solution of the problem, we account equations (4.7)quasistatically, assuming that u2j−1 = 0, j = 1, 2, . . . . Then we have

u2j−1 = (−1)j+1 2Λ

Mp22j−1

, j = 1, 2, ... ,

162 VI. Application Lagrange Multipliers

and therefore

u|x=l =∞∑

j=1

(−1)j+1u2j−1 =2Λ

M

∞∑

j=1

1

p22j−1

=Λ

c, c =

ES

l.

It follows that equation of constraints (4.5) can be rewritten as

Λ

c−

π2

4l

∞∑

k=1

k2q2k

= 0 .

Saved in the infinite sum one addend only, in first-order approximation wehave

Λ(1) =cπ2

4l

(q(1)1

)2.

Substituting this value of generalized reactions into the first equation of sys-

tem (4.6), we obtain for determining the function q(1)1 the Duffing equation

q(1)1 + ω2

1q(1)1 + µ

(q(1)1

)3= 0 , µ =

Eπ4

4ρl4. (4.8)

In the work [250] the same equation was generated by using another method.We obtain a second-order approximation, taking into account in system

(4.7) the first equation dynamically and the rest of them quasistatically. Then

for determining the functions q(2)1 and u

(2)1 we have the following system of

two equations

q(2)1 + ω2

1q(2)1 = −

π2

Mlq(2)1 Λ(2) ,

u(2)1 + p2

1u(2)1 =

2

MΛ(2) ,

where

Λ(2) =cc1

c1 − c

(π2

4l

(q(2)1

)2− u

(2)1

), c1 = m1p

21 =

π2ES

8l, c =

ES

l.

The construction of approximate solutions of equations of nonlinear os-cillations, using Gaussian principle, is considered in Appendix C. Recall thatthe explanation of the Rietz and Bubnov–Galerkin methods, using the inte-gral variational principles, can be found in the work of G. Yu. Dzhanelidzeand A. I. Lur’e [56].

E x a m p l e VI .3 . We study in more detail Duffing’s equation (4.8),describing the lateral oscillations of bar in the case when the longitudinal

displacements of supports are lacking. Here q(1)1 is a first-order approxima-

tion of Lagrangian coordinate q1. In studying the forced oscillations under theperturbation force P sin νt, where the constant amplitude P is taken per unit

4. Lateral vibration of a beam with immovable supports 163

mass, uniform Duffing’s equation (4.8) is replaced by the following nonuni-form equation (the indices of Lagrangian coordinates and ω are omitted):

q + ω2q + µq3 = P sin νt . (4.9)

In addition, we consider an inelastic resistance for lateral oscillations of bar.For the energy dissipation in material in the case of oscillations to be

accounted there are many propostions (see, for example, the book: PanovkoYa. G. Internal friction in oscillating elastic systems. M. 1960; Pisarenko G. S.The energy dissipation for mechanical oscillations. M. 1960; Skudrzyk E.Simple and complex vibratory systems. The Pennsylvania State UniversityPress University Park and London. 1968; Sorokin E. S. Inelastic resistanceof materials of construction in oscillating. M. 1954). Now we apply one ofthem. The resistance forces are usually displaced in phase by the value π/2per unit elastic forces. The existence of elastic forces does not violate thelaw of sinusoidal vibrations. Therefore by the form of elastic forces we canconstruct the forces of inelastic resistance, replacing q(t) by the quantity q(t),what means that a phase is shifted by π/2. In addition, we multiply then theobtained relation by the coefficient ϕ = η/ν, where η is a loss coefficient. So,we assume that to the elastic force

ω2q + µq3 = ω2q(1 + µq2/ω2)

corresponds the resistance force equal to

ϕω2q(1 + µq2/ω2) .

Since the oscillations are sinusoidal, we can replace in the latter relationthe quantity q2 by q2/ν2. Thus, to determine the function q(t), in place ofequation (4.9) we have

q + ω2q + µq3 + ϕω2q + ϕµq3/ν2 = P sin νt . (4.10)

We shall seek the steady-state vibrations in the system by the Bubnov–Galerkin method, assuming (see Appendix C) that

q(t) = a1 cos νt + a2 sin νt . (4.11)

In accordance with the Bubnov–Galerkin method, the virtual work of theforces of elasticity, resistance, inertia and of the perturbing force for thevirtual displacement of system δq = δa1 cos νt + δa2 sin νt in a period 2π/ν

of forced oscillations must be equal to zero, i. e. by (4.10) we have

δa1

2π/ν∫

0

(q + ω2q + µq3 + ϕω2q + ϕµq3/ν2 − P sin νt) cos νt dt+

+δa2

2π/ν∫

0

(q + ω2q + µq3 + ϕω2q + ϕµq3/ν2 − P sin νt) sin νt dt = 0 .

(4.12)

164 VI. Application Lagrange Multipliers

The variations δa1 and δa2 are independent and therefore the coefficientsof them must vanish. Substituting into formula (4.12) the law of motion(4.11) and integrating, we obtain the following nonlinear system of algebraicequations in unknowns a1 and a2:

a1(ω2−ν2)+

3

4µa3

1+3

4µa1a

22+ϕω2νa2+

3

4ϕµνa2

1a2+3

4ϕµνa3

2 = 0 ,

a2(ω2−ν2) +

3

4µa3

2+3

4µa2a

21−ϕω2νa1−

3

4ϕµνa2

2a1−3

4ϕµνa3

1 = P .

Hence9

16µ2(1 + ϕ2ν2) a6 +

3

2µ(ω2 − ν2 + ϕ2ν2ω2) a4+

+[(ω2 − ν2)2 + ϕ2ω4ν2] a2 = P 2 ,

(4.13)

where a =√

a21 + a2

2.The amplitude-frequency characteristics of system, given by equation

(4.13), are computed numerically and are shown in Fig. VI. 1 by solid lines.We observe that in this case the form of amplitude-frequency characteris-tic is very sensitive to the value of η. The computation are performed forω2 = 0.7172 · 106 s−2, µ = 0.1414 · 106 cm−2·s−2, P = 0.083 ν2 cm·s−2. Thiscorresponds to the steel bar with built-in ends of length 78 cm, thickness0.42 cm, width 10 cm, initial axial stress 0.2760·103 N·cm−2. The amplitudeof oscillations of supports is assumed to be equal to 0.05 cm.

Fig.VI. 1

5. Normal frequencies and oscillation modes of system of bars 165

The curves constructed demonstrate that, in fact, the form of amplitude-frequency characteristic responds highly actively to a small change of a losscoefficient, in which case the amplitude break occurs for small values of η

only. In this case for η = 0.07 the amplitude break yet occurs but for η = 0.1it is already absent.

A cubic term in the resistance force has a great influence on amplitude-frequency characteristics. Neglecting this term, in place of equation (4.13) weobtain

9

16µ2 a6 +

3

2µ(ω2 − ν2) a4 + [(ω2 − ν2)2 + ϕ2ω4ν2] a2 = P 2 . (4.14)

The amplitude-frequency characteristics, computed by formula (4.14) for thesame values of ω2, µ, P, are shown in Fig. VI. 1 by dotted lines. Note thatthe amplitude-frequency characteristics for the former values of η have moresharp breaks and jumps of amplitudes, in which case the distance betweenthe points of break and jump is larger than that for the curves, constructedby means of equation (4.13).

Thus, the Duffing equation with linear resistance describes the solutionof the formulated problem in the first approximation. When studying a sim-ilar equation numerically, P. E. Tovstik and T. M. Tovstik [425] have found,depending on the excitation level, possible arising of strange attractors andperiodical solutions with a period multiple to the period of excitation. Onthese interesting peculiarities of the Duffing equation see in detail AppendixG. The theory of strange attractors is presented, for example, in the treatiseby G. A. Leonov [426].

§ 5. The application of Lagrange’s equations of the first kind

to the determination of normal frequencies and

oscillation modes of system of bars

In § 3 of this Chapter a new method for the study of normal oscillationsof mechanical systems with distributed parameters, which is based on theapplication of Lagrange’s equations of the first kind, was suggested. Thismethod is the most effective one for applying to the elastic systems, consistingof lumped masses, bars, rings, and plates, which can be connected rigidly toeach other or by means of linear compliances. As an additional example wemake use of the suggested method to study the oscillations of a system of barswith lateral and longitudinal oscillations. Considering this example, we showhow a holonomic, i. e. rigid, constraint between elements of system becomeselastic.

In Fig. VI. 2 is shown an elastic system, consisting of three uniform straightbars and one linear compliance δ = 1/c. We assume that the bars lie in thesame plane and for small oscillations of system, bar 1 has the longitudinaloscillations and bars 2 and 3 the bending oscillations.

166 VI. Application Lagrange Multipliers

Fig.VI. 2

From the principle of releasability of system it follows that by relation(2.1) the oscillations of bars can be represented as

u(x1, t) =∞∑

σ=1

q1σ(t)X1σ(x1) , X1σ(x1) = sin(2σ − 1)πx1

2l1,

y2(x2, t) =

∞∑

σ=1

q2σ(t)X2σ(x2) ,

y3(x3, t) = η(t) +(x3 −

l3

2

)ϕ(t) +

∞∑

σ=1

q3σ(t)X3σ(x3) ,

0 xi li , i = 1, 2, 3 .

(5.1)

Here X2σ(x2) and X3σ(x3) are the beam functions of a cantilever and afree bar, respectively [12, 227]. The first two addends in relation for y3(x3, t)correspond to the motion of bar 3 as rigid body. The quantity η is equal to adisplacement of the center of mass C of bar 3 in the axis y3 and ϕ is an angleof its rotation. We consider also the displacement ξ of the center of massC of bar 3 in the axis x3 and regard the quantities ξ, η, ϕ, qνσ (ν = 1, 2, 3;σ = 1, 2, . . .) as generalized Lagrangian coordinates.

Let be δ = 0. Then all constraints between the coordinates introducedare holonomic and are given by the following equations

f1 = u(l1, t) − y2(a2, t) =

∞∑

σ=1

q1σX1σ(l1) −

∞∑

σ=1

q2σX2σ(a2) = 0 ,

f2 = y2(l2, t) − ξ =

∞∑

σ=1

q2σX2σ(l2) − ξ = 0 ,

f3 = y3(a3, t) = η +(a3 −

l3

2

)ϕ +

∞∑

σ=1

q3σX3σ(a3) = 0 ,

(5.2)

5. Normal frequencies and oscillation modes of system of bars 167

f4 =∂y2

∂x2

∣∣∣∣x2=l2

+∂y3

∂x3

∣∣∣∣x3=a3

=

∞∑

σ=1

q2σX ′

2σ(l2) + ϕ+

∞∑

σ=1

q3σX ′

3σ(a3) = 0 .

By relations (2.3) and (2.4) the kinetic energy of bars and potential energyof their deformation can be represented as [12, 227]

T =m3(ξ

2 + η2)

2+

m3l23ϕ

2

24+

n∑

ν=1

∞∑

σ=1

Mνσ q2νσ

2,

Π =

n∑

ν=1

∞∑

σ=1

Mνσω2νσ

q2νσ

2, M1σ =

m1

2,

Mµσ =mµ

lµ

lµ∫

0

X2µσ

(x) dx =mµX2

µσ(lµ)

4,

n = 3 , σ = 1, 2, . . . , ν = 1, 2, 3 , µ = 2, 3 .

(5.3)

In these formulas ωνσ are normal frequencies of bars when the constraintsare absent, mν are their masses.

We shall use further Lagrange’s equations of the first kind in the gener-alized coordinates:

d

dt

∂L

∂qρ

−∂L

∂qρ

=

k∑

i=1

Λi

∂fi

∂qρ

, L = T − Π . (5.4)

Here k is the number of constraints and q1, q2, . . . is a system of all Lagrangiancoordinates introduced above. Equations (5.4) give

m3ξ = −Λ2, m3η = Λ3,m3l

23

12ϕ = Λ3

(a3 −

l3

2

)+ Λ4 ,

M1σ(q1σ + ω21σ

q1σ) = Λ1X1σ(l1) ,

M2σ(q2σ + ω22σ

q2σ) = −Λ1X2σ(a2) + Λ2X2σ(l2) + Λ4X′

2σ(l2) ,

M3σ(q3σ + ω23σ

q3σ) = Λ3X3σ(a3) + Λ4X′

3σ(a3) .

(5.5)

The generalized reactions Λ1,Λ2,Λ3 are equal to the forces of interactionof bars at the points of their connection to each other and Λ4 is equal to themoment of interaction between bars 2 and 3. From the principle of releasabil-ity from constraints, it follows that equations (5.5) can also be regarded asthe equations of dynamics for the elements of system acted by the forcesΛ1,Λ2,Λ3 and the moment Λ4 that are due to constraints no matter whatthe constraints are rigid or elastic. Then we make use of Lagrange’s equationsof the first kind (5.4) not only in the case of rigid constraints but also in thecase when all or certain constraints are elastic. For this purpose we assumefirst that all constraints are rigid and write then equations (5.4). In this casein the equations of the elastic constraints the reactions are introduced corre-sponding to these constraints. In the considered example the first constraint

168 VI. Application Lagrange Multipliers

is elastic. The reaction Λ1 is equal to the force of stretching (contraction) ofspring with the compliance δ, in which case Λ1 > 0 if the spring is stretched.Therefore the first equation of system (5.2) takes the form

f1 =

∞∑

σ=1

q1σX1σ(l1) −

∞∑

σ=1

q2σX2σ(a2) + Λ1δ = 0 . (5.6)

Note that if we multiply this relation by minus unity, i. e. represent it in theform of f∗

1 = −f1 = 0, then the new Lagrange multiplier Λ∗

1 is, obviously,such that Λ∗

1 = −Λ1. Hence the quantity Λ∗

1δ enters into the equation f∗

1 = 0with plus sign.

Thus, if the i-th constraint is elastic and its compliance is equal to δi,

then, assuming first that it is holonomic, we proceed to an elastic constraint

by adding the quantity Λiδi.

Suppose, the elastic system considered oscillates with the sought normalfrequency p. Then the reactions Λi and the coordinates qρ can be representedas

Λi = Λi cos(p t + α) , qρ = qρ cos(p t + α) . (5.7)

Then, taking into account equations (5.5), we obtain

ξ =Λ2

m3p2, η = −

Λ3

m3p2,

ϕ = −12

l23

(a3 − l3/2)Λ3 + Λ4

m3p2,

q1σ =X1σ(l1)Λ1

M1σ(ω21σ

− p2),

q2σ =−X2σ(a2)Λ1 + X2σ(l2)Λ2 + X ′

2σ(l2)Λ4

M2σ(ω22σ

− p2),

q3σ =X3σ(a3)Λ3 + X ′

3σ(a3)Λ4

M3σ(ω23σ

− p2).

(5.8)

Substituting relations (5.7) and formulas (5.8) into the equations of con-straints (5.2), (5.6), we find

4∑

j=1

αij(p2)Λj = 0 , αij = αji , i = 1, 4 . (5.9)

Here index i corresponds to the constraint number.This elastic system has zero values of Λj , j = 1, 4, only for the normal

form, of oscillations with the frequency p, for which the forces of interactionbetween all elements of system are absent. The system considered has nosuch forms of oscillations. Therefore in accordance with system (5.9) all itsnormal frequencies must satisfy the following equation

det[αij(p2)] = 0 . (5.10)

5. Normal frequencies and oscillation modes of system of bars 169

It is useful to represent the coefficients αij as

αii = δi + βii + γii ,

αij = βij + γij , i = j .(5.11)

Here δi is a compliance of the i-th constraint in the case when it is elastic.The quantities βij , which are inversely proportional to the quantity p2, canbe called the coefficients of compliance of inertia forces. For this problem wehave

β11 = β12 = β13 = β14 = β23 = β24 = 0 , βij = βji ,

β22 = −1

m3p2, β33 = −

1 + 12(a3 − l3/2)2/l23m3p2

,

β34 = −12(a3 − l3/2)

m3l23p

2, β44 = −

12

m3l23p

2.

These relations are used to construct the functions αij(p2) (i, j = 1, 4) with

the help of formulas (5.11).The coefficients γij = γji are infinite sums of simple fractions:

γ11(p2) =

∞∑

σ=1

X21σ

(l1)

M1σ(ω21σ

− p2)+

∞∑

σ=1

X22σ

(a2)

M2σ(ω22σ

− p2),

γ12(p2) = −

∞∑

σ=1

X2σ(a2)X2σ(l2)

M2σ(ω22σ

− p2), γ13(p

2) = 0 ,

γ14(p2) = −

∞∑

σ=1

X2σ(a2)X′

2σ(l2)

M2σ(ω22σ

− p2),

γ22(p2) =

∞∑

σ=1

X22σ

(l2)

M2σ(ω22σ

− p2), γ23(p

2) = 0 , (5.12)

γ24(p2) =

∞∑

σ=1

X2σ(l2)X′

2σ(l2)

M2σ(ω22σ

− p2), γ33(p

2) =

∞∑

σ=1

X23σ

(a3)

M3σ(ω23σ

− p2),

γ34(p2) =

∞∑

σ=1

X3σ(a3)X′

3σ(a3)

M3σ(ω23σ

− p2),

γ44(p2) =

∞∑

σ=1

[X ′

2σ(l2)

]2

M2σ(ω22σ

− p2)+

∞∑

σ=1

[X ′

3σ(a3)

]2

M3σ(ω23σ

− p2).

For p2 = 0 the quantities γij take the form

γij(0) =∂2Π

∂Λi ∂Λj

. (5.13)

170 VI. Application Lagrange Multipliers

Here Π is a total potential energy of the deformation of elements of systemunder the generalized reactions Λi. To verify the validity of formula (5.13),we consider relation (5.3) for the potential energy of deformation of bars andLagrange’s equations (5.5). For quasistatic account of all forms of normaloscillations of elements of system, i. e. in the case when qνσ = 0 (ν = 1, 2, 3;σ = 1, 2, . . .) we have

Π =1

2

∞∑

σ=1

[Λ1X1σ(l1)

]2

M1σω21σ

+

+1

2

∞∑

σ=1

[− Λ1X2σ(a2) + Λ2X2σ(l2) + Λ4X

′

2σ(l2)

]2

M2σω22σ

+

+1

2

∞∑

σ=1

[Λ3X3σ(a3) + Λ4X

′

3σ(a3)

]2

M3σω23σ

.

(5.14)

Taking into account the above relations and formulas (5.12), we concludethat relations (5.13) are valid.

The substantial fact is that the potential energy of deformation of barscan be represented not only in the form of infinite series (5.14) but also inclosed form:

Π = Π1 + Π2 + Π3 , Π1 =Λ2

1l1

2E1S1,

Πµ =1

2

lµ∫

0

M2µ(x) dx

EµJµ

, µ = 2, 3 .

(5.15)

Here E is Young’s modulus, J is a moment of inertia of a cross sectionof bar, S is a cross section area. The bending moments M2(x),M3(x) arelinear functions of the generalized reactions Λi. We remak that for numericalcomputation of M3(x) the force Λ3 and the moment Λ4, applied to bar 3, arecounteracted in the case of its quasistatic deformation by the inertia forcesof translational and angular motions. Using formulas (5.13) and (5.15), weobtain

γ11(0) =l1

E1S1+

a32

3E2J2, γ12(0) = −

a22(3l2 − a2)

6E2J2,

γ13(0) = 0 , γ14(0) = −a22

2E2J2, γ22(0) =

l323E2J2

,

γ23(0) = 0 , γ24(0) =l22

2E2J2, γ33(0) =

f33(z)l33E3J3

,

γ34(0) =f34(z)l23E3J3

, γ44(0) =f44(z)l3E3J3

+l2

E2J2, (5.16)

f33 =1

105−

11z

105+

13z2

35−

z3 + z4

3+

3z5 − z6

5,

5. Normal frequencies and oscillation modes of system of bars 171

f34 = −11

210+

13z

35−

z2

2−

2z3

3+

3z4

2−

3z5

5,

f44 =13

35− z + 2z3 − z4 , z =

a3

l3.

For approximate numerical computation of frequencies with provision forequation (5.10), we make use of the approximate approach stated in § 3. Recallthat it is based on the dynamic account of N first normal forms of oscillationsof elements of system and on the quasistatic account of the rest of normalforms. According to this approach the relations γij(p

2) can approximately becomputed by formulas

γ11(p2) =

N∑

σ=1

X21σ

(l1)

M1σ(ω21σ

− p2)+

N∑

σ=1

X22σ

(a2)

M2σ(ω22σ

− p2)+

+γ11(0) −N∑

σ=1

X21σ

(l1)

M1σω21σ

−N∑

σ=1

X22σ

(a2)

M2σω22σ

,

γ12(p2) = −

N∑

σ=1

X2σ(a2)X2σ(l2)

M2σ(ω22σ

− p2)+ γ12(0)+

+

N∑

σ=1

X2σ(a2)X2σ(l2)

M2σω22σ

, γ13(p2) = 0 ,

γ14(p2) = −

N∑

σ=1

X2σ(a2)X′

2σ(l2)

M2σ(ω22σ

− p2)+ γ14(0)+

+

N∑

σ=1

X2σ(a2)X′

2σ(l2)

M2σω22σ

,

γ22(p2) =

N∑

σ=1

X22σ

(l2)

M2σ(ω22σ

− p2)+ γ22(0)−

−N∑

σ=1

X22σ

(l2)

M2σω22σ

, γ23(p2) = 0 , (5.17)

γ24(p2) =

N∑

σ=1

X2σ(l2)X′

2σ(l2)

M2σ(ω22σ

− p2)+ γ24(0) −

N∑

σ=1

X2σ(l2)X′

2σ(l2)

M2σω22σ

,

γ33(p2) =

N∑

σ=1

X23σ

(a3)

M3σ(ω23σ

− p2)+ γ33(0) −

N∑

σ=1

X23σ

(a3)

M3σω23σ

,

γ34(p2) =

N∑

σ=1

X3σ(a3)X′

3σ(a3)

M3σ(ω23σ

− p2)+ γ34(0) −

N∑

σ=1

X3σ(a3)X′

3σ(a3)

M3σω23σ

,

172 VI. Application Lagrange Multipliers

γ44(p2) =

N∑

σ=1

[X ′

2σ(l2)

]2

M2σ(ω22σ

− p2)+

N∑

σ=1

[X ′

3σ(a3)

]2

M3σ(ω23σ

− p2)+

+γ44(0) −

N∑

σ=1

[X ′

2σ(l2)

]2

M2σω22σ

−

N∑

σ=1

[X ′

3σ(a3)

]2

M3σω23σ

.

Recall that the static coefficients γij(0) are given by formulas (5.16). Usingrelations (5.17) and formulas (5.11), we can obtain the approximate repre-sentations of the functions αij(p

2) (i, j = 1, 4).

Now we compute eigenfunctions. Denote by Λρj the quantities Λj , sat-isfying system (5.9) for the normal frequencies pρ. From relations (5.1) and(5.8) it follows that the normal forms of oscillations of the system consideredare described by the functions

uρ(x1) =

∞∑

σ=1

X1σ(l1)Λρ1

M1σ(ω21σ

− p2ρ)X1σ(x1) , 0 x1 l1 ,

yρ2(x2) =

∞∑

σ=1

−X2σ(a2)Λρ1 + X2σ(l2)Λρ2 + X ′

2σ(l2)Λρ4

M2σ(ω22σ

− p2ρ)

X2σ(x2) ,

0 x2 l2 ,

(5.18)

yρ3(x3) =∞∑

σ=1

X3σ(a3)Λρ3 + X ′

3σ(a3)Λρ4

M3σ(ω23σ

− p2ρ)

X3σ(x3) , 0 x3 l3 .

Thus, we find the representation of normal forms of oscillations of the origi-nal compound elastic system by the normal forms of its unit cells (separateelements).

It is convenient to consider the functions ustρ

(x1), ystρµ

(xµ) (µ = 2, 3) defin-ing the deformation of bars in the quasistatics under the generalized reac-tions Λρj . These functions can be found in closed form by the methods ofstrength of materials and can be represented by formulas (5.1) and (5.8) inthe form of infinite series

ustρ

(x1) =∞∑

σ=1

X1σ(l1)Λρ1

M1σω21σ

X1σ(x1) , 0 x1 l1 ,

ystρ2(x2) =

∞∑

σ=1

−X2σ(a2)Λρ1 + X2σ(l2)Λρ2 + X ′

2σ(l2)Λρ4

M2σω22σ

X2σ(x2) ,

0 x2 l2 ,

ystρ3(x3) =

∞∑

σ=1

X3σ(a3)Λρ3 + X ′

3σ(a3)Λρ4

M3σω23σ

X3σ(x3) , 0 x3 l3 .

6. Transformation of the frequency equation 173

Then relations (5.18) imply that the sought forms of oscillations can be rep-resented as

uρ(x1) = ustρ

(x1) +

∞∑

σ=1

X1σ(l1)Λρ1p2ρ

M1σ(ω21σ

− p2ρ)ω2

1σ

X1σ(x1) ,

yρ2(x2) = ystρ2(x2)+

+∞∑

σ=1

[− X2σ(a2)Λρ1 + X2σ(l2)Λρ2 + X ′

2σ(l2)Λρ4

]p2

ρ

M2σ(ω22σ

− p2ρ)ω2

2σ

X2σ(x2) ,

yρ3(x3) = ystρ3(x3) +

∞∑

σ=1

[X3σ(a3)Λρ3 + X ′

3σ(a3)Λρ4

]p2

ρ

M3σ(ω23σ

− p2ρ)ω2

3σ

X3σ(x3) , (5.19)

0 xi li , i = 1, 2, 3 .

The frequencies of longitudinal oscillations ω1σ increase with σ and thefrequencies of lateral oscillations ω2σ and ω3σ increase with σ2. The quantitiesX ′

2σ(l2) and X ′

3σ(a3) increase with σ. Therefore the series, entering into the

first formula of (5.19), converges as 1/σ4 and the other two sums as 1/σ7.Such a fast convergence of series is explained by that in solution (5.19) thereare separated the quasistatic forms of deformations of elements of system.

§ 6. Transformation of the frequency equation

to a dimensionless form and determination

of minimal number of parameters governing

a natural frequency spectrum of the system

For numerical implementation of the suggested new method for determi-nation of a natural frequency spectrum of the system of bars it is necessaryto transform the frequency equation (5.10) to a dimensionless form. For thispurpose it is necessary first of all to choose the central bar of this system. Inthe example shown in Fig. VI. 2, it is the second bar.

Natural frequencies of the second bar in the case of its disconnection toother bars are as follows [12]:

ω22σ

= λ42σ

k22 , k2

2 =1

m2

E2J2

l32, (6.1)

where λ2σ are roots of the equation

ch λ cos λ = −1 . (6.2)

We have

λ21 = 1.875 , λ22 = 4.694 , λ23 = 7.855 , λ24 = 10.996 ,

λ25 = 14.137 , λ2σ =π

2(2σ − 1) , σ > 5 .

174 VI. Application Lagrange Multipliers

The quantities λ22σ

= ω2σ/k2 give the dimensionless natural frequency spec-trum of the considered cantilever.

Now we shall find the natural frequency spectrum p∗ of the mechanicalthree-bar system under consideration in the form

p∗ =p

k2. (6.3)

If we mentally separate the first and the third bars from the system, thentheir natural frequencies can be represented as [12]:

ω21σ

= λ41σ

k21 , k2

1 =1

m1

E1J1

l1, λ2

1σ=

(2σ − 1)π

2,

ω23σ

= λ43σ

k23 , k2

3 =1

m3

E3J3

l33.

(6.4)

Here λ3σ are roots of the equation

ch λ cos λ = 1 , (6.5)

whereλ31 = 4.7300 , λ32 = 7.8532 , λ33 = 10.9956 ,

λ34 = 14.137 , λ3σ =π

2(2σ + 1) , σ > 4 .

Natural vibration modes of the bars of this system are as follows [12]:

X1σ(x1) = sin(2σ − 1)π

2ξ , ξ =

x1

l1,

X2σ(x2) = sinλ2σξ − sh λ2σξ + A2σ(ch λ2σξ − cos λ2σξ) , ξ =x2

l2,

X3σ(x3) = sinλ3σξ + shλ3σξ − A3σ(cos λ3σξ + ch λ3σξ) , ξ =x3

l3.

(6.6)

Here

A2σ =sh λ2σ + sinλ2σ

ch λ2σ + cos λ2σ

, A3σ =sh λ3σ − sin λ3σ

ch λ3σ − cos λ3σ

,

A21 = 1.3622 , A22 = 0.98187 , A23 = 1.000777 ,

A24 = 0.999965 , A25 = 1.0000015 , A2σ = 1 , σ > 5 ,

A31 = 1.0178 , A32 = 0.999223 , A33 = 1.0000335 ,

A34 = 0.9999986 , A35 = 1.0000001 , A2σ = 1 , σ > 5 .

(6.7)

By using expressions (6.6), (6.7) and equations (6.2) and (6.5), we shallrepresent the reduced masses Mµσ, µ = 1, 2, 3 , specified by the formulas(5.3), in the form

Mµσ = mµA2µσ

, µ = 1, 2, 3 , A21σ

=1

2. (6.8)

6. Transformation of the frequency equation 175

Formulas (6.1), (6.3), (6.4), (6.8) allow us to write the quantities Mµσ(ω2µσ

−p2), entering the expressions (5.12), in the following manner

Mµσ(ω2µσ

− p2) = mµA2µσ

(ω2µσ

− p2) =

= m2mµ2A2µσ

k22(kµ2λ

4µσ

− p2∗) , µ = 1, 2, 3 .

(6.9)

Here

mµ2 =mµ

m2, kµ2 =

k2µ

k22

, µ = 1, 2, 3 . (6.10)

From expressions (5.12) and (6.9) it follows that all quantities γij(p2∗), i, j =

1, 4, contain the factor1

m2k22

=l32

E2J2.

Thus we shall multiply all elements of determinant (5.10) by m2k22. As this

takes place the quantities

αij(p2∗) =

E2J2

l32αij(p

2) , i, j = 1, 2, 3 ,

become dimensionless, because the functions Xµσ are dimensionless.The expressions γi4 , i = 1, 2, 3, contain derivatives X ′

2σ(l2), X

′

3σ(a3), and

γ44 contains those derivatives squared. We have:

X ′

2σ(l2) =

dX2σ(x2)

dx2

∣∣∣∣x2=l2

=1

l2

dX2σ

dξ

∣∣∣∣ξ=1

=1

l2X ′

ξ,2σ(1) ,

X ′

3σ(a3) =

1

l3

dX3σ

dξ

∣∣∣∣ξ=

a3

l3

=1

l3X ′

ξ,3σ

(a3

l3

).

(6.11)

Therefore we shall multiply the fourth row and the fourth column of deter-minant (5.10) by l2, in this case all αij will become dimensionless.

As follows from formulas (6.9), (6.10), the quantities αij(p2∗), i, j = 1, 4,

depend on four dimensionless parameters

m12 =m1

m2, m32 =

m3

m2, k12 =

k21

k22

, k32 =k23

k22

.

Other four dimensionless parameters will be added to them:

l2

l3,

a2

l2,

a3

l3,

δ

δ∗, δ∗ =

l32E2J2

.

Hence we have eight dimensionless parameters in total.

176 VI. Application Lagrange Multipliers

The relations l2/l3, a3/l3, δ/δ∗ appear in the quantities α11, β33, β34, β44

in the following manner:

α11 = m2k22(δ + γ11) =

=E2J2

l32δ + m2k

22γ11 =

δ

δ∗+ m2k

22γ11 ,

β33 = m2k22β33 = −

m2

m3

(1 + 12

(a3

l3−

1

2

)2)1

p2∗

,

β34 = m2k22l2β34 = −12

m2

m3

l2

l3

(a3

l3−

1

2

)1

p2∗

,

β44 = m2k22l

22β44 = −12

m2

m3

(l2

l3

)21

p2∗

.

The parameters a2/l2 and a3/l3 will also enter determinant (5.10) in termsof the functions X2σ(a2), X3σ(a3) and X ′

3σ(a3). We note that the quantities

γ34 and γ44, as it follows from formulas (5.12), (6.11), are proportional tol2/l3 and (l2/l3)

2 respectively.When using the Nth approximation we should bear in mind that the

quantities γij(0) are the functions of the parameters intriduced above. pa-rameters. It is necessary to express γij(0) in terms of them. As this takesplace we should have in view that

m1k21 =

E1S1

l1, m2k

22 =

E2J2

l32, m3k

23 =

E3J3

l33.

Then we have:m2k

22

m1k21

=E2J2

l32·

l1

E1S1=

1

m12k12,

m2k22

m3k23

=1

m32k32=

E2J2

E3J3

(l3

l2

)3

.

Having taken those expressions into account, we obtain:

γ11(0) =1

m12k12

1

3ζ3 , ζ =

a2

l2, z =

a3

l3,

γ12(0) = −a22(3l2 − a2)

6l32= −

ζ2(3 − ζ)

6,

γ14(0) = −1

2ζ2 , γ22(0) =

1

3, γ24(0) =

1

2,

γ33(0) =f33(z)

m32k32, γ34(0) =

f34(z)

m32k32

(l2

l3

),

γ44(0) =f44(z)

m32k32

(l2

l3

)2

+ 1 .

6. Transformation of the frequency equation 177

When considering the Nth approximation we shall have:

γN

11(p2∗) =

N∑

σ=1

2

m12(k12λ41σ

− p2∗)

+

N∑

σ=1

X22σ

(a2)

A22σ

(λ42σ

− p2∗)+

+1

m12k12

(1 −

8

π2

N∑

σ=1

1

(2σ − 1)2

)+

(1

3ζ3 −

N∑

σ=1

X22σ

(a2)

A22σ

λ42σ

),

γN

12(p2∗) = −

N∑

σ=1

X2σ(a2)X2σ(l2)

A22σ

(λ42σ

− p2∗)

−ζ2(3 − ζ)

6+

N∑

σ=1

X2σ(a2)X2σ(l2)

A22σ

λ42σ

,

γN

14(p2∗) = −

N∑

σ=1

X2σ(a2)X′

ξ,2σ(1)

A22σ

(λ42σ

− p2∗)

−ζ2

2+

N∑

σ=1

X2σ(a2)X′

ξ,2σ(1)

A22σ

λ42σ

,

γN

22(p2∗) =

N∑

σ=1

4

(λ42σ

− p2∗)

+1

3−

N∑

σ=1

4

λ42σ

,

γN

24(p2∗) =

N∑

σ=1

X2σ(l2)X′

ξ,2σ(1)

A22σ

(λ42σ

− p2∗)

+1

2−

N∑

σ=1

X2σ(l2)X′

ξ,2σ(1)

A22σ

λ42σ

,

γN

33(p2∗) =

N∑

σ=1

X23σ

(a3)

m32A23σ

(k32λ42σ

− p2∗)

+f33(z)

m32k32−

N∑

σ=1

X23σ

(a3)

m32A23σ

k32λ42σ

,

γN

34(p2∗) =

( N∑

σ=1

X3σ(z)X ′

ξ,3σ(z)

m32A23σ

(k32λ43σ

− p2∗)

+f34(z)

m32k32−

N∑

σ=1

X3σ(z)X ′

ξ,3σ(z)

m32A232k32λ

43σ

)l ,

γN

44(p2∗) =

N∑

σ=1

(X ′

ξ,2σ(1))2

A22σ

(λ42σ

− p2∗)

+ l2N∑

σ=1

(X ′

ξ,3σ(z))2

m32A23σ

(k32λ43σ

− p2∗)+

+f44(z)

m32k32l2 + 1 −

N∑

σ=1

(X ′

ξ,2σ(1))2

A22σ

λ42σ

− l2N∑

σ=1

(X ′

ξ,3σ(z))2

m32A23σ

k32λ43σ

.

Here: l = l2/l3,

X2σ(a2) = sin(λ2σζ) − sh(λ2σζ) + A2σ(ch(λ2σζ) − cos(λ2σζ)) , ζ = a2/l2 ,

X2σ(l2) = sin λ2σ − sh λ2σ + A2σ(ch λ2σ − cos λ2σ) ,

X ′

ξ,2σ(1) = λ2σ(cos λ2σ − ch λ2σ + A2σ(sh λ2σ + sinλ2σ)) ,

X3σ(a3) = sin(λ3σz) + sh(λ3σz) − A3σ(cos(λ3σz) + ch(λ3σz)) , z = a3/l3 ,

X ′

ξ,3σ(z) = X ′

z,3σ(z) = λ3σ(cos(λ3σz) + ch(λ3σz) − A3σ(sh(λ3σz) − sin(λ3σz)) .

We note that when using the given formulas one should remember thatthe values of X2σ(l2) have changing (alternating) signs (the plus sign — forodd σ and the minus sign — for even σ), and

∣∣∣∣X2σ(l2)

2

∣∣∣∣ = A2σ , A2σ > 0 .

178 VI. Application Lagrange Multipliers

So the frequency equation (5.10) can be written in the dimensionless form

det[αij(p2∗)] = 0 . (6.12)

Its coefficientsαii = δi + βii + γii ,

αij = βij + γij , i = j ,

depend on the eight dimensionless parameters

m1

m2,

m3

m2,

k21

k22

,k23

k22

,l2

l3,

a2

l2,

a3

l3,

δ

δ∗.

The dimensionless frequencies p∗ of the system under investigation relatedto the required dimension frequencies p by formula (6.3) may be found fromequations (6.12).

Further development of ideas of the method presented in § 3, § 5, and § 6is given in Appendix F. The method suggested there allows us to determinethe first frequency of elastic systems to a high accuracy. This can be usedfor testing the complex programs used for analysis of vibration of the elasticsystems.

§ 7. A special form of equations of the dynamics

of system of rigid bodies

For many bodies, the equations of motion for a system of rigid bodies,represented in the form of Lagrange’s equations of the second kind, are mostcomplicated [120, 406] and very difficult not only for their integration buteven for their writing. Therefore the question is actual how to represent theseequations in the form convenient for computer calculation, what, in turn, isreduced to finding new forms of representation for equations of motion of onebody.

As is known, the kinetic energy of free rigid body with six degrees offreedom, cannot be represented as the sum, involving only the squares ofgeneralized velocities multiplied by the constant values. In the independentgeneralized coordinates its kinetic energy has a rather complicated form.This explains the difficulties, connected with the application of Lagrange’sequations of the second kind even to one rigid body. Taking into account thisfact, we make use of Lagrange’s equations of the first kind (2.22) of Chapter Isince the kinetic energy of body in dependent coordinates has very simpleform (1.1).

Suppose, the active forces Fν are applied to the body at the points Nν =(xν , yν , zν). Then the possible elementary work is as follows

δA =∑

ν

Fν · (δρρρ + xνδi + yνδj + zνδk) =

= Qρρρ · δρρρ + Qi · δi + Qj · δj + Qk · δk ,

(7.1)

7. A special form of equations of the dynamics 179

whereQρρρ =

∑

ν

Fν , Qi =∑

ν

xνFν ,

Qj =∑

ν

yνFν , Qk =∑

ν

zνFν .(7.2)

The form of kinetic energy (1.1), equations of constraint (1.2), and pos-sible elementary work (7.1) result in that in this case in order to write equa-tions (2.22) of Chapter I, it is convenient to use the vector form of Lagrange’sequations of the first kind (1.3). Using the rules of application of this formuladescribed in § 1, we have

d

dt

∂T

∂ρρρ−

∂T

∂ρρρ= Qρρρ , κ = 1, 6 ,

d

dt

∂T

∂ i−

∂T

∂i= Qi + Λκ

∂fκ

∂i≡ Qi + 2Λ1i + Λ4j + Λ6k ,

d

dt

∂T

∂ j−

∂T

∂j= Qj + Λκ

∂fκ

∂j≡ Qj + 2Λ2j + Λ5k + Λ4i ,

d

dt

∂T

∂k−

∂T

∂k= Qk + Λκ

∂fκ

∂k≡ Qk + 2Λ3k + Λ6i + Λ5j .

In this case from relations (1.1) and (7.2) it follows that for the rigid bodyvector Lagrange’s equations of the first kind take the form

Mρρρ =∑

ν

Fν ,

Ix i =∑

ν

xνFν + 2Λ1i + Λ4j + Λ6k , (7.3)

Iy j =∑

ν

yνFν + 2Λ2j + Λ5k + Λ4i ,

Izk =∑

ν

zνFν + 2Λ3k + Λ6i + Λ5j .

We eliminate the unknown multipliers Λκ , κ = 1, 6, from vector La-grange’s equations of the first kind. For this purpose we differentiate twice intime the equations of constraint (1.2):

i2 = −i · i , j2 = −j · j , k2 = −k · k ,

2i · j + i · j + i · j=0 , 2j · k + j · k + j · k=0 ,

2k · i + k · i + k · i=0 .

(7.4)

180 VI. Application Lagrange Multipliers

Substituting into these relations the second derivatives in time from equa-tions (7.3), we obtain the formulas for the Lagrange multipliers Λκ :

2Λ1 = −∑

ν

xνFν · i − Ix i2 ,

2Λ2 = −∑

ν

yνFν · j − Iy j2 ,

2Λ3 = −∑

ν

zνFν · k − Izk2 ,

Λ4 = −2IxIy

Ix + Iy

i · j −Iy

Ix + Iy

∑

ν

xνFν · j −Ix

Ix + Iy

∑

ν

yνFν · i ,

Λ5 = −2IyIz

Iy + Iz

j · k −Iz

Iy + Iz

∑

ν

yνFν · k −Iy

Iy + Iz

∑

ν

zνFν · j ,

Λ6 = −2IzIx

Iz + Ix

k · i −Ix

Iz + Ix

∑

ν

zνFν · i −Iz

Iz + Ix

∑

ν

xνFν · k .

(7.5)

Substituting then relations (7.5) into system (7.3), we find

Mρρρ =∑

ν

Fν ,

i = −i2i −2Iy

Ix + Iy

(i · j)j −2Iz

Iz + Ix

(k · i)k +Lz

Ix + Iy

j −Ly

Iz + Ix

k ,

j = −j2j −2Iz

Iy + Iz

(j · k)k −2Ix

Ix + Iy

(i · j)i +Lx

Iy + Iz

k −Lz

Ix + Iy

i ,

k = −k2k −2Ix

Iz + Ix

(k · i)i −2Iy

Iy + Iz

(j · k)j +Ly

Iz + Ix

i −Lx

Iy + Iz

j .

(7.6)

Here Lx, Ly, Lz are a projection of the principal moment of active forces

L =∑

ν

(xν i + yνj + zνk) × Fν . (7.7)

E x a m p l e VI .4 . We shall show that equations (7.6) result in dynamicEuler’s equations with respect to the projections p, q, r of the vector of anglevelocity ωωω on the axes x, y, z. We have

i = ωωω × i = rj − qk , i · j = −p q , k · i = −r p ,

−i2i = −(q2 + r2)i ,

i = ωωω × i + ωωω × i = rj − qk − (q2 + r2)i + p qj + p rk .

Then the projection of the second equation of system (7.6) on x-axis givesthe identity and on y-axis the following relation

r + p q =2Iy

Ix + Iy

p q +Lz

Ix + Iy

.

8. The study of certain problems of robotics 181

Taking into account that A = Iy + Iz, B = Iz + Ix, C = Ix + Iy, we obtainthe third dynamic equation

Cr − (A − B) p q = Lz .

The projecting of the same vector equation on z-axis gives the second Euler’sequation

Bq − (C − A) r p = Ly .

Similarly, from the third equation of system (7.6) we can obtain the thirdand first Euler’s equations and from the fourth equation of system (7.6) thesecond and first dynamic Euler’s equations.

Note, that if on the vectors i, j,k constraints (1.2) are not imposed thena body considered is pseudorigid according to the terminology of J. Casey[417. 2004]. Basing on the apparatus of the monograph by Truesdell [418],J. Casey describes the dynamics of this continuum by the Lagrange equationsthe number of which is equal to twelve.

The theory of vector Lagrange equations of the first kind is used in thenext section when studying some problems of robotics.

§ 8. The application of special form

of equations of dynamics to the study

of certain problems of robotics

Consider a motion control of a platform of dynamic stand [66]. We assumethat the motion are controlled by the six bars of variable lengths (hydrauliccylinders). One end point of each bar is connected by spherical joints with afixed point and the other end point of each bar with one of the points Nν , ν =1, 6, of platform. Such large dynamic stands are employed by the leading airconstruction companies, in which case one stand is constructed approximatelyfor ten aeroplanes. The cabin of aeroplane and the platform of stand are fixedand, using the control devices of aeroplane, the pilot moves a stand by varyingthe lengths of bars. In this case the pilot has a full illusion of a real motion inspace together with aeroplane. The stands are used to train pilots, includingthe training of an accurate behavior in extremal situations and the practiceof landing the aeroplane in concrete wold’s airports, to maintain good flyingform, and so on.

We introduce the fixed system of coordinates Oξηζ and the system Cxyz

with the unit vectors i, j, k, which is rigidly fixed with the platform of stand

and directed in the lines of its principal central axes of inertia. Let be ρρρ =−−→OC.

Then the position of platform as a rigid body is defined by the vectors ρρρ, i,j, k.

The assumption that the body is rigid, as is remarked in § 1, can be re-garded as the imposing of ideal constraints. These constraints are given by

182 VI. Application Lagrange Multipliers

equations (1.2). Then vector Lagrange’s equations of the first kind, describ-ing the motion of the platform of stand, can be represented in the form ofequations (7.3).

The force Fν , applied to the platform by virtue of the bar, can be repre-sented as

Fν = Fν lν/|lν | , lν = ρρρ + xν i + yνj + zνk ,

where Fν = uν is a control parameter. In this case in system (7.3) the sum-ming is over all ν from 1 to 6.

Equations of constraint (1.2) yield relations (7.4). By these relations, fromequations (7.3) we can obtain formulas (7.5) for the coefficients Λ1,Λ2, . . . ,Λ6.Eliminating the Lagrange multipliers from system (7.3), we arrive to differ-ential system (7.6) with respect to the vectors ρρρ, i, j, k, which involves sixcontrol parameters uν = Fν , ν = 1, 6, being the strains of bars. In this systemLx, Ly, Lz are the projections of principal moment of active forces (7.7).

We remark that system (7.6) found can be computed since it is solvablefor the second time derivatives of unknown vectors.

Considering the problem of dynamic stand, we need to pay attention tothe following fact, which is not connected directly with the previous contents.The given law of varying the lengths of bars (hydraulic cylinders) lν(t) weshall regard as nonstationary constraints fν ≡ l2

ν− l2

ν(t) = 0, ν = 1, 6, i. e.

we shall study the motion control by constraints. We introduce the Lagrangemultipliers Λν . In this case the reaction of the ν-th constraint has the form

Rlν = Λν

∂l2ν

∂lν= 2Λν lν ≡ Fν .

It follows that

2Λν =Fν

lν=

uν

lν.

If the position of body is given by six generalized coordinates being thelengths of bars, then the theorem of the holonomic mechanics of Chapter IIIbecomes descriptive. According to this theorem the motion such that oneof generalized coordinates is a given function of time can be obtained if weintroduce one additional force corresponding to this coordinate.

Consider now the system of rigid bodies, connected sequentially to eachother by spherical joints. Similar mechanical systems often occur in therobotics. Suppose, the number of joints s is equal to the number of movingbodies. The friction in joints is assumed to be negligible, i. e., by assumption,the constraints are ideal. Suppose also that the joint with number σ connectsthe bodies (σ − 1) and σ. Then the equations of constraint have the form

ρρρσ + xσ

σiσ + yσ

σjσ + zσ

σkσ−

−ρρρσ−1 − xσ

σ−1iσ−1 − yσ

σ−1jσ−1 − zσ

σ−1kσ−1 = 0 ,

σ = 1, s .

(8.1)

9. Application of the generalized Gaussian principle 183

Here the vectors ρρρσ, iσ, jσ, kσ, corresponding to the body σ, have the samesense as above; xσ

ρ, yσ

ρ, zσ

ρare the coordinates of the joint with number σ in

the system Cρxρyρzρ. A fixed body is assumed to be zero. Denote by Rσ theforce caused by the body (σ − 1) and applied by means of the joint to thebody σ.

We now make use of the releasability principle. In this case the equationof motion of body σ involves the reactions Rσ and Rσ+1 . Note that thebody s is under one reaction Rs . We differentiate twice equations (8.1) intime and eliminate then the second derivatives, using for each bodies theobtained special form of equations of its motion. In this case we obtain thesystem of s equations with respect to s unknown reactions Rσ . The equa-tion corresponding to arbitrary σ, which is not equal to 1 and s, involves thereactions Rσ−1, Rσ, Rσ+1 . For σ = 1 and σ = s we have the equations inunknowns R1, R2 and Rs−1, Rs , respectively. This implies that this systemof equations has a structure convenient for solving it with the help of thecomputer by the method of sequential elimination of the sought reactions.Determining these reactions and substituting them into the equations of mo-tion, we obtain a system of differential equations of motion of the consideredchain of bodies. This system is solvable for the second derivatives, i. e. it canbe used for numerical integration by computer.

§ 9. Application of the generalized Gaussian principle

to the problem of suppression

of mechanical systems oscillations

Introduction. This Section shows possibility and expedience of the em-ployment of the generalized Gaussian principle presented in § 3 of Chapter IV,for studying the problems of control of mechanical systems oscillations. Sim-ilar problems have been analyzed in studies [419. 1980] in details, where thefundamental approach to their solution is the method that is based on theminimization of the functional of control force squared. This Section showsthat application of the Gaussian generalized principle here proves to be veryeffective.

Consideration is being given to the mechanical system with finite quantityof degrees of freedom. The presence of one control force acting within sometime interval is assumed. This force, which is necessary to provide the move-ment of the system from a specified position to another one within the finitetime, is being determined. In particular, if the final position of the systemis to be the one of stable equilibrium then the problem under considerationbecomes the problem of vibration suppression. It is shown that this problemcan be solved with the help of Gaussian generalized principle. For a certainmechanical system with two degrees of freedom the problem was solved inthe studies [419. 1980] using the minimization of the functional of controlforce squared. It turns out that the solution developed in [419. 1980] within a

184 VI. Application Lagrange Multipliers

certain range of variation of system dimensionless parameters slightly differsfrom the solution obtained by using the method proposed.

Oscillation suppression of a trolley with a pendulum. The prob-

lem statement. Let us analyze the following problem [419. 1980]. The loadof mass m2 is suspended with the cable of length l attached to the cranetrolley of mass m1 running on horizontal rails (see Fig. VI. 3).

F x

ϕ

Fig.VI. 3

It is required to move the suspended load by the given distance a froma state of rest to another state of rest in the fixed time T by choosing thehorizontal force F (t), applied to the trolley.

The equations of motion of the system under consideration for small-amplitude oscillations will take the form

(m1 + m2)x − m2lϕ = F ,

x − lϕ = gϕ .(9.1)

In order to provide termination of free oscillation of the load at the timet = T , the control force F (t) should be such that the boundary conditions

ϕ(0) = ϕ(T ) = 0 , ϕ(0) = ϕ(T ) = 0 ,

x(0) = x(0) = x(T ) = 0 , x(T ) = a(9.2)

are satisfied.Let us introduce the principal coordinates and go to dimensionless vari-

ables ϕ, ξ, τ, γ, u by the formulas

ξ =m1 + m2

m1l

(x −

m2lϕ

m1 + m2

), τ = γt ,

γ2 =(m1 + m2)g

m1l, u =

F

γ2m1l.

Here ξ is a dimensionless displacement of the system center of mass, γ is anatural frequency, u is a control. Now instead of the differential equationssystem (9.1) we obtain two independent equations

ϕ + ϕ = u , ξ = u , (9.3)

9. Application of the generalized Gaussian principle 185

in which the derivatives correspond to the dimensionless time τ . For the sakeof simplicity the letter t will stand for the dimensionless time τ too. Boundaryconditions (9.2) will be rewritten as

ϕ(0) = ϕ(T ) = 0 , ϕ(0) = ϕ(T ) = 0 ,

ξ(0) = ξ(0) = ξ(T ) = 0 , ξ(T ) = 1 , T = γT .(9.4)

The system of equations (9.1) is linear. Hence, the solution of the bound-ary problem (9.1), (9.2) will depend linearly on the quantity a. Therefore,when examining the boundary problem (9.3), (9.4) for the sake of simplicitythe value of a can be accepted such that ξ(T ) = 1.

Analysis of the results following from the method that is based

on the minimization of the functional of control force squared. Tosolve the formulated problem (9.3), (9.4) it is necessary to add one conditionmore. It should express the principle, which forms the basis for choosingthe force F (t) from the entire set of the forces such that this problem has asolution. In the treatise [419. 1980] it is shown that if the choice of the controlu is subject to the condition of minimality of the functional

J =

∫T

0

u2(t)dt , (9.5)

and the maximum principle of Pontryagin [420] is used, then the control u

will be as follows [419. 1980, p. 328]

u(t) = C1 + C2t + C3 sin t + C4 cos t . (9.6)

Here Ck, k = 1, 4, are arbitrary constants. By choosing these constants sothat boundary conditions (9.4) are satisfied, we shall uniquely determine therequired control u(t). From the form of expression (9.6) it follows that thefunction u(t) is a general solution of the differential equation

....u + u = 0 ,

which can be represented in the form

d2

dt2

(d2

dt2+ 1

)u = 0 ,

which is directly connected with the initial system (9.3).Note, that if we considered the system of equations

xσ + ω2σxσ = u , σ = 1, s , (9.7)

then the control u, minimizing the functional (9.5), would satisfy the differ-ential equation

(d2

dt2+ ω2

1

)(d2

dt2+ ω2

2

). . .

(d2

dt2+ ω2

s

)u = 0 , (9.8)

186 VI. Application Lagrange Multipliers

The solution of this equation takes the form

u(t) =

s∑

σ=1

(Aσ cos ωσt + Bσ sin ωσt) . (9.9)

The problem of suppression of small-amplitude oscillations of the mechan-ical system in the time T , i. e. the following boundary problem

xσ(0) = x0σ

, xσ(0) = x0σ

,

xσ(T ) = xσ(T ) = 0 ,

σ = 1, s ,

(9.10)

may be solved by the choice of arbitrary constants Aσ and Bσ.Hence, as follows from expression (9.9), the minimization of functional

(9.5) is achieved by search of the required control u(t) in the form of theseries in resonance frequencies.

Oscillation suppression by the minimization of the functional of

control force squared as an example of the mixed problem of me-

chanics. The system of equations (9.7), representing small-amplitude oscilla-tions of the mechanical system under the action of control force u(t), is writ-ten in principal coordinates xσ, σ = 1, s. In initial coordinates qσ, σ = 1, s,this system will take the form

s∑

τ=1

(aστ qτ + cστqσ) = bσu(t) , σ = 1, s . (9.11)

Here aστ , cστ , bσ, σ, τ = 1, s, are given constant values. These constants aresuch that when going to the principal coordinates the system (9.11) takes(9.7).

In system (9.11) any of coefficients bσ may always be supposed equal toone. Let us assume that, for example, b1 = 1. Having substituted the controlforce u(t), set by the first equation of system (9.11), into equation (9.8), weobtain the differential equation of the order (2s + 2) relative to generalizedcoordinates qσ, σ = 1, s. Let us represent this equation in the form

s∑

σ=1

(a2s+2,σ

(2s+2)

qσ + a2s,σ

(2s)

qσ + · · · + a0,σqσ) = 0 , (9.12)

where a2n,σ, n = 0, s + 1, σ = 1, s, are constants that are found in the processof calculation.

Hence, as applied to system (9.11) the minimization of functional (9.5)means the obedience of oscillations of the system to constraint equation(9.12).

Let us assume that system (9.11) and the constraint (9.12) are specified.The problem of determining the control force u(t), providing satisfaction of

9. Application of the generalized Gaussian principle 187

the constraint (9.12), is a particular case of the so called mixed dynamicproblem.

Thus, the presence of constraint (9.12) following from the minimization offunctional (9.5) makes it possible to consider the problem of determining thecontrol force u(t), which provides oscillation suppression, as a certain mixeddynamic problem.

Application of the generalized Gaussian principle to the problem

of oscillation suppression. The theory of mixed dynamic problems appliedto system (9.11) implies the following fact. Let the motion of the systemconsidered be subject to the constraint given in the form

s∑

σ=1

a2s+2,σ(t, q, q, ... ,(2s+1)

q )(2s+2)

qσ + a2s+2,0(t, q, q, ... ,(2s+1)

q ) = 0 ,

σ = 1, s , κ = 1, k , k s ,

(9.13)

where a2s+2,σ, σ = 0, s, are some functions of variables indicated. Then forcertain restrictions to the relation between coefficients a2s+2,σ and bσ, σ =0, s, a 2s-order differential equation with respect to the control force u(t)can be derived in accordance with the algorithm developed in Chapter V.Note that in the general case this equation includes the time, generalizedcoordinates, and generalized velocities.

It is essential that when minimizing functional (9.5), from the set of theequations of form (9.13) there is chosen that subset to all elements of whichone single equation (9.8) corresponds. Note specially that its structure is de-fined only by the spectrum of natural frequencies of the system and doesn’tdepend on the choice of generalized coordinates. The following question nat-urally arises: if the equation of constraints (9.13) could be subject to anothercondition that also entails one single 2s-order equation with constant coeffi-cients.

This alternative condition can be found as follows. As in § 1 of Chapter IV,let us introduce the tangent space and represent the system of equations(9.11) therein as one vector equation

MW = Y + u(t)b ,

where

MW =

s∑

σ,τ=1

aστ qτeσ , Y = −

s∑

σ,τ=1

cστqτeσ , b =

s∑

σ=1

bσeσ .

The vectors of a reciprocal basis in this case do not depend on time andcoordinates qσ, σ = 1, s, i. e. are constant.

In accordance with the generalized Gaussian principle the constraint isideal if the quantity

−→R

2

2s=

(M

(2s)

W −(2s)

Y

)2

= ((2s)u b)2 (9.14)

188 VI. Application Lagrange Multipliers

is minimal. We choose from constraints (9.13) that subset, for elements of

which the quantity−→R

2

2sis equal to its lower boundary, which equals zero. As

follows from expression (9.14), all these elements are described by one singleequation

(2s)u = 0 . (9.15)

Therefore, the alternative solution of the problem of vibration suppressionwith one control force can be derived on the basis of the generalized Gaussprinciple.

The general solution of equation (9.15) is

u(t) =

2s∑

k=1

Cktk−1 . (9.16)

Contrary to the control u(t), set by formula (9.9), the control that is foundin the polynomial form (9.16), will not have oscillations corresponding toall natural system frequencies. The sought function u(t) will be sufficientlysmooth, which is its definite advantage.

Vibration suppression of n physical pendulums. As an example letus consider the following boundary problem

x1 = u , xi(0) = xi(0) = xi(T ) = 0 , i = 1, s , s = n + 1 ,

xj + ω2j−1xj = u , x1(T ) = a , xj(T ) = 0 , j = 2, s .

(9.17)

Natural frequencies of the system are assumed to be such that

ω1 = 1 , 1 < ω2 < ... < ωn .

Note that problem (9.17) is, in particular, the problem of vibration sup-pression of n physical pendulums suspended to a trolley, whis is to be movedwith the sought acceleration u(t) in time T by the given distance a. The sys-tem is supposed to be in the state of equilibrium at the initial and terminaltime instants [419. 1980, p. 340].

Formulas (9.16) and (9.17) imply that functions xi(t), i = 1, s, can berepresented as

xi(t) =

2s∑

k=1

Ckξik(t) , i = 1, s ,

where

ξ1k(t) =

∫t

0

τk−1(t − τ)dt , k = 1, 2s ,

ξjk(t) =

∫t

0

τk−1 sin ωj−1(t − τ)

ωj−1dt , j = 2, s .

9. Application of the generalized Gaussian principle 189

Constants Ck for different frequencies ωk, k = 1, n, are uniquely determinedfrom the solution of the system of equations

2s∑

k=1

ajkCk = aδ1j, j = 1, 2s , (9.18)

whereδ1j

= 1 , j = 1 , δ1j

= 0 , j = 1 ,

ajk = ξjk(T ) , as+j,k = ξjk(T ) ,

j = 1, s , k = 1, 2s .

Let us show that the functions u(t) and xi(t), i = 1, s, in problem (9.17)are such that

u(t) = −u(T − t) ,

x1(t) = a − x1(T − t) ,

xj(t) = −xj(T − t) .

(9.19)

Introduce into consideration the following functions

x1(t) = x1(T − t) − a ,

xj(t) = xj(T − t) ,

j = 2, s .

(9.20)

They are such that¨xi(t) = xi(T − t) , i = 1, s .

This and expressions (9.20) imply that problem (9.17) appears in new func-tions as

¨x1 = u(t) , xi(0) = ˙xi(0) = ˙xi(T ) = 0 , i = 1, s ,

¨xj + ω2j−1xj = u(t) , xj(T ) = 0 , j = 2, s , x1(T ) = −a .

(9.21)

Hereu(t) = u(T − t) . (9.22)

The solution of the system of linear algebraic equations is proportional tothe value of a, hence, comparing (9.17) with (9.21) results in

xi(t) = −xi(t) , i = 1, s ,

u(t) = −u(t) .

It follows from the latter formulas and expressions (9.20) and (9.22) thatrelations (9.19) are really satisfied.

When minimizing functional (9.5) the control of this problem will besought as

u(t) =n∑

k=1

(Ck cos ωkt + Cn+k sin ωkt) + C2n+1 + C2n+2t .

190 VI. Application Lagrange Multipliers

In this case constants Ck, k = 1, 2s, s = n+1, are also found from the systemof form (9.18), therefore, relations (9.19) remain.

Computing was done for n = 2 and a = 1. In this case the solutiondepends on two parameters

T

T2,

T2

T1, T1 = 2π , T2 =

2π

ω2.

The results of calculations are shown in Fig. VI. 4 and VI. 5. The Figuresshow diagrams obtained by the generalized Gaussian principle by solid curvesand those obtained by the minimization of functional (9.5) — by dashedcurves.

Thus, it is shown that in the problem of transfer of the system from onephase state to another one with one control force the generalized Gauss prin-ciple can be invoked along with the minimization of the functional of thecontrol force squared. Comparision of these two approaches when studyingthe motion of a trolley with two pendulums shows that for some values ofdimensionless parameters the solutions obtained by these two methods practi-cally coincide (see Fig. VI. 4). But there are also such ranges of dimensionlessparameters for which these solutions essentially differ from one another (seeFig. VI. 5). The presence of large-amplitude oscillations in the solution con-structed by the minimization of the functional of the control force squaredcan be explained by the fact that in this case the control is sought as asum of harmonics tending to cause a resonance of the system. In contrast,

Fig.VI. 4

9. Application of the generalized Gaussian principle 191

Fig.VI. 5

when using the generalized Gauss principle the control force is sought as apolynomial, which provides a smooth change of all functions required.

The spatial motion of a load on a cable attached to a controlled trolley isalso considered in the works by B. Simeon [434] and M. A. Chuev [252. 2008].

C h a p t e r VII

EQUATIONS OF MOTION

IN QUASICOORDINATES

In the present chapter it is shown that all known types of equations of

motion of nonholonomic systems are equivalent since they can be obtained

from the invariant vector form of the law of motion of mechanical system

with ideal constraints. The nonholonomicity of constraints, which does not

allow for the equations of motion to be represented in the form of Lagrange’s

equations of the second kind, turns out to be most clearly if the equations

of motion of nonholonomic system are written in quasicoordinates. In the

case of linear constraints these equations are generated here by three different

methods. This permits us to consider the problem of nonholonomicity from

three different points of view.

§ 1. The equivalence of different forms of equations

of motion of nonholonomic systems

Vector equation (3.10) of Chapter IV gives the law of motion of bothholonomic and nonholonomic systems such that for the ideal constraints thegeneralized accelerations qσ, σ = 1, s, satisfy system of equations (1.5) andthe vectors εεεl+κ, κ = 1, k, satisfy the conditions (1.10) of Chapter IV. Theimportant fact is that this equation has a vector form invariant under thechoice of the system of coordinates, in which the motion is described and theequations of constraints are given. Therefore in the present section we obtainfrom this equation all main types of the equations of motion of nonholonomicsystems and show thus their equivalence.

Projecting equation (3.10) of Chapter IV on a system of the vectors εεελ,λ = 1, l, which make up the basis of L–space, we obtain the following systemof scalar equations

MW · εεελ = Y · εεελ , λ = 1, l . (1.1)

Let the vectors εεελ, λ = 1, l, be the functions of the variables t, q, qσ, σ = 1, s.Then, supplementing equations (1.1) by equations (1.6) of Chapter IV, weobtain the closed system of equations such that the law of motion takes theform

W = F(t, q, q) .

V. S. Novoselov writes [169, p. 28] that the reduction of the problem to thisequation can be regarded as "the reduction of the problem of nonholonomicmechanics to the conditional problem of mechanics of holonomic systems".

A concrete form of equations (1.1) depends on both the representation ofsystem of the vectors εεελ, λ = 1, l, and the form of expansion of the scalarproducts MW · εεελ, λ = 1, l. Consider the main forms of equations (1.1).

193

194 VII. Equations of Motion in Quasicoordinates

We assume that the integralable differential constraints and the linearfirst-order nonholonomic constraints are the special case of the constraints,given by the following equations

fκ

1 (t, q, q) = 0 , κ = 1, k .

By assumption, the vectors εεεl+κ = ∇∇∇′fκ

1 , κ = 1, k, satisfy condition (1.10) ofChapter IV and therefore the equations of constraints imply that for the givenvalues of the variables t and qσ, σ = 1, s, the generalized velocities qσ, σ =1, s, can be expressed in terms of the independent variables vλ

∗, λ = 1, l. In

the works of V. S. Novoselov [169] they are called kinematical characteristics

and in the works [149, 203, 229, 247, 248], devoted to the Poincare–Chetaevequations, the Poincare parameters. The variables vλ

∗, λ = 1, l, are given by

the functionsvλ

∗= fλ

∗(t, q, q) , λ = 1, l .

Supplementing them by the relations

vl+κ

∗= f l+κ

∗(t, q, q) = fκ

1 (t, q, q) , κ = 1, k ,

we obtainqσ = qσ(t, q, v∗) , σ = 1, s . (1.2)

Suppose that at least one of the relations fσ∗

dt, σ = 1, s, is not a totaldifferential and cannot be reduced to it. In this case, as is known, the variables

πσ =

t∫

t0

vσ

∗(t) dt , σ = 1, s ,

cannot be regarded as a new system of Lagrangian coordinates. Thereforethey are called quasicoordinates and the quantities πσ = vσ

∗, σ = 1, s, qua-

sivelocities. For linear constraints the generalized velocities and quasiveloci-ties are related as

vρ

∗= aρ

σ(t, q) qσ + a

ρ

0(t, q) , qσ = bσ

τ(t, q) vτ

∗+ bσ

0 (t, q) ,

ρ, σ, τ = 1, s ,

or in short form

vα

∗= aα

βqβ , qα = bα

βvβ

∗, α, β = 0, s ,

q0 = t , q0 = v0∗

= 1 , a0β

= b0β

= δ0β

.(1.3)

Index "1"of the coefficients al+κ

α, κ = 1, k, α = 0, s, entering into the equa-

tions of constraint, is omitted for short.Applying the variables v

ρ

∗ , ρ = 1, s, we can introduce the vectors

εεερ =∂v

ρ

∗

∂qσeσ , εεετ =

∂qσ

∂vτ∗

eσ , ρ, σ, τ = 1, s ,

1. The equivalence of different forms of equations 195

such thatεεερ · εεετ = δρ

τ, ρ, τ = 1, s . (1.4)

In this case the system of the vectors εεελ, λ = 1, l, makes up the basis ofL–space since we have

εεεl+κ =∂vl+κ

∗

∂qσeσ = ∇∇∇′fκ

1 , εεεl+κ · εεελ = 0 , κ = 1, k , λ = 1, l .

The account of the equations of constraints, based on the representationof generalized velocities in the form

qσ = F σ (t, q, v1∗, . . . , vl

∗) , σ = 1, s ,

means, as V. V. Rumyantsev [203, p. 3] writes, that "there is performed theparametrization of constraints imposed on the system . . . ". In this case thebasis of L–space is known and given by the following relation

εεελ =∂F σ

∂vλ∗

eσ , λ = 1, l , σ = 1, s .

Thus, the partition of tangent space into the subspaces K and L by theequations of constraints can be made using their parametrization. In thiscase the basis, of L–space, necessary for the passage to the concrete form ofequations (1.1) is known.

If the constraints are linear, then by relations (1.3) their parametrizationcan take the form

qσ = bσ

λ(t, q) vλ

∗+ bσ

0 (t, q) , σ = 1, s , λ = 1, l . (1.5)

From formulas (1.5) we have

εεελ = bσ

λ(t, q) eσ .

Relations (3.1) and (3.2) of Chapter IV yield that the vectors MW, en-tering into equations (1.1), can be represented as

MW =d(MV)

dt, (1.6)

where

MV =∂T

∂qσeσ .

Since

eσ =∂qσ

∂vρ

∗

εεερ , ρ, σ = 1, s ,

we get

MV =∂T

∂qσ

∂qσ

∂vρ

∗

εεερ .

196 VII. Equations of Motion in Quasicoordinates

The generalized velocities qσ are regarded as the functions of all variables vρ

∗ ,ρ, σ = 1, s, and only in the final relations, taking into account the equationsof constraints, we assume that vl+κ

∗= 0, κ = 1, k. Then we obtain

MV =∂T

∂qσ

∂qσ

∂vρ

∗

εεερ =∂T ∗

∂vρ

∗

εεερ , (1.7)

where T ∗ = T ∗(t, q, v∗) is a function of variables t, qσ, vσ∗, σ = 1, s, which

is found by means of the substitution of relations (1.2) into the functionT = T (t, q, q).

Relations (1.6), (1.7) give

MW =

(d

dt

∂T ∗

∂vρ

∗

)εεερ +

∂T ∗

∂vρ

∗

εεερ .

Then

MW · εεελ =d

dt

∂T ∗

∂vλ∗

+∂T ∗

∂vρ

∗

εεερ · εεελ .

Taking into account relations (1.4), we obtain

εεερ · εεελ = −εεερ · εεελ ,

and therefore

MW · εεελ =d

dt

∂T ∗

∂vλ∗

− MV · εεελ , λ = 1, l . (1.8)

Since

εεελ =

(d

dt

∂qσ

∂vλ∗

)eσ +

∂qσ

∂vλ∗

eσ

and accordance with relations (3.3) and (3.4) of Chapter IV

eσ =∂V

∂qσ, eσ =

∂V

∂qσ,

we have

MV · εεελ =∂T

∂qσ

(d

dt

∂qσ

∂vλ∗

)+

∂T

∂qτ

∂qτ

∂vλ∗

.

Taking into account that

∂T ∗

∂qτ=

∂T

∂qτ+

∂T

∂qσ

∂qσ

∂qτ,

we obtain

MV · εεελ =∂T

∂qσ

(d

dt

∂qσ

∂vλ∗

−∂qτ

∂vλ∗

∂qσ

∂qτ

)+

∂qτ

∂vλ∗

∂T ∗

∂qτ, λ = 1, l . (1.9)

1. The equivalence of different forms of equations 197

From relations (1.8) and (1.9) it follows that equations (1.1) take the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ−

∂T

∂qσT σ

λ= Qλ , λ = 1, l . (1.10)

Here the following quantities

Qλ = Qσ

∂qσ

∂vλ∗

, λ = 1, l , σ = 1, s ,

are generalized forces, corresponding to the Poincare parameters (quasiveloc-ities) vλ

∗, λ = 1, l, and

T σ

λ=

d

dt

∂qσ

∂vλ∗

−∂qσ

∂πλ.

Here and in equations (1.10) we use the notation

∂

∂πλ=

∂qτ

∂vλ∗

∂

∂qτ.

Taking into account that

∂T ∗

∂vρ

∗

∂vρ

∗

∂qσ=

∂T

∂qσ,

equations (1.10) take the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+

∂T ∗

∂vρ

∗

Wρ

λ= Qλ , λ = 1, l ρ = 1, s . (1.11)

Here

Wρ

λ= −

∂vρ

∗

∂qσT σ

λ, λ = 1, l , ρ, σ = 1, s .

Equations (1.10) and (1.11), as follows from their generation, can be usedfor holonomic and nonholonomic systems with the linear and nonlinear invelocities ideal constraints. In the case when a time enters, in explicit form,into neither kinetic energy, nor the equations of constraints, equations (1.10)and (1.11) were obtained by G. Hamel in 1938 [314] and in the general case byV. S. Novoselov [169] in 1957. Therefore in § 2 of Chapter II equations (1.11)are called the Hamel–Novoselov equations. For their generation in the works[169] there are used general equation of mechanics (2.7) of Chapter IV andthe definition of variations of coordinates

δqσ =∂qσ

∂vρ

∗

δπρ , ρ, σ = 1, s , (1.12)

based on the analysis of the postulate of N. G. Chetaev (2.6) of Chapter IV.V. S. Novoselov calls equations (1.10) the equations of the type of S. A.

Chaplygin since under the assumptions introduced by S. A. Chaplygin, they

198 VII. Equations of Motion in Quasicoordinates

result in the Chaplygin’s equations. For the same reason equations (1.11) arecalled by V. S. Novoselov the equations of the type of Voronets–Hamel andthe coefficients W

ρ

λare called the Voronets–Hamel coefficients of the first

kind. It is shown that these coefficients can be transformed into the relations

Wρ

λ=

∂qσ

∂vλ∗

(d

dt

∂vρ

∗

∂qσ−

∂vρ

∗

∂qσ

), λ = 1, l , ρ, σ = 1, s . (1.13)

It follows that the quasivelocity is a true velocity in the case when the appli-cation of the Lagrange operator to the function v

ρ

∗(t, q, q) gives a zero.In the case of linear uniform stationary constraints from relations (1.3)

and (1.13) we conclude that the coefficients Wρ

λ, λ = 1, l, ρ = 1, s, take the

form

Wρ

λ= c

ρ

λµvµ

∗, c

ρ

λµ=

(∂aρ

σ

∂qτ−

∂aρτ

∂qσ

)bσ

λbτ

µ,

λ, µ = 1, l , ρ, σ, τ = 1, s .

(1.14)

Then equations (1.11) are the following

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+ c

ρ

λµvµ

∗

∂T ∗

∂vρ

∗

= Qλ ,

λ, µ = 1, l , ρ = 1, s .(1.15)

For l = s these equations and the relations for the coefficients cρστ

, ρ, σ, τ =1, s, as V. S. Novoselov [169, p. 55] remarks, "are obtained first by P. V.Voronets in 1901 [41] and then by G. Hamel [313] in 1904". Further V. S.Novoselov writes: "It should be remarked that before the work of Voronetswas published, in 1901 in "Comptes rendus"the note of Poincare [373] wasprinted, where he obtains the equations highly close to equation"(1.15). ThePoincare equations correspond to the case when in equations (1.15) for l = s

the coefficients cρστ

, ρ, σ, τ = 1, s, are constant and the forces are expressedvia the forcing function U :

Qτ = bσ

τ

∂U

∂qσ, σ, τ = 1, s .

Thus, equations (1.15) can be represented in the form proposed byPoincare [149]:

d

dt

∂L∗

∂vτ∗

= cρ

στvσ

∗

∂L∗

∂vρ

∗

+ Xτ L∗ , ρ, σ, τ = 1, s . (1.16)

Here L∗ (q, v∗) = T ∗ + U is the Lagrange function and the quantities

Xτ = bσ

τ

∂

∂qσ, σ, τ = 1, s , (1.17)

1. The equivalence of different forms of equations 199

are linear differential operators. They, as L. M. Markhashov [149, p. 43] writes,". . . make up the basis of a certain s-dimensional Lie algebra . . . "with thecommutator

[Xσ, Xτ ] = Xσ Xτ − Xτ Xσ = cρ

στXρ , ρ, σ, τ = 1, s . (1.18)

The coefficients cρστ

, ρ, σ, τ = 1, s, which appeared in the commutator, arecalled the structural constants of Lie algebra. In the same work on the nextpage L. M. Markhashov remarks: "the arbitrary chosen system, of s operatorsacting in s-dimensional space, for which only the condition det [bσ

τ(q)] = 0

is valid, does not make up the Lie algebra . . . "since in this case in relations(1.18) the coefficients cρ

στare the functions of qσ, σ = 1, s.

Using a tangent space and the vectors εεερ, εεετ , ρ, τ = 1, s, in it, we canrepresent relations (1.17) and (1.18), respectively, as

Xτ = εεετ · ∇∇∇ ,[Xσ, Xτ

]= εεεσ · ∇∇∇ (εεετ · ∇∇∇) − εεετ · ∇∇∇ (εεεσ · ∇∇∇) = cρ

στεεερ · ∇∇∇ = cρ

στXρ .

From this representation of the operators Xτ and their commutator it followsthat they make up the closed system of operators [203] even if the coefficientscρστ

are variable.Consider the contravariant components of the vector δy relative to the

basis εεετ, denoting them by δ′vρ

∗ , i. e. assuming that

δ′vρ

∗= δy · εεερ .

In this case we have

δy = δ′vτ

∗εεετ = δ′vτ

∗bσ

τeσ = δqσ eσ ,

and thereforeδqσ = bσ

τδ′vτ

∗, σ, τ = 1, s .

Comparing these relations with relations (1.12), one can observe that in theworks of V. S. Novoselov the quantities δ′vτ

∗are denoted by δπτ while in the

works, devoted to the Poincare–Chetaev equations (see, for example, [203]),we have

δ′vτ

∗= ωτ , τ = 1, s .

Let r (t, q) be a radius-vector of arbitrary point of mechanical system.Then we have

δr =∂r

∂qσδqσ = bσ

τ

∂r

∂qσδ′vτ

∗= δ′vτ

∗Xτr . (1.19)

Thus, using the operators Xτ , we can represent the virtual displacements δr,entering into the general equation of mechanics, in the form (1.19). Poincaremade extraordinary discovery. He states that there exist mechanical systemssuch that their tangential space has a remarkable property. The introduced

200 VII. Equations of Motion in Quasicoordinates

in this space basis εεετ = bστeσ, corresponding to quasivelocities, is given by

the functions bστ

of generalized coordinates, for which in commutator (1.18)the coefficients cρ

στare constant. As is remarked earlier, in this case the

operators Xτ make up the basis of the Lie algebra. A distinctive example ofmechanical system with such remarkable property of tangent space is a rigidbody, which goes round a fixed point. In this case the Poincare parametersare, in particular, the projections of the vector of instantaneous angle velocityon the principal axes of inertia of body and Poincare equations (1.16) becomethe dynamic Euler equations (see, for example, [203]).

Consider now the case when linear transformations (1.3) are nonuniformand nonstationary. In this case from relations (1.13) and (1.14) it follows thatequations (1.11) under both potential and nonpotential forces have the form

d

dt

∂L∗

∂vλ∗

−∂L∗

∂πλ= c

ρ

µλvµ

∗

∂L∗

∂vρ

∗

+ cρ

0λ

∂L∗

∂vρ

∗

+ Qλ ,

λ, µ = 1, l , ρ, σ = 1, s .

(1.20)

Here

cρ

ατ= aρ

σ

(bβ

α

∂bστ

∂qβ− bβ

τ

∂bσα

∂qβ

)=

(∂aρ

γ

∂qβ−

∂aρ

β

∂qγ

)bγ

αbβ

τ,

α, β, γ = 0, s , ρ, σ, τ = 1, s .

(1.21)

Two different representations of the coefficients cρατ

result from that

aρ

γbγ

β= δ

ρ

β.

Equations (1.20) are called the equations of nonholonomic systems in the

Poincare–Chetaev variables [149, 203, 229] or the equations of motion of non-

holonomic systems in quasicoordinates [28, 166].N. G. Chetaev generalized Poincare equations (1.16) to the case when the

number of Lagrangian coordinates exceeded the number of the independentPoincare parameters, i. e., making use of the Poincare approach, he obtainedequations (1.15) provided that the coefficients cl+κ

λµ= 0, κ = 1, k, and the

coefficients cν

λµ, λ, µ, ν = 1, l, were constant. He remarked, however, that the

equations found make a sense also for the variable coefficients cν

λµ, λ, µ, ν =

1, l [248]. This generalization of the Poincare equations can be found also inthe works of L. M. Markhashov, V. V. Rumyantsev, and Fam Guen [149, 203,229].

Finally, we consider two simplest forms of expansion of scalar products inequations (1.1), proposed by Appell and Maggi.

Introducing the Appell function

T1 =MW2

2,

we have

MW = MWσeσ =∂T1

∂qσeσ =

∂T ∗

1

∂vρ

∗

εεερ .

2. The Poincare–Chetaev–Rumyantsev approach 201

Then, using equations (1.1), we obtain Appell’s equations

∂T ∗

1

∂vσ∗

= Qλ , λ = 1, l .

Maggi’s equations(

d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vλ∗

= 0 , λ = 1, l ,

are obtained from equations (1.1) if we take into account (1.4) and the fol-lowing relation

εεελ =∂qσ

∂vλ∗

eσ , λ = 1, l .

The connection of Maggi’s equations with the Poincare–Chetaev ones isconsidered in the work of L. M. Markhashov [149]. In this work he writes(p. 46): "Poincare equations are obtained almost together with the mainforms of equations of motion of nonholonomic systems. In spite of a greatlikeness for a long time the both theories were developed independently.The generalized Poincare–Chetaev equations proper for both holonomic andnonholonomic systems are obtained in the work . . . " [229]. In the work ofV. V. Rumyantsev [203] the application of the Poincare–Chetaev equationsto nonholonomic dynamics is considered from the new point of view. Weespecially stress that in the work [203] V. V. Rumyantsev extends first the ap-proach of Poincare–Chetaev to nonlinear nonholonomic constraints and there-fore equations (1.20) should be called the equations of Poincare–Chetaev–

Rumyantsev. Recall that these equations and equations (1.10) and (1.11) in§ 3 of Chapter II are obtained from Maggi’s equations.

§ 2. The Poincare–Chetaev–Rumyantsev approach

to the generation of equations of motion

of nonholonomic systems

In the previous section the Poincare–Chetaev–Rumyantsev equations(1.20) are obtained on the base of the vector form of the law of motion ofmechanical systems with ideal constraints. Thus, there was given their geo-metric interpretation. However, the particular approach, used by the authorsfor generating these equations, was not brought to light. This approach de-serves additional attention since it permits us to explain from the new pointof view the reason why the equations of motion of nonholonomic systemscannot be represented in the form of Lagrange’s equations of the second kindwithout multipliers. Consider briefly this approach.

Let on the motion of mechanical system the linear nonholonomic con-straints be imposed which are given by the equations

fκ

1 ≡ al+κ

σ(t, q) qσ + al+κ

0 (t, q) = 0 ,

κ = 1, k , σ = 1, s , l = s − k .(2.1)

202 VII. Equations of Motion in Quasicoordinates

Suppose, equations (2.1) are such that, using them for the introduction ofthe quasivelocities v

ρ

∗ , ρ = 1, s, by formulas

vλ

∗= aλ

σ(t, q) qσ + aλ

0 (t, q) , λ = 1, l ,

vl+κ

∗= al+κ

σ(t, q) qσ + al+κ

0 (t, q) , κ = 1, k , σ = 1, s ,

we obtainqσ = bσ

τ(t, q) vτ

∗+ bσ

0 (t, q) , σ, τ = 1, s , (2.2)

or in compact form

vα

∗= aα

β(t, q) qβ , qβ = bβ

α(t, q) vα

∗,

q0 = t , v0∗

= q0 = 1 , a0β

= b0β

= δ0β

, α, β = 0, s .(2.3)

Here δ0β

are the Kronecker symbols.To generate the equations of motion of nonholonomic system we apply

the generalized D’Alembert–Lagrange principle

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δqσ = 0 , σ = 1, s ,

in which the quantities δqσ under constraints (2.1) must satisfy the conditionsof N. G. Chetaev

al+κ

σδqσ = 0 , κ = 1, k , σ = 1, s .

Making use of the through numeration µ = 1, 2, 3, ... for the notationsof the Cartesian coordinates of the points of system and the projections ofactive forces applied to these points, we have

mµ xµ

∂xµ

∂qσ=

d

dt

∂T

∂qσ−

∂T

∂qσ, Qσ = Xµ

∂xµ

∂qσ, σ = 1, s .

This implies that the generalized D’Alembert–Lagrange principle can be rep-resented as

(mµ xµ − Xµ)∂xµ

∂qσδqσ = 0 . (2.4)

We remark that if in the system there exist the rigid and elastic bodies, thenthe summing over all µ goes to integration.

The quantities vρ

∗ , ρ = 1, s, introduced by formulas (2.3), are called thePoincare–Chetaev parameters. These parameters in differential form are in-troduced in the following way

δ′vρ

∗= aρ

σδqσ , δqσ = bσ

ρδ′vρ

∗, ρ, σ = 1, s .

In this case from equations of constraint (2.1) we have

δ′vl+κ

∗= al+κ

σδqσ = 0 , κ = 1, k ,

2. The Poincare–Chetaev–Rumyantsev approach 203

which are the conditions of N. G. Chetaev.These conditions imply that

δqσ = bσ

λδ′vλ

∗, σ = 1, s , λ = 1, l .

Substituting the above relations into equation (2.4) and taking into accountthat the quantities δ′vλ

∗,λ = 1, l, are arbitrary, we obtain

(mµ xµ − Xµ)∂xµ

∂qσbσ

λ= 0 , λ = 1, l .

In the case when there exist both potential and nonpotential forces theseequations take the form

(mµ xµ − Xµ −

∂U

∂xµ

)∂xµ

∂qσbσ

λ= 0 , λ = 1, l . (2.5)

Here U is a forcing function.Introduce the following notations

∂

∂πλ= Xλ = bσ

λ

∂

∂qσ, Qλ = Xµ

∂xµ

∂qσbσ

λ, σ = 1, s , λ = 1, l ,

and represent equations (2.5) as

mµ xµ

∂xµ

∂πλ=

∂U

∂πλ+ Qλ , λ = 1, l . (2.6)

Compute the time derivative of the function f(t, q) according to formulas(2.2), (2.3). We obtain

df

dt=

∂f

∂t+

∂f

∂qσqσ = vα

∗Xαf = vα

∗

∂f

∂πα,

α = 0, s , π0 = q0 = t , σ = 1, s ,

(2.7)

where∂

∂πα= Xα = bβ

α

∂

∂qβ, α, β = 0, s . (2.8)

We remark that

∂

∂πτ= Xτ = bσ

τ

∂

∂qσ, σ, τ = 1, s , (2.9)

since b0τ

= 0, τ = 1, s.Relations (2.7) yield, in particular, that

xµ = vα

∗Xαxµ = vα

∗

∂xµ

∂πα, α = 0, s , (2.10)

204 VII. Equations of Motion in Quasicoordinates

and therefore∂xµ

∂vρ

∗

=∂xµ

∂πρ= Xρxµ , ρ = 1, s . (2.11)

Substituting the velocities xµ, expressed via the Poincare–Chetaev param-eters, into the relation for the kinetic energy of system mµx2

µ/2, we obtain

the function T ∗ of variables t, qσ, vσ∗, σ = 1, s. This function is such that

∂T ∗

∂vρ

∗

= mµxµ

∂xµ

∂vρ

∗

= mµxµ

∂xµ

∂πρ, ρ = 1, s , (2.12)

∂T ∗

∂πλ= mµxµ

∂xµ

∂πλ, λ = 1, l . (2.13)

Taking into account relations (2.12), the left-hand side of equations (2.6) hasthe form

mµxµ

∂xµ

∂πλ=

d

dt

(mµxµ

∂xµ

∂πλ

)− mµxµ

d

dt

∂xµ

∂πλ=

=d

dt

∂T ∗

∂vλ∗

− mµxµ

d

dt

∂xµ

∂πλ.

(2.14)

Below we shall show that

d

dt

∂xµ

∂πτ=

∂xµ

∂πτ+ cρ

ατvα

∗

∂xµ

∂vρ

∗

, ρ, τ = 1, s , α = 0, s . (2.15)

Here cρατ

are certain unknown functions of the variables t and qσ, σ = 1, s.Relations (2.13)–(2.15) imply that, finally, equations (2.6) are the following

d

dt

∂L∗

∂vλ∗

−∂L∗

∂πλ= c

ρ

µλvµ

∗

∂L∗

∂vρ

∗

+ cρ

0λ

∂L∗

∂vρ

∗

+ Qλ ,

L∗ = T ∗ + U , λ, µ = 1, l , ρ, σ = 1, s .

(2.16)

We shall show that relations (2.15) are valid and find the entering intothem coefficients cρ

ατ, α = 0, s, ρ, τ = 1, s. From relations (2.7), (2.10), and

(2.11) we conclude that relations (2.15) are valid if

∂2xµ

∂πα ∂πτ=

∂2xµ

∂πτ ∂πα+ cρ

ατ

∂xµ

∂πρ,

i. e. in the case when[Xα, Xτ

]xµ = XαXτ xµ − XτXα xµ = cρ

ατXρ xµ ,

α = 0, s , ρ, τ = 1, s .(2.17)

By formulas (2.8) and (2.9), we obtain

[Xα, Xτ

]xµ = bβ

α

∂

∂qβbσ

τ

∂xµ

∂qσ− bσ

τ

∂

∂qσbβ

α

∂xµ

∂qβ.

2. The Poincare–Chetaev–Rumyantsev approach 205

Since

bβ

αbσ

τ

∂2xµ

∂qβ ∂qσ= bσ

τbβ

α

∂2xµ

∂qσ ∂qβ,

b0τ

= 0 ,∂b0

α

∂qσ=

∂δ0α

∂qσ= 0 , α, β = 0, s , σ, τ = 1, s ,

we have[Xα, Xτ

]xµ =

(bβ

α

∂bστ

∂qβ− bβ

τ

∂bσα

∂qβ

)∂xµ

∂qσ. (2.18)

Represent the coefficients of ∂xµ/∂qσ in relation (2.18) as

bβ

α

∂bστ

∂qβ− bβ

τ

∂bσα

∂qβ= cρ

ατbσ

ρ. (2.19)

Then from relations (2.18) and (2.9) we obtain that relations (2.17) and,therefore, relations (2.15) are valid.

The coefficients aρσ

are the elements of a matrix inverse to the matrix withthe elements bσ

ρ, ρ, σ = 1, s. Therefore from (2.19) we have

cρ

ατ= aρ

σ

(bβ

α

∂bστ

∂qβ− bβ

τ

∂bσα

∂qβ

), ρ, σ, τ = 1, s , α, β = 0, s . (2.20)

By relations (2.3), we obtain

aρ

σbσ

τ= δρ

τ, aρ

γbγ

α= δρ

α, ρ, σ, τ = 1, s , α, γ = 0, s .

Then

aρ

σ

∂bστ

∂qγ= −bσ

τ

∂aρσ

∂qγ, aρ

γ

∂bγα

∂qβ= −bγ

α

∂aργ

∂qβ,

ρ, σ, τ = 1, s , α, γ = 0, s .

Taking into account that

∂b0α

∂qβ= b0

τ= 0 , α, β = 0, s , τ = 1, s ,

we have

aρ

σ

∂bστ

∂qγ= −bβ

τ

∂aρ

β

∂qγ, aρ

σ

∂bσα

∂qβ= −bγ

α

∂aργ

∂qβ,

ρ, σ, τ = 1, s , α, β, γ = 0, s .

(2.21)

Replacing in formulas (2.20) in the first double sum over all σ and β thedummy index β by γ and applying then relations (2.21), we obtain

cρ

ατ=

(∂aρ

γ

∂qβ−

∂aρ

β

∂qγ

)bγ

αbβ

τ, ρ, τ = 1, s , α, β, γ = 0, s . (2.22)

206 VII. Equations of Motion in Quasicoordinates

Comparing equations (1.20) with equations (2.16) and relations (1.21)with relations (2.20) and (2.22), we conclude that equations (2.16) coincidewith the Poincare–Chetaev–Rumyantsev equations from the previous section.In addition, they are obtained here with the usage of the technique suggestedby these authors.

The basic formulas for generating equations (2.16) are expressions (2.15),connected directly with commutator (2.17) introduced by Poincare. As wasshown by Lagrange, in the case when the quantities πτ are the true coordi-nates qτ

∗, τ = 1, s, the following relations

d

dt

∂xµ

∂qτ∗

=∂xµ

∂qτ∗

, τ = 1, s ,

are satisfied. In the case of quasicoordinates these Lagrange identities areviolated and it appears a correction, which is accounted by means of thecoefficients cρ

ατ. For their computation Yu. I. Neimark and N. A. Fufaev apply

the so-called permutable relations [163, 166].We assume, following the works of V. V. Dobronravov, V. S. Novoselov and

Yu. I. Neimark, N. A. Fufaev, that

δ′vρ

∗= δπρ , δ′qσ = δqσ , ρ, σ = 1, s .

Then, using relations (2.3), represented in differential form, we obtain

δqσ = bσ

ρδπρ , δπρ = aρ

σδqσ , ρ, σ = 1, s .

By definition, we have

π0 = q0 = t , δπ0 = δq0 = δt = 0 , dπ0 = dq0 = dt , dπρ = aρ

γdqγ ,

ρ = 1, s , γ = 0, s ,

and

δ dπρ =∂aρ

γ

∂qβδqβ dqγ + aρ

γδ dqγ , d δπρ =

∂aρ

β

∂qγdqγ δqβ + a

ρ

βd δqβ ,

ρ = 1, s , β, γ = 0, s .

Consider the difference

δ dπρ − d δπρ , ρ = 1, s ,

and substitute into it the quantities δqβ and dqγ , given in the form

δqβ = bβ

τδπτ , dqγ = bγ

αdπα , τ = 1, s , α, β, γ = 0, s .

Then we obtain the following relations

δ dπρ − d δπρ = cρ

ατdπα δπτ + aρ

γδ dqγ − a

ρ

βd δqβ ,

ρ, τ = 1, s , α, γ = 0, s .(2.23)

3. The approach of J. Papastavridis 207

Here the quantities cρατ

are given by formulas (2.22). Note that Yu. I. Neimarkand N. A. Fufaev believe that the procedure of computation of the coefficientscρατ

by means of the generation of permutable relations (2.23) is more simplethan their direct computation by formulas (2.20) or (2.22).

§ 3. The approach of J. Papastavridis

to the generation of equations

of motion of nonholonomic systems

At present, J. Papastavridis is one of the leading specialists in the fieldof nonholonomic mechanics. In his works [370] a new original approach togenerating the equations of motion of nonholonomic systems is proposed.Consider briefly this approach, assuming, for the sake of generality, that allor certain equations of constraints

ϕκ(t, q, q) = 0 , κ = 1, k ,

depend nonlinearly on velocities. We introduce quasivelocities by formulas

vλ

∗= ϕλ

∗(t, q, q) , λ = 1, l , l = s − k ,

vl+κ

∗= ϕl+κ

∗(t, q, q) = ϕκ(t, q, q) , κ = 1, k .

Assuming that

det

[∂v

ρ

∗

∂qσ

]= 0 , ρ, σ = 1, s ,

we haveqσ = qσ(t, q, v∗) , σ = 1, s .

The reasoning of J. Papastavridis, as that of the other scholars, is based onthe D’Alembert–Lagrange principle (2.4) and the conditions of N. G. Chetaev

∂ϕκ

∂qσδqσ = 0 , σ = 1, s , σ = 1, s .

In accordance with the conditions of N. G. Chetaev the relation between qua-sivelocities and the generalized velocities, represented in differential form, isas follows

δ′vρ

∗=

∂ϕρ

∗

∂qσδqσ , δqσ =

∂qσ

∂vρ

∗

δ′vρ

∗, ρ, σ = 1, s .

In this case from the equations of constraints we have

δqσ =∂qσ

∂vλ∗

δ′vλ

∗, λ = 1, l , σ = 1, s .

208 VII. Equations of Motion in Quasicoordinates

Substituting these relations into the generalized D’Alembert–Lagrange prin-ciple in the form (2.4) and taking into account that the quantities δ′vλ

∗,

λ = 1, l, are arbitrary, we obtain

(mµxµ − Xµ

) ∂xµ

∂qσ

∂qσ

∂vλ∗

= 0 , λ = 1, l , σ = 1, s . (3.1)

In these equations the quantities x3ν−2, x3ν−1, x3ν are the Cartesian co-ordinates of the point, the position of which is given by the radius-vector

rν = x3ν−2 i1 + x3ν−1 i2 + x3ν i3 .

This point has the mass mν = mµ, µ = 3ν − 2, 3ν − 1, 3ν , and the activeforce, applied to it, is as follows

Fν = X3ν−2 i1 + X3ν−1 i2 + X3ν i3 .

Taking into account the above relation, equations (3.1) become

(mν rν − Fν

)·∂rν

∂qσ

∂qσ

∂vλ∗

= 0 , λ = 1, l , σ = 1, s .

Replacing the summing over ν by the integration, we obtain

∫ (r dm − dF

)·

∂r

∂qσ

∂qσ

∂vλ∗

= 0 , λ = 1, l , σ = 1, s . (3.2)

Here r = r(t, q) is a radius-vector of elementary mass dm, acted by theactive force dF. We remark that in equations (3.2) the notations follow tothe surveys of J. Papastavridis [370. 1998]. Following this work, we introducethe vectors

eσ =∂r

∂qσ, εεετ =

∂qσ

∂vλ∗

eσ , σ, τ = 1, s , (3.3)

which belong to not tangential space but to the usual Euclidean space, inwhich the motion of the mechanical system is considered.

Introduce the notation

∂

∂πτ=

∂qσ

∂vτ∗

∂

∂qσ, σ, τ = 1, s ,

and represent the vectors εεετ as

εεετ =∂r

∂πτ, τ = 1, s .

Then equations (3.2) take the form

∫r ·

∂r

∂πλdm = Qλ , λ = 1, l , (3.4)

3. The approach of J. Papastavridis 209

where

Qλ =

∫∂r

∂πλ· dF = Qσ

∂qσ

∂vλ∗

, Qσ =

∫∂r

∂qσ· dF , λ = 1, l , σ = 1, s .

The functions under the integral in equations (3.4) can be represented inthe following way

r ·∂r

∂πλ=

d

dt

(r ·

∂r

∂πλ

)− r ·

d

dt

∂r

∂πλ, λ = 1, l . (3.5)

Taking into account that

r =∂r

∂qαqα = qα eα , q0 = t , e0 =

∂r

∂t, α = 0, s ,

we obtain

eσ =∂r

∂qσ, ˙eσ =

∂r

∂qσ= qα

∂eα

∂qσ,

∂r

∂πλ= qα

∂eα

∂qσ

∂qσ

∂vλ∗

+∂qσ

∂qρ

∂qρ

∂vλ∗

eσ =∂qσ

∂vλ∗

˙eσ +∂qσ

∂πλeσ ,

α = 0, s , σ = 1, s , λ = 1, l .

On the other hand, we have

d

dt

∂r

∂πλ=

d

dt

(∂qσ

∂vλ∗

eσ

)=

∂qσ

∂vλ∗

˙eσ +

(d

dt

∂qσ

∂vλ∗

)eσ , λ = 1, l , σ = 1, s .

Therefore

d

dt

∂r

∂πλ=

∂r

∂πλ+ T σ

λ

∂r

∂qσ, λ = 1, l , σ = 1, s ,

where

T σ

λ=

d

dt

∂qσ

∂vλ∗

−∂qσ

∂πλ. (3.6)

Then, from the relations

∂r

∂vλ∗

=∂qσ

∂vλ∗

∂r

∂qσ=

∂r

∂πλ, λ = 1, l ,

it follows that relations (3.5) have the form

r ·∂r

∂πλ=

d

dt

∂(r2/2)

∂vλ∗

−∂(r2/2)

∂πλ− T σ

λ

∂(r2/2)

∂qσ,

λ = 1, l , σ = 1, s .

(3.7)

Substituting these relations into equations (3.4), we obtain

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ−

∂T

∂qσT σ

λ= Qλ , λ = 1, l , σ = 1, s . (3.8)

210 VII. Equations of Motion in Quasicoordinates

Here T ∗ is a kinetic energy of system expressed in terms of quasivelocities.Taking into account relations (3.3), we represent the sum

T σ

λ

∂r

∂qσ= T σ

λeσ

in the following way

T σ

λeσ = T σ

λ

∂vρ

∗

∂qσεεερ .

Since∂r

∂vρ

∗

=∂qσ

∂vρ

∗

∂r

∂qσ=

∂qσ

∂vρ

∗

eσ = εεερ ,

we obtain

T σ

λeσ = −W

ρ

λ

∂r

∂vρ

∗

.

Here

Wρ

λ= −

∂vρ

∗

∂qσT σ

λ. (3.9)

Finally, relations (3.7) are the following:

r ·∂r

∂πλ=

d

dt

∂(r2/2)

∂vλ∗

−∂(r2/2)

∂πλ+ W

ρ

λ

∂(r2/2)

∂vρ

∗

, λ = 1, l , ρ = 1, s ,

and equations (3.4) take the form

d

dt

∂T ∗

∂vλ∗

−∂T ∗

∂πλ+

∂T ∗

∂vρ

∗

Wρ

λ= Qλ , λ = 1, l , ρ = 1, s . (3.10)

We shall show that relations (3.9) can be represented as

Wρ

λ=

∂qσ

∂vλ∗

(d

dt

∂vρ

∗

∂qσ−

∂vρ

∗

∂qσ

). (3.11)

Formulas (3.9) and (3.6) yield the relation

Wρ

λ= −

∂vρ

∗

∂qσ

(d

dt

∂qσ

∂vλ∗

−∂qσ

∂πλ

). (3.12)

Since∂v

ρ

∗

∂qσ

∂qσ

∂vλ∗

= δρ

λ,

we obtain (d

dt

∂vρ

∗

∂qσ

)∂qσ

∂vλ∗

= −∂v

ρ

∗

∂qσ

d

dt

∂qσ

∂vλ∗

. (3.13)

Since the function vρ

∗(t, q, q(t, q, v∗)) is identically equal to vρ

∗ , we find

∂vρ

∗

∂qσ+

∂vρ

∗

∂qτ

∂qτ

∂qσ= 0 , ρ, σ, τ = 1, s ,

3. The approach of J. Papastavridis 211

and therefore

∂vρ

∗

∂qσ

∂qσ

∂πλ=

∂vρ

∗

∂qσ

∂qσ

∂qτ

∂qτ

∂vλ∗

=∂v

ρ

∗

∂qτ

∂qτ

∂qσ

∂qσ

∂vλ∗

= −∂v

ρ

∗

∂qσ

∂qσ

∂vλ∗

. (3.14)

From relations (3.12)–(3.14) it follows that, in fact, the coefficients Wρ

λcan

be represented in the form (3.11).Equations (3.8) and (3.10) coincide with equations (1.10) and (1.11), re-

spectively. Recall that in the case when a time does not enter, in explicitform, into both the kinetic energy and the equations of constraints equa-tions (1.10) and (1.11) have been obtained by G. Hamel [314] in 1938 and inthe general case by V. S. Novoselov [169] in 1957. In 1998 V. V. Rumyantsev[203] obtained these equations, having generalized the Poincare and Chetaevequations. He states [203, p. 57] that these equations ". . . can be regarded asthe general equations of the classical mechanics, involving as special cases allknown equations of motion".

Equations (1.10) and (1.11) pass to one another and they are representedin the first and second forms in quasicoordinates. Therefore they can be calledequations of motion in quasicoordinates. For linear nonholonomic constraintsand for potential and nonpotential forces, these equations, as was shownabove, pass to the Poincare–Chetaev–Rumyantsev equations (1.20). Sinceequations (1.10), (1.11) coincide with equations (3.8), (3.10), respectively, andequations (1.20) coincide with equations (2.16), we can say that in the presentchapter the Poincare–Chetaev–Rumyantsev equations (1.20) were obtainedby three different methods. Their generation is based on the vector form of thelaw of motion with ideal constraints in § 1 and on the D’Alembert–Lagrangeprinciple and the conditions of N. G. Chetaev in § § 2 and 3. In addition, in§ 2 the technique of Poincare–Chetaev–Rumyantsev was used and in § 3 thetechnique of Papastavridis.

A P P E N D I C E S

A P P E N D I X A

THE METHOD OF CURVILINEAR COORDINATES

In Appendix A the kinematics of point in curvilinear coordinates is consid-

ered. The formulas obtained are extended to the motion of any of mechanical

systems. The theory, given in the Appendix, is widely used in studying the

base material of the monograph.

§ 1. The curvilinear coordinates of point.

Reciprocal bases

Suppose, the position of the point M in three-dimensional space is definedby the radius-vector r = r(q1, q2, q3), i. e. the Cartesian coordinates of pointx1, x2, x3 are uniquely represented via the quantities q1, q2, q3:

xk = Fk(q1, q2, q3) , k = 1, 2, 3 . (A.1)

If

D(x1, x2, x3)

D(q1, q2, q3)=

∣∣∣∣∣∣∣∣∣∣∣

∂x1

∂q1

∂x1

∂q2

∂x1

∂q3

∂x2

∂q1

∂x2

∂q2

∂x2

∂q3

∂x3

∂q1

∂x3

∂q2

∂x3

∂q3

∣∣∣∣∣∣∣∣∣∣∣

= 0 ,

then system of equations (A.1) is solvable for q1, q2, q3:

qσ = fσ(x1, x2, x3) , σ = 1, 2, 3 , (A.2)

and the quantities q1, q2, q3 are called the curvilinear coordinates of point inspace.

From relations (A.2) it follows directly that, equating any curvilinear co-ordinate qσ to the constant quantity Cσ, we obtain the equation of coordinate

surface

fσ(x1, x2, x3) = Cσ , σ = 1, 2, 3 .

The crossing of two coordinate surfaces gives a coordinate line, along whichone coordinate is varied only. For example, the crossing of the coordinatesurfaces q1 = f1(x1, x2, x3) = C1 and q2 = f2(x1, x2, x3) = C2 gives acoordinate line, which the coordinate q3 varies along (Fig. A. 1).

213

214 Appendix A

Fig. A. 1

The crossing of the coordinate lines q1, q2, q3 is the point M . If throughthis point we construct the tangents to the coordinate lines in ascending orderof the quantities q1, q2, q3, then we obtain the axes of curvilinear coordinates,which can make up as orthogonal (for example, the axes of spherical or cylin-drical coordinates) as nonorthogonal systems. For the motion to be given incurvilinear coordinates it is necessary that the quantities q1, q2, q3 are givenas time functions:

qσ = qσ(t) , σ = 1, 3 . (A.3)

These functions are called equations of motion of point.Taking into account that the radius-vector r = r(q1, q2, q3) of the point

M is a differentiable function, we obtain

dr =

3∑

σ=1

∂r

∂qσdqσ .

Denoting

eσ =∂r

∂qσ, σ = 1, 3 , (A.4)

we have

dr =

3∑

σ=1

eσdqσ . (A.5)

Note that |∂r/∂qσ| = |eσ| = Hσ, where Hσ are scale factors Lame. Using

formulas (A.4), we obtain

Hσ =

√(∂x1

∂qσ

)2

+

(∂x2

∂qσ

)2

+

(∂x3

∂qσ

)2

, σ = 1, 3 . (A.6)

Formula (A.5) gives a decomposition of the vector dr using the axes of thecurvilinear system of coordinates qσ

with the basis eσ. The quantities

Appendix A 215

dqσ from relations (A.5) are called contravariant components of the vectordr. A set of the vectors eσ is called a natural or fundamental basis of thecurvilinear system of coordinates qσ

at the point M .The tangential planes to the coordinate surfaces at the point M are called

coordinate planes. They pass through the corresponding vectors of basis. Forexample, the tangential plane to the surface q3 = C3 passes through thevectors e1 and e2.

Denote by eτ a certain vector collinear to a vector of normal to the coor-dinate surface qτ = Cτ at the point M . Obviously, the system of all vectorseτ

also makes up a certain basis. For the definition of that the basis eτ

is unique we need

eτ· eσ = δτ

σ=

1 , σ = τ ,

0 , σ = τ .(A.7)

Here δτσ

are the Kronecker symbols.The basis eτ

is called a reciprocal or dual basis relative to the funda-mental one. The reciprocal basis can also be introduced by using the gradientoperation (see the next section).

Note that for any fundamental basis there exists a unique reciprocal ba-sis and if the fundamental basis is orthonormal, then the reciprocal basiscoincides with the fundamental one.

§ 2. The relation between a reciprocal basis

and gradients of scalar functions

We assume that we have the certain function f(x1, x2, x3) in Cartesiancoordinates of a point and this function can be represented in the curvilinearcoordinates: f(q1, q2, q3).

The differential of this function in Cartesian coordinates is as follows

df =

3∑

k=1

∂f

∂xk

dxk (A.8)

and in curvilinear coordinates

df =

3∑

σ=1

∂f

∂qσdqσ . (A.9)

The gradient of the function f is the vector

grad f =

3∑

k=1

∂f

∂xk

ik .

216 Appendix A

If we introduce Hamiltonian operator nabla

∇∇∇ =

3∑

k=1

∂

∂xk

ik , (A.10)

then the gradient of the function f takes the form grad f = ∇∇∇f .Taking into account that dr =

∑3k=1 dxkik, relation (A.8) can be repre-

sented as the scalar product

df = ∇∇∇f · dr . (A.11)

The question arises how in place of the formulae (A.10), which is validfor Cartesian coordinates, to find a relation for the vector ∇∇∇ in curvilinearcoordinates in such a way that the derivative df can be represented in theform (A.11) with dr in the form (A.5)?

Substituting relation (A.5) into (A.11) and comparing with (A.9), weobtain the relation

∇∇∇f · eσ =∂f

∂qσ. (A.12)

It is easily checked that relation (A.12) is valid if

∇∇∇f =

3∑

τ=1

∂f

∂qτeτ . (A.13)

Representation (A.13) is convenient to obtain the vectors of reciprocalbasis. Really, using the concrete coordinate surface of the form (A.2) andtaking into account relation (A.13), we have

∇∇∇fσ = grad fσ =

3∑

τ=1

∂fσ

∂qτeτ = eσ .

§ 3. Covariant and contravariant

components of vector

To the curvilinear system of coordinates qσ, σ = 1, 2, 3, correspond boththe fundamental basis eσ = ∂r/∂qσ, σ = 1, 2, 3, and the reciprocal basiseτ = ∇∇∇fτ , τ = 1, 2, 3. Any of the vectors a can be decomposed into asfundamental as reciprocal bases, i. e. can be represented in the form

a =

3∑

σ=1

aσeσ , a =

3∑

τ=1

aτeτ . (A.14)

Here aσ are contravariant components of the vector a and aτ are covariant

components of the vector a in the basis eτ.

Further, we make use of the rule of dummy index, summation of repeatedindices in the corresponding limits is implied. Then from formulas (A.14) and(A.7) we obtain

a · eσ = aτeτ · eσ = aτδσ

τ= aσ , a · eσ = aτe

τ· eσ = aτδτ

σ= aσ .

Appendix A 217

Thus, we find the simple formulas to obtain the components of the arbitraryvector a decomposed into considered bases, respectively:

aσ = a · eσ , aσ = a · eσ . (A.15)

Relations (A.14) and (A.15) yield the rules of raising an index and missing

an index:

aσ = a · eσ = aτeτ· eσ = gτσaτ , aσ = a · eσ = aτeτ · eσ = gτσaτ . (A.16)

Here gστ = gτσ = eσ · eτ , σ, τ = 1, 2, 3, are elements of basic metric form orbasic metric tensor and gστ = gτσ = eσ

· eτ , σ, τ = 1, 2, 3, are components ofcomplementary metric form or complementary metric tensor. Applying themetric tensor, it is not difficult to obtain the transition formulas from thefundamental basis to the reciprocal one and vice versa:

eσ = (eσ · eτ )eτ = gστeτ , eσ = (eσ

· eτ )eτ = gστeτ .

Note that if two vectors, represented in the same bases, are multipliedscalarly by each other, then the obtained relation turns out rather lengthy(nine addends):

a · b = gστaσbτ = gστaσbτ .

If the vectors are decomposed into the different bases, then the scalar productinvolves only three addends:

a · b = aσbτeσ · eτ = aσbτδτ

σ= aσbσ = aσbσ .

From the chain of relations

aσ = a · eσ = |a||eσ| cos ϕ = |eσ|preσa ,

aτ = a · eτ = |a||eτ| cos ψ = |eτ

|preτ a

we obtain the formulas for computing the projections of the vector a on thevectors of fundamental and reciprocal bases:

preσa =

aσ

|eσ|

, preτ a =aτ

|eτ|

. (A.17)

§ 4. Covariant and contravariant

components of velocity vector

Let now the motion of the point M be considered in the curvilinear sys-tem of coordinates, in which case equations of motion (A.3) are known. Bydefinition, the velocity is given by the vector v = dr/dt and therefore we have

v =dr

dt=

∂r

∂qσqσ = qσeσ .

218 Appendix A

At the same time the velocity vector in fundamental basis can be representedas v = vσeσ. Then, comparing with the previous formula, we obtain thefollowing representations for the contravariant components of velocity vector:

vσ = qσ , σ = 1, 2, 3 . (A.18)

For the orthogonal fundamental basis eσ the relation for the modulus ofvelocity vector has the form

v = |v| =√

(vσeσ)2 =√

(H1q1)2 + (H2q2)2 + (H3q3)2 , (A.19)

where, as we have shown earlier, Hσ are scale factors.According to formulas (A.16) and (A.18) the covariant components of

velocity vector are the following

vσ = gστvτ = gστ qτ , σ = 1, 2, 3 . (A.20)

Consider another possible representation of the components vσ. For thispurpose it is convenient to introduce the function

T1 =v2

2, (A.21)

which can be regarded as the kinetic energy of point with unit mass, what ismarked by index "1". Function (A.21) can be rewritten as

T1 =1

2v · v =

1

2qσeσ · qτeτ =

1

2gστ qσ qτ . (A.22)

By relation (A.22), formula (A.20) can be represented now in the form

vσ =∂T1

∂qσ. (A.23)

As will be shown below, the function T1, given by formula (A.21), plays animportant role in computing the covariant components of acceleration vectorof point.

§ 5. Christoffel symbols

In § 1 we introduce the vectors of fundamental basis

eσ =∂r

∂qσ, σ = 1, 2, 3 ,

which show the changes of radius-vector of point versus the changes of gener-alized coordinates. Let us study now the effect of coordinates qτ , τ = 1, 2, 3,on the vector eσ, σ = 1, 2, 3. For this purpose we consider the derivatives

∂eσ

∂qτ, σ, τ = 1, 2, 3 . (A.24)

Appendix A 219

Since the vector can be represented in one of the forms of (A.14), where thecovariant and contravariant components are computed by formulas (A.15),for the sought vectors we obtain

∂eσ

∂qτ=

(∂eσ

∂qτ· eρ

)eρ ,

∂eσ

∂qτ=

(∂eσ

∂qτ· eρ

)eρ ,

ρ, σ, τ = 1, 2, 3 .

The covariant and contravariant components of vectors (A.24) in the aboverelation are called the Christoffel symbols of the first and second kinds andare denoted by Γρ,στ and Γρ

στ, respectively. Thus, we have

Γρ,στ =∂eσ

∂qτ· eρ, Γρ

στ=

∂eσ

∂qτ· eρ .

Finally, the previous relations take the form:

∂eσ

∂qτ= Γρ,στe

ρ ,∂eσ

∂qτ= Γρ

στeρ ,

ρ, σ, τ = 1, 2, 3 .

Obviously, by formulas (A.16) the Christoffel symbols are related as

Γρ

στ= gρπΓπ,στ , Γρ,στ = gρπΓπ

στ,

π, ρ, σ, τ = 1, 2, 3 .(A.25)

Represent the Christoffel symbols of the first kind via the elements of basicmetric tensor. Assuming that the mixed second derivatives are continuous inthe coordinates of radius-vector of point, we obtain the chain of relations

∂eσ

∂qτ=

∂2r

∂qσ∂qτ=

∂2r

∂qτ∂qσ=

∂eτ

∂qσ.

Applying this formula twice, we can perform the following transformations:

Γρ,στ =∂eσ

∂qτ· eρ =

1

2

(∂eσ

∂qτ· eρ +

∂eτ

∂qσ· eρ

)=

=1

2

(∂(eσ · eρ)

∂qτ+

∂(eτ · eρ)

∂qσ−

∂eρ

∂qτ· eσ −

∂eρ

∂qσ· eτ

)=

=1

2

(∂(eσ · eρ)

∂qτ+

∂(eτ · eρ)

∂qσ−

∂eτ

∂qρ· eσ −

∂eσ

∂qρ· eτ

).

This implies the formula for computing the Christoffel coefficients of the firstkind:

Γρ,στ =1

2

(∂gρσ

∂qτ+

∂gρτ

∂qσ−

∂gστ

∂qρ

). (A.26)

220 Appendix A

The Christoffel coefficients of the second kind can be determined, in turn, byformulas (A.25).

§ 6. Covariant and contravariant

components of acceleration vector.

The Lagrange operator

Represent the acceleration vector in the following way:

w =dv

dt=

d

dt(vρeρ) = vρeρ + vρ

deρ

dt.

We havedeρ(q)

dt=

∂eρ

∂qσqσ ,

and therefore this formula can be represented as

w = vρeρ + vρvσ∂eρ

∂qσ. (A.27)

Multiplying scalarly the above relation by the vectors eπ, we obtain thecontravariant components of acceleration vector:

wπ = qπ + Γπ

ρσqρqσ .

Multiplying scalarly (A.27) by the vectors eπ, we obtain the covariant com-ponents of acceleration vector:

wπ = gπρqρ + Γπ,ρσ qρqσ . (A.28)

Now we proceed to the obtaining of the second representation of covariantcomponents of acceleration. We write the acceleration vector as:

w =dv

dt=

d

dt(vτe

τ ) = vτeτ + vτ

deτ

dt.

However, sincedeτ (q)

dt=

∂eτ

∂qσqσ ,

this formula takes the form

w = vτeτ + vτvσ

∂eτ

∂qσ.

Multiplying scalarly this relation on the vectors eρ, we obtain

wρ = vρ + vτvσ∂eτ

∂qσ· eρ . (A.29)

Appendix A 221

Consider the last scalar product in formula (A.29). By the property of thevectors of reciprocal bases (A.7), we have

eτ· eρ = δτ

ρ= const .

Then∂(eτ

· eρ)

∂qσ= 0 .

Hence we obtain∂eτ

∂qσ· eρ = −

∂eρ

∂qσ· eτ = −Γτ

ρσ.

It follows that formula (A.29) can be rewritten as

wρ = vρ − Γτ

ρσvτvσ . (A.30)

However, by the rule of raising an index (A.16) we have

vτ = gτπvπ, Γπ,ρσ = gπτΓτ

ρσ.

Therefore formula (A.30) becomes

wρ = vρ − Γπ,ρσvπvσ . (A.31)

Relations (A.26) give

Γπ,ρσvπvσ =1

2

(∂gπρ

∂qσ+

∂gπσ

∂qρ−

∂gρσ

∂qπ

)vπvσ .

If in the right-hand side of this relation in the first double sum we interchangethe summation indices π and σ, then this sum coincides with the last doublesum, given with the minus sign. In this case, collecting terms, we obtain

Γπ,ρσvπvσ =1

2

∂gπσ

∂qρvπvσ =

∂T1

∂qρ, (A.32)

where T1 is a function, introduced by formula (A.21). Recall that accordingto (A.23) the covariant components of velocity vector are also represented bythis function. Therefore we have

vρ =d

dt

∂T1

∂qρ. (A.33)

Using relations (A.32) and (A.33), from formula (A.31) we find the fi-nal second representation of a covariant component of acceleration vector ofpoint, namely

wρ =d

dt

∂T1

∂qρ−

∂T1

∂qρ. (A.34)

222 Appendix A

Fig. A. 2

Introducing the Lagrange operator

Lρ =d

dt

∂

∂qρ−

∂

∂qρ,

we rewrite representation (A.34) as

wρ = Lρ(T1) .

The projections of acceleration on the vectors of fundamental basis can befound by formulas (A.17):

preρw =

Lρ(T1)

Hρ

.

§ 7. The case of cylindrical system of coordinates

As an example of application of the found formulas we consider the cylin-drical system of coordinates q1 = ρ, q2 = ψ, q3 = z (Fig. A. 2). The Carte-sian coordinates of point x, y, z are represented via the cylindrical coordinatesin the following way:

x = ρ cos ψ , y = ρ sin ψ , z = z . (A.35)

To the constant values of the generalized coordinates

ρ = C1 , ψ = C2 , z = C3

correspond the coordinate surfaces, passing through the point M(C1, C2, C3)(Fig. A. 2): a vertical cylinder of radius ρ; a vertical plane, which makes the

Appendix A 223

angle ψ with the plane Oxz; a horizontal plane, raised off Oxy by z. Thecrossing of these coordinate surfaces at the point M gives the coordinatelines: the horizontal straight line O1M , the vertical straight line NM , andthe circle of radius ρ centered at the point O1.

The vectors of fundamental basis, which in accordance with formulas (A.4)and (A.35) have the form

e1 = eρ =∂r

∂ρ= cos ψi + sin ψj ,

e2 = eψ =∂r

∂ψ= −ρ sin ψi + ρ cos ψj , (A.36)

e3 = ez =∂r

∂z= k ,

are directed along the tangents to the coordinate curves in ascending orderof the corresponding curvilinear coordinates.

These vectors, as is shown in Fig. A. 2, make up an orthogonal but unnor-malized system since by formulas (A.6) and (A.35) we have

Hρ = 1 , Hψ = ρ, , Hz = 1 . (A.37)

The orthogonality of fundamental basis can also be established analytically.Really, formulas (A.36) imply that

e1 · e2 = −ρ cos ψ sin ψ + ρ sin ψ cos ψ = 0 ,

e1 · e3 = 0, e2 · e3 = 0 .

The reciprocal basis eρ,eψ,ez coincides in directions with the fundamentalone and has the lengths

|eρ| = 1 , |eψ

| =1

ρ, |ez

| = 1 .

The found bases permit us to construct the matrix of basic metric tensor

(gστ ) =

1 0 00 ρ2 00 0 1

, (A.38)

and the matrix of complementary metric tensor

(gστ ) =

1 0 00 1/ρ2 00 0 1

. (A.39)

It is easily seen that the product of matrix (A.38) by matrix (A.39) is theunit matrix. For the computation of the Christoffel symbols of the first kind

224 Appendix A

we make use of formula (A.26). Since in matrix (A.38) the variable elementis g22 = gρρ = ρ2 only, then the only nonzero symbols are the following

Γ2,21 = Γ2,12 = −Γ1,22 = ρ . (A.40)

The Christoffel symbols of the second kind can be computed now by formulas(A.25), using the elements of matrix (A.39).

Formulas (A.18) and (A.19) give

v1 = vρ = ρ , v2 = vψ = ψ , v3 = vz = z ,

v = |v| =

√ρ2 + (ρψ)2 + z2 , (A.41)

therefore

T1 =1

2(ρ2 + (ρψ)2 + z2) . (A.42)

If contravariant components of velocity (A.41) are known, the covariant com-ponents can be obtained by formulas (A.20):

v1 = g1τvτ = v1 = ρ ,

v2 = g2τvτ = ρ2v2 = ρ2ψ ,

v3 = g3τvτ = v3 = z .

(A.43)

The covariant components of accelerations can be found by means of rep-resentations (A.28). Since for the cylindrical system of coordinates the nonze-ro Christoffel symbols of the first kind are the symbols, given by formulas(A.40) only, we have

w1 = g1σ qσ + Γ1,στ qσ qτ = q1 + Γ1,22q2q2 = ρ − ρψ2 ,

w2 = g2σ qσ + Γ2,στ qσ qτ = ρ2q2 + Γ2,12q1q2 + Γ2,21q

2q1 = ρ2ψ + 2ρψρ ,

w3 = g3σ qσ + Γ3,στ qσ qτ = q3 = z .

(A.44)Note that it is rather convenient to determine the covariant components

of velocity and acceleration, applying the functions T1. Really, using formu-las (A.23) with provision for (A.42) we can compute at once the covariantcomponents of velocity (A.43) obtained above:

v1 = vρ =∂T1

∂ρ= ρ ,

v2 = vψ =∂T1

∂ψ= ρ2ψ ,

v3 = vz =∂T1

∂z= z ,

Appendix A 225

and by formulas (A.34) the covariant components of acceleration, which co-incide with relations (A.44):

w1 = wρ =d

dt

∂T1

∂ρ−

∂T1

∂ρ=

d

dtρ − ρψ2 = ρ − ρψ2 ,

w2 = wψ =d

dt

∂T1

∂ψ−

∂T1

∂ψ=

d

dt(ρ2ψ) = ρ2ψ + 2ρψρ ,

w3 = wz =d

dt

∂T1

∂z−

∂T1

∂z= z .

The projections of velocity and acceleration are obtained by formulas (A.17),taking into account lengths (A.37) of the vectors of fundamental basis:

preρv = ρ , preψ

v = ρψ , prezv = z ,

preρw = ρ − ρψ2 , preψ

w = ρψ + 2ψρ , prezw = z .

§ 8. Covariant components of acceleration vector

for nonstationary basis

Consider now a more general case when the radius-vector r depends notonly on q = (q1, q2, q3) but on time t, i. e. it is the function of the formr = r(t, q). In particular, this is possible in the case when the curvilinear co-ordinates qσ give the position of points relative to the system of coordinatesOx1x2x3, which has a given motion relative to the stationary (absolute) sys-tem of coordinates O1ξ1ξ2ξ3. In this case even for the fixed values of qσ theradius-vector r varies in time in virtue of the translational motion of systemOx1x2x3.

The absolute velocity v is computed by formula

v = r =∂r

∂t+

∂r

∂qσqσ . (A.45)

Introducing, for short, the notation q0 = t (therefore q0 = 1), we can expressvelocity (A.45) in the following way:

v =∂r

∂qαqα , α = 0, 3 . (A.46)

We emphasize that such a representation is introduced only for short andtherefore we do not need to consider the problem in four-dimensional space.The coordinate vectors are, as before, only the vectors

eσ(t, q) =∂r

∂qσ, σ = 1, 3 .

226 Appendix A

Thus, the nonstationary basis varies not only with the change from point topoint but at each point in a time.

Compute a covariant component of the acceleration w:

wπ = w · eπ =dv

dt·

∂r

∂qπ=

d

dt

(v ·

∂r

∂qπ

)− v ·

d

dt

∂r

∂qπ. (A.47)

Differentiating first relation (A.45) with respect to qπ and then with respectto qπ (π = 1, 2, 3), we have

∂v

∂qπ=

∂r

∂qπ,

∂v

∂qπ=

∂2r

∂t∂qπ+

∂2r

∂qσ∂qπqσ =

d

dt

∂r

∂qπ.

It follows that the addends, entering into relation (A.47), take the form

v ·

∂r

∂qπ= v ·

∂v

∂qπ=

1

2

∂v2

∂qπ=

∂T1

∂qπ,

v ·

d

dt

∂r

∂qπ= v ·

∂v

∂qπ=

1

2

∂v2

∂qπ=

∂T1

∂qπ.

Finally, for wπ we obtain

wπ =d

dt

∂T1

∂qπ−

∂T1

∂qπ, T1 =

v2

2, π = 1, 3 . (A.48)

Thus, Lagrange’s form of representation of the covariant component wπ doesnot change also in the case of nonstationary basis.

According to representation (A.46) the kinetic energy T1 of a point withunit mass is as follows

T1 =v2

2=

1

2

∂r

∂qα·

∂r

∂qβqαqβ =

1

2gαβ qαqβ , α, β = 0, 3 . (A.49)

If in relation (A.49) we discriminate the addends, involving explicitly ∂r/∂q0

= ∂r/∂t, then we have

T1 = T(2)1 + T

(1)1 + T

(0)1 ,

T(2)1 =

1

2

∂r

∂qρ·

∂r

∂qσqρqσ =

1

2gρσ qρqσ ,

T(1)1 =

∂r

∂t·

∂r

∂qσqσ = g0σ qσ ,

T(0)1 =

1

2

(∂r

∂t

)=

1

2g00 .

(A.50)

Note that in formulas (A.50) the metric coefficients are the quantities gρσ,

ρ, σ = 1, 3, entering into the relation T(2)1 only.

Appendix A 227

By formulas (A.48), (A.49) the covariant components of acceleration vec-tor in expanded form are the following

wπ = gπρqρ + Γπ,αβ qαqβ , π, ρ = 1, 3 , α, β = 0, 3 . (A.51)

This formula is the extension of the first representation of covariant compo-nent of acceleration (A.28) to the case of nonstationary basis.

We emphasize that like the previous remark, in formula (A.51) theChristoffel symbols ourselves are only the following

Γπ,ρσ =∂eρ

∂qσ· eπ , π, ρ, σ = 1, 3 ,

and by the use of the vector e0 = ∂r/∂t, the quantities Γπ,ρ0, Γπ,00 denoteonly the functions

Γπ,ρ0 =∂eρ

∂t· eπ =

∂2r

∂qρ∂t· eπ ,

Γπ,00 =∂e0

∂t· eπ =

∂2r

∂t2· eπ ,

π, ρ = 1, 3 .

They are introduced here for brevity of notation and allow us to obtain inthe case of nonstationary basis the formulas similar to those in the stationarycase.

§ 9. Covariant components

of a derivative of vector

In Chapter IV the relations for the covariant components of derivatives ofvector are used. We obtain here the corresponding formulas for the vector a ofarbitrary physical structure. Recall that in § 6 of this Appendix they alreadyhave been obtained as a result of the differentiation of velocity vector.

Consider the representation of the vector a in reciprocal basis:

a = aτeτ .

Find the vector b, which is a derivative of the vector a:

b = a = aτeτ + aτ

deτ

dt.

Since we havedeτ (t, q)

dt=

∂eτ

∂qαqα ,

the previous formula takes the form

b = aτeτ + aτ qα

∂eτ

∂qα.

228 Appendix A

Multiplying this relation scalarly by the vectors eρ, we get

bρ = aρ + aτ qα∂eτ

∂qα· eρ .

Arguing as in § 6, we find

∂eτ

∂qα· eρ = −Γτ

ρα,

and therefore finally we have

bρ = aρ − Γτ

ραaτ qα . (A.52)

The particular case of this formula is relation (A.30).Formula (A.52) is often used in Chapter IV. Note that in Chapter IV it is

also obtained more general formulas.The formulas, found above, can be used to describe motion of represen-

tation point in the curvilinear coordinates q = (q1, ... , qs). In this case theindices π, ρ, σ, τ are varied from 1 to s = 3N and for nonstationary system,α and β from 0 to s = 3N .

In Chapter IV, using a tangent space, the formulas of this Appendix areextended to mechanical systems, consisting of not only the mass points butthe rigid and elastic bodies. In this case the covariant and contravariant com-ponents of the velocity vectors v and the acceleration vectors w of mechanicalsystem, as well as for one point, are represented by the following function

T1 =T

M=

1

2gαβ qαqβ , α, β = 0, s ,

where M is a mass of total system and T is its kinetic energy.

A P P E N D I X B

STABILITY AND BIFURCATION

OF STEADY MOTIONS

OF NONHOLONOMIC SYSTEMS

Appendix B contain a brief survey of the works, devoted to questions of

the existence, stability, and branching of a steady motion of conservative non-

holonomic systems. This Appendix is the plenary report of A.V.Karapetyan

with the same title, which was spoken in the International science conference

on mechanics "The third Polyakhov readings" (St.Petersburg, February 4-6,

2003).

In studying the questions of existence, stability, and branching of steadymotions of conservative nonholonomic systems two approaches [97, 98, 101,333, 334] are usually applied. In the general case when steady motions ofconservative nonholonomic systems correspond to the symmetries, to whichthe linear first integrals do not correspond (unlike the conservative holonomicsystems), the methods of Lyapunov–Malkin and Andronov–Hopf (see [91, 94,97, 99. 1985, 101, 333]) are used. These methods are based on the analysisof equations of perturbed motion and on the characteristic equation of thelinearized equations of perturbed motion. The latter has always zero roots,the number of which is not so less as the dimension of a family of steadymotions, which unperturbed steady motion belongs to. If the number of zeroroots is equal to the above-mentioned dimension and the rest of roots havenegative real parts, then the unperturbed motion is stable, in which case anyperturbed motion sufficiently close to the unperturbed one tends asymptot-ically to a steady motion of the considered family but, generally speaking,not unperturbed motion (according to the Lyapunov–Malkin theory). On theboundary of domain of stability (in the space of parameters of problem) thecharacteristic equation has either zero root, either a pair of pure imaginaryroots. In the first case another families of steady motions are branched offunperturbed steady motion and in the second case the families of periodicmotions (the Andronov–Hopf bifurcation occurs).

The described approach to the study of steady motions of conservativenonholonomic systems is also applied in the case when the nonholonomic con-straints have a so-called "dissipative" effect [94, 99. 1981, 1985]. The secondapproach to the study of questions of existence, stability, and branching ofsteady motions of nonholonomic systems is based on the modified theory ofRouth–Salvadori, Poincare–Chetaev, and Smale (see [97, 98, 99. 1983, 100.1994, 2000, 101, 333, 334]). It can be applied to the cases when to the sym-metries of system correspond not only steady motions but also the linear firstintegrals. Consider this case in more detail. At first we consider the case whenthe linear integrals, corresponding to the symmetries of system, are given inexplicit form.

229

230 Appendix B

Let

H = H (v, r) =1

2(A (r)v · v) + (a (r) · v) + a (r) = h (B.1)

be a total mechanical energy of system and

K = K (v; r) = BT (r)v + b (r) = k = const (B.2)

be a k-dimensional vector of linear integrals (the sign "T" means a transpo-sition).

Here v is an n-dimensional vector of quasivelocities (in particular, of im-pulses or generalized velocities), r ∈ M is an m-dimensional vector of de-termining coordinates such that the n × n-matrix A (r) of positive definitequadratic form, the n-dimensional vector a (r), and the scalar function a (r),entering together into a total mechanical energy, and also the n × k-matrixB (r) and the k-dimensional vector b (r) of the coefficients of the first inte-grals depend on these coordinates. Denote by M a configuration space of thesystem dimM n.

According to the Routh theory, on the fixed levels of the first integralsK = k to the critical points of the functions H correspond steady motions, inwhich case to the minimum points correspond stable steady motions. Takinginto account a structure of function (B.1) and the first integrals (B.2), theproblem of obtaining the critical points of this function on the fixed levels ofthese integrals can be solved in two stages. At the first stage we determine asingle minimum of the function H on the fixed levels k of the first integralsK = const with respect to the variables v (in this case the variables r areregarded as parameters):

minv

H

∣∣∣∣K=k

= H(vk(r); r

),

H(vk(r); r

)= a(r) +

1

2

[(C(r)ck · ck

)−

−

(A−1(r)a(r) · a(r)

)]= Wk(r) ,

ck = ck(r) = k − b(r) + BT(r)A(r)a(r) ,

C(r) =(BT(r)A−1(r)B(r)

)−1

,

vk(r) = A−1(r)B(r)C(r)ck − A−1(r)a(r) .

(B.3)

Here and below we assume that rankB (r) = k, ∀ r ∈ M, i. e. the integrals areindependent of a whole configuration space. The function Wk (r) is called aneffective potential, which depends, obviously, on the variables r ∈ M and theparameters k ∈ Rk. Then the problem of study of steady motions of systemis reduced to the problem of the analysis of effective potential.

Appendix B 231

Theorem 1. If the effective potential takes a nondegenerate stationary

value at the point r0 ∈ M, then the relation

r = r0, v = v0 = vk (r0)

describes a steady motion.

The point r0, at which the effective potential has a stationary value, de-pends on the constants k of the first integrals. This means that the pointsr0 (k), which are stationary in configuration space, make up k-parametricfamilies in the space

k ∈ Rk, r ∈ M

. The same families in the space

k ∈ Rk, r ∈ M , v ∈ Rn

make up the points r = r0 (k), v = v0 = vk (r0),which are stationary in the phase space, i. e. make up steady motions.

Even for the fixed values of the constants k, the effective potential Wk (r)can take stationary values not only at the point r0 but, generally speaking,at the certain another points r1, r2, . . . . These points also depend on theconstants k. In the general case for certain values of k∗ the families r0 (k),r1 (k), r2 (k), . . . can have common points. Such values of k∗ are called bifur-cational by Poincare. Obviously, the corresponding steady motions r = r0 (k),v = v0 = vk (r0) have common points if and only if the families r0 (k), r1 (k),r2 (k), . . . have common points (see (B.3)). In addition, by construction ofeffective potential, for indices we have the following relation

ind δ2H (v0, r0) |(2) = ind δ2Wk (r0) .

The latter permits us to simplify substantially the construction of the bifur-cational diagrams of Poincare–Chetaev and to restrict ourself by constructingthe families r0 (k) ∪ r1 (k) ∪ r2 (k) ∪ . . . in the space k, r only.

Consider the set

Σh,k =h ∈ R,k ∈ Rk : h = hs (k) , s = 0, 1, 2, ...

(B.4)

of the space h,k, where

h = hs (k) = H (vk (r) ; r) ; r = rs (k) , s = 0, 1, 2, ... .

Set (B.4) is called bifurcational by Smale: in this set we have the crossplot-tings of topological types of domains, of motions in configuration space, whichare defined by the relation Wk (r) h, r ∈ M.

Theorem 2. If the effective potential takes locally a strictly minimal sta-

tionary value for the fixed values k0 of the constants k at the point r0

(k0

),

then r = r0

(k0

), v = v0

(k0

)is a stable steady motion.

Theorem 3. If the index of the second variation of effective potential is

odd at the point r0

(k0

), then r = r0

(k0

), v = v0

(k0

)is an unstable steady

motion.

Theorems 1 – 3 follow from the Routh–Salvadori theory [97, 98, 334] andcorrespond to the special form of the first integrals (B.1), (B.2).

232 Appendix B

Remark. If for the certain r0∈ M we have rankB (r) < k, then to

construct an effective potential in the neighborhood of the point r0, we needin additional consideration [332].

The results considered are applied to the study of questions of existence,stability, and branching of steady motions of a heavy nonuniform dynamicallysymmetric ball on absolutely roughened horizontal plane [100. 1994, 127.1999].

Note that the approach described is applied to the analysis of steadymotions of the conservative nonholonomic systems of Chaplygin, which allowthe linear integrals, as known as unknown, [99. 1983, 100. 2000] in explicitform. The point is that in the case when the k-parametric groups of symmetryexist and the "dissipative" effect is lacking (otherwise the linear integralsdo not exist in abstracto) the equations of motions of such systems can bereduced to the form

d

dt

∂T

∂r=

∂T

∂r+ Gr −

DW

Dr, p = Γr . (B.5)

Here r is an m-dimensional vector of determining coordinates, p is an k-dimensional vector of impulses of pseudocyclic coordinates, 2T = (Dr · r),where D = D (r) is a symmetric m×m-matrix of positive definite quadraticform, G = G (r, r,p) is an alternate m × m-matrix, W = W (r,p) is an"effective" potential, Γ = Γ (r,p) is a k × m-matrix, depending linearly onp,

D

Dr=

∂

∂r+ ΓT ∂

∂p.

It is easily seen that equations (B.5) allow the generalized integral ofenergy

H = T + W = const . (B.6)

Suppose, γα = γα (r,p) is an m-dimensional vector, composed of theelements of the α-th row of the matrix Γ, where α = 1, k. If

(Dγα

Dr

)T

=Dγα

Dr, α = 1, k ,

then the system of km equations in partial derivatives

∂p

∂r= Γ (r,p)

is completely integrable and has the family of solutions p = Φ (r)k, whichdepends on the k arbitrary constants k, and the determinant of the k × k-matrix Φ (r) is not equal to zero. The latter means that system (B.5) exceptfor generalized integral of energy (B.6) allows the k linear integrals

K = Φ−1 (r)p = k = const . (B.7)

Appendix B 233

Though the explicit form of these integrals is unknown the general theoryof Routh–Salvadori permits us to assert that the stationary values of integral(B.6) on fixed levels of integrals (B.7) correspond to the steady motions

r = r0, r = 0, p = p0 (B.8)

of system (B.5), in which case the locally strictly minimal values correspondto the stable steady motions. Obviously, steady motions (B.8) make up ak-parametric family since the k + m constants r0 and p0 in (B.8) satisfy thesystem of m equations

DW

Dr= 0 . (B.9)

The function H has a minimum on steady motion (B.8) under conditions(B.7) if the function W under these conditions has a minimum at the point(r0,p0). The latter occurs if all the eigenvalues of the matrix

D2W

Dr2(B.10)

are positive at the point (r0,p0). If the determinant of matrix (B.10) is neg-ative at the point (r0,p0), then steady motion (B.8) is unstable. Obviously,to generate equations (B.9) and matrix (B.10) it is not required to know anexplicit form of the first integrals (B.7). The existence and structure of theseintegrals permit us to make use of the Routh–Salvadori theory and to affirmreasonably that steady motion (B.8) is stable for all positive eigenvalues ofmatrix (B.10), which is symmetric under the condition that the matrices

Dγα

Dr, α = 1, k ,

are symmetric. However for the bifurcational diagrams of Poincare–Chetaevand Smale to be constructed it is necessary to know the solution of sys-tem (B.7)–(B.9) in the form of r = r0 (k) and the quantities h = h (k) =W (r0 (k) ,p0 (k)), respectively, i. e. it is necessary to obtain the first integrals(B.7) in explicit form (though in terms of special functions, not necessarily interms of the elementary ones as in the problem on the motion of dynamicallysymmetric ball on absolutely roughened plane). In the problem on the motionof circular disk on absolutely roughened plane these first integrals are knownin the form of hypergeometric Gaussian series. This makes it possible [127.1999, 2001] to study completely the problem on a steady rolling of disk onhorizontal plane.

A P P E N D I X C

THE CONSTRUCTION OF APPROXIMATE SOLUTIONS

FOR EQUATIONS OF NONLINEAR OSCILLATIONS

WITH THE USAGE OF THE GAUSS PRINCIPLE

The Gauss principle is applied to the construction of approximate solu-

tions of equations of nonlinear oscillations, in particular, of the solution by

the Bubnov–Galerkin method.

If the motion of mechanical system is incompletely defined, then it isrational to construct the equations, which permit us to determine completelythis motion, using Gauss’ principle represented in the form

δ′′Z = 0 , (C.1)

where the function Z is given by formula (3.8) of Chapter IV. Two accentsof the symbol δ are used to emphasize that the second time derivatives ofgeneralized coordinates are varied only.

We make use of this principle to find the approximate solutions of thenonlinear equation

mx = F (t, x, x) , (C.2)

where m is a mass of mass point, x is its coordinate in the case of linearmotion, F is a projection of the force acting on the point.

Suppose, we seek the motion of mass point in the interval [0, τ ] in theform

x(t) =n∑

ν=1

aνfν(t) , (C.3)

where fν(t) are linearly independent functions, aν are the sought parameters.The function x(t), given in the form (C.3), does not satisfy, generally

speaking, differential equation (C.2) and therefore, substituting it into thisequation, we obtain

mx − F (t, x, x) = R , (C.4)

where R is a residual. From the mechanical point of view this residual isregarded as a force, under which the motion of point exactly satisfies thelaw (C.3).

We shall assume that the motion in the form (C.3) is incompletely givenin the sense that the parameters aν are not known. In order to find theseparameters, it is necessary that the average value of the square force R in theinterval [0, τ ] is minimal by virtue of the varying of the accelerations only (asin the Gauss principle) i. e. the following relation

δ′′

τ∫

0

(mx − F (t, x, x)

)2dt = 0

235

236 Appendix C

is satisfied. In other words, we shall seek the coefficients aν under the as-sumption that the error of mean square on the interval [0, τ ] is minimum.

Taking into account that in the Gauss principle the accelerations arevaried only, we have

τ∫

0

(mx − F (t, x, x)

)δxdt = 0 .

Substituting relation (C.3) into this equation, we obtain

n∑

ν=1

δaν

τ∫

0

(m

n∑

ν=1

aν fν− F

(t,

n∑

ν=1

aνfν ,

n∑

ν=1

aν fν

))fνdt = 0 . (C.5)

The quantities δaν are arbitrary and independent. Therefore from equation(C.5) it follows that

τ∫

0

(m

n∑

ν=1

aν fν− F

(t,

n∑

ν=1

aνfν ,

n∑

ν=1

aν fν

))fνdt = 0 , ν = 1, n . (C.6)

The conditions, under which this system of algebraic equations has solu-tions different from zero, depend on the form of as the function F (t, x, x), asthe functions fν(t), ν = 1, n.

The quantity R, introduced by formula (C.4), was regarded above as aforce. Now we regard it as an error, which occurs for the function x(t), givenin the form (C.3), to be satisfied equation (C.2). According to this approachthe system of algebraic equations (C.6) with respect to the parameters aν

becomes a system, which under certain assumptions permit us to find a partialapproximate solution of equations (C.2) in the form (C.3).

We apply now this method to determine the approximate periodic so-lutions of equation (C.2). For the sake of simplicity, we seek the periodicsolutions in the form

x(t) = a1 cos ωt + a2 sin ωt . (C.7)

Then system (C.6), in which the time τ is assumed to be equal to the period2π/ω, can be represented in the following way:

2π/ω∫

0

(− mω2(a1 cos ωt + a2 sin ωt)−

−F (t, a1 cos ωt + a2 sin ωt ,−a1ω sin ωt + a2ω cos ωt))cos ωtdt = 0 ,

2π/ω∫

0

(− mω2(a1 cos ωt + a2 sin ωt)−

−F (t, a1 cos ωt + a2 sin ωt ,−a1ω sin ωt + a2ω cos ωt))sin ωtdt = 0 .

(C.8)

Appendix C 237

Equations (C.8) are used to construct approximately the solution of equation(C.2) in the form (C.7) by the Bubnov–Galerkin method. They are usuallydeduced from the fundamental equation of dynamics. Recall that, as is shownfor Example VI. 3 considered in § 4 of Chapter VI, the clarification of themethods of Ritz and Bubnov–Galerkin by means of the integral variationalprinciples, can be found in the work of G. Yu. Dzhanelidze and A. I. Lur’e [56].

The above method for obtaining the approximate solutions of equations(C.2) can easily be extended to the case of arbitrary mechanical system withs degrees of freedom. In this case the Gauss principle (C.1) is used in integralform, i. e. we assume that

τ∫

0

(MW − Y)δ′′Wdt = 0 . (C.9)

Recall that we have

MW − Y =

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)eσ , δ′′W = δ′′qσeσ .

Here T is a kinetic energy of system, Qσ is a generalized force, correspondingto the generalized coordinate qσ, eσ and eσ are the vectors of fundamentaland reciprocal bases, respectively. Therefore equation (C.9) can be rewrittenas

τ∫

0

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)δ′′qσdt = 0 .

It follows that the functions qσ(t), given as

qσ(t) =n∑

ν=1

aσ

νfν(t) , σ = 1, s , (C.10)

can be regarded as an approximate solution of Lagrange’s equations if theparameters aσ

νsatisfy the following equations

τ∫

0

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)fνdt = 0 , σ = 1, s , ν = 1, n . (C.11)

Here the functions qσ(t) are assumed to be given in the form (C.10).The applying of formulas (C.11) to the solution of nonlinear system of

differential equations, which describes the steady-state oscillations of a certainelectromechanical system, using the Bubnov–Galerkin method, can be foundin the work [262].

A P P E N D I X D

THE MOTION OF NONHOLONOMIC SYSTEM

WITH OUT REACTIONS

OF NONHOLONOMIC CONSTRAINTS

In Appendix D the motion of nonholonomic systems in the case when

the reactions of constraints are lacking is considered. By the Mei Fengxiang

terminology such a motion is called a free motion of nonholonomic system.

The free motion of the Chaplygin sledge is studied. A realizing of the free

motion of nonholonomic systems acted by external forces is discussed.

§ 1. Existence conditions for "free motion"

of nonholonomic system

The motion of nonholonomic system is defined by forces, constraints, andthe initial data. In the work of Mei Fengxiang [362. 1994] the notion of free

motion of nonholonomic system, which is regarded as a motion under zerovalues of reactions of nonholonomic constraints, is introduced. In this work,in particular, the free motion of the Chaplygin sledge is considered.

In the work [362] existence conditions for a free motion of nonholonomicsystem are given. Below we obtain these conditions.

The motion of mechanical system with the ideal nonholonomic constraints

ϕκ(t, q, q) = 0 , q = (q1, . . . , qs) , κ = 1, k , (D.1)

is described by Lagrange’s equations of the first kind in curvilinear coordi-nates

d

dt

∂T

∂qσ−

∂T

∂qσ= Qσ + Rσ , σ = 1, s . (D.2)

Here the generalized reactions of nonholonomic constraints take the form

Rσ = Λκ

∂ϕκ

∂qσ, σ = 1, s , κ = 1, k .

Equations (D.2) can be written by using the Christoffel symbols of the firstkind

M(gστ qτ + Γσ,αβ qαqβ) = Qσ + Λκ

∂ϕκ

∂qσ,

σ, τ = 1, s , α, β = 0, s , q0 = t , q0 = 1 .

This system can be solved as the algebraic system with respect to qτ , τ = 1, s:

qτ =∆στ

∆

(Qσ + Λκ

∂ϕκ

∂qσ− MΓσ,αβ qαqβ

). (D.3)

239

240 Appendix D

Here ∆ is a determinant of the matrix (Mgστ ), ∆στ is an algebraic comple-ment with (σ, τ) number.

We differentiate equations of constraints (D.1) with respect to time

dϕκ

dt≡

∂ϕκ

∂t+

∂ϕκ

∂qτqτ +

∂ϕκ

∂qτqτ = 0 , κ = 1, k , τ = 1, s , (D.4)

and substitute solutions (D.3) into formulas (D.4). Then we obtain

∂ϕκ

∂t+

∂ϕκ

∂qτqτ +

∆στ

∆

∂ϕκ

∂qτ

(Qσ + Λκ

∂ϕκ

∂qσ− MΓσ,αβ qαqβ

)= 0 ,

κ = 1, k , σ, τ = 1, s , α, β = 0, s .

(D.5)

The Lagrange multipliers Λκ, κ = 1, k, can be determined from this systemif the corresponding determinant is not equal to zero. Assuming Λκ = 0,κ = 1, k, from relations (D.5) we obtain necessary and sufficient conditionsfor the existence of free motion of nonholonomic system. In the case whenthe constraints are stationary and the kinetic energy is independent of timethey have the form

∂ϕκ

∂qτqτ +

∆στ

∆

∂ϕκ

∂qτ

(Qσ − MΓσ,ρτ qρqτ

)= 0 ,

κ = 1, k , ρ, σ, τ = 1, s .

Just these conditions under (7) number are given in the work [362].If in place of Lagrange’s equations of the first kind we take Maggi’s equa-

tions(

d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vλ∗

= 0 , λ = 1, l , σ = 1, s , (D.6)

then in the second group of equations for a free motion of nonholonomicsystem we have zeros:

(d

dt

∂T

∂qσ−

∂T

∂qσ− Qσ

)∂qσ

∂vl+κ

∗

= 0 , κ = 1, k , σ = 1, s . (D.7)

§ 2. Free motion of the Chaplygin sledge

Consider the case when the center of mass of the Chaplygin sledge issituated above a runner. Let x, y be coordinates of the center of mass C ofsledge in a horizontal plane and θ be an angle of its rotation. Then the kineticenergy of system is as follows

T =M

2(x2 + y2) +

J

2θ2 ,

Appendix D 241

where M is a mass of sledge, J is a moment of inertia of sledge about thevertical axis, passing through the center of mass. On the motion of sledge itis imposed the nonholonomic constraint

ϕ ≡ x sin θ − y cos θ = 0 . (D.8)

Denoteq1 = x , q2 = θ , q3 = y

and introduce the quasivelocities

v1∗

= x , v2∗

= θ , v3∗

= x sin θ − y cos θ .

This implies that

x = v1∗, θ = v2

∗, y = v1

∗tg θ −

v3∗

cos θ.

We generate the Maggi’s equations:

Mx − Qx + (My − Qy) tg θ = 0 ,

Jθ − Qθ = 0 ,

(My − Qy)(−

1

cos θ

)= Λ ,

(D.9)

where Qx, Qy, Qθ are generalized exterior forces. Condition (D.7) takes theform

(My − Qy)(−

1

cos θ

)= 0 . (D.10)

Differentiating equation of constraint (D.8) in time, we obtain

x − y ctg θ = −θ(x ctg θ + y) . (D.11)

Consider the motion of the Chaplygin sledge under the conditions

Qx = Qy = Qθ = 0 . (D.12)

Then the first equation of system (D.9) is as follows

x = −y tg θ . (D.13)

Substituting (D.13) into relation (D.11), we have

y = θ(x + y tg θ) cos2 θ .

Now we represent condition (D.10) (assuming Qy = 0) as

θ(x + y tg θ) cos θ = 0 . (D.14)

242 Appendix D

Taking into account the equation of constraints y = x tg θ and assuming thatcos θ = 0, equation (D.14) becomes

θx = 0 . (D.15)

We notice that the obtained condition (D.15) of a free motion of theChaplygin sledge imposes a restriction on the choice of initial data. Really, if

x∣∣t=0

= x0 , y∣∣t=0

= y0 , θ∣∣t=0

= θ0 ,

then, according to formula (D.15), the following relation

θ0x0 = 0 (D.16)

is satisfied. Restriction (D.16) allows the following choice of initial data:

x0 = 0 , θ0 = 0 . (D.17)

Since nonholonomic constraint (D.8) are satisfied, this implies the initial con-dition for y:

y0 = 0 . (D.18)

To the initial data (D.17), (D.18) corresponds a motion such that the centerof mass of sledge rests and the sledge uniformly rotates round it. In thiscase the force, preventing the displacement of sledge in transverse directionrelative to a runner, is lacking and we say that with initial data (D.17), (D.18)the sledge moves (rotates) to be free.

Condition (D.16) allows another choice of the initial data:

θ0 = 0 , x0 = 0 , y0 = 0 , x0 sin θ0 − y0 cos θ0 = 0 .

In this case the center of mass of sledge has a linear and uniform motionalong the initial orientation of runner and the sledge does not rotate.

Condition (D.16) also allows the following obvious choice of the initialdata:

x0 = y0 = θ0 = 0 .

This corresponds to the rest of sledge if the exterior forces are lacking.If the equation of constraint is obtained under condition (D.14), then in

place of relations (D.15) we have

θy = 0 . (D.19)

The investigation of possible motions under condition (D.19) leads to thesame three free motions of the Chaplygin sledge.

Thus, if conditions (D.12) and (D.16) are satisfied, then the Chaplyginsledge moves free in the above-mentioned sense. If not the sledge has a stan-dard motion proper for nonholonomic system. In this case we need to find a

Appendix D 243

reaction of constraint in order to check wether this nonholonomic constraintis nonretaining.

§ 3. The possibility of free motion

of nonholonomic system

under active forces

By nonholonomic constraints (D.1) the law of system motion in L–spaceand in K–space can be represented, respectively, as

MWL = YL + RL , MWK = YK + RK . (D.20)

In the case of ideal constraints RL = 0 and in studying a free motion ofnonholonomic system we have, in addition, RK = 0. Therefore equations(D.20) take the form

MWL = YL , MWK = YK . (D.21)

Equations (D.21) imply that in the case of free motion the component WK

of the vector of acceleration of system W is the function of variables t, q, q,given in the following way

WK =YK(t, q, q)

M. (D.22)

On the other hand, as is shown in § 1 of Chapter IV, the vector-functionWK(t, q, q) is uniquely defined by equations of constraint (D.1). Then theforce YK(t, q, q), entering into relation (D.22), can be called the control forceYK

control, under which the incomplete program of motion, given in the form(D.1), is realized. Thus, by formulas (D.21) a free motion of nonholonomicsystem can be regarded as a motion such that the active force Y has a com-ponent, belonging to L–space only, and the control force Ycontrol belongingto K–space only. Applying this approach to a free motion of nonholonomicsystem we obtain that in accordance with the theory of constrained motionthe control force Ycontrol = YK

control has the form YK

control = Λcontrolκ

∇′ϕκ .

Here Λcontrolκ

is a generalized control force, under which the constraint withκ number is realized. Note that the same approach is usedused in Chapter IIIto solve the problem of flight dynamics on the directing of a mass point on atarget by the curve of pursuit.

The concept of study of free motion of nonholonomic systems, developedin the work of Mei Fengxiang [362. 1994], can also be of another practicalimportance. For example, in the treatise [226] it is considered a controllablemotion of nonholonomic systems. The control is chosen from the conditionthat the nonholonomic system has a given program motion. In this case thecontrol forces also provide generation of forces equal to the corresponding re-actions of nonholonomic constraints. Under these control forces the reactions

244 Appendix D

of nonholonomic constraints are equal to zero and therefore by the terminol-ogy of the work [362. 1994] such a controllable motion is a free motion ofnonholonomic system. In the book [226] the possibility of small deviations ofthe obtained generalized coordinates and velocities from the required ones istaken into account. This is gone along with the occurrence of small reactionsof nonholonomic constraints, which are regarded in the considered problemas disturbances. Finally, the original problem is reduced to the conditionalproblem of adaptive control with unknown disturbances. The algorithms ofcontrol and also the estimation of so far as a program motion of system isrealized for a given accuracy of stabilization are given.

A P P E N D I X E

THE TURNING MOVEMENT OF A CAR

AS A NONHOLONOMIC PROBLEM

WITH NONRETAINING CONSTRAINTS

The turning movement of a car with its possible sideslip is considered as a

nonholonomic problem with nonretaining constraints. The four possible types

of the car motion are studied.

§ 1. General remarks

The complete theory of the motion of a car with deformable wheels isdeveloped by N. A. Fufaev and detailed in his book [130]. The treatise byV. F. Zhuravlev and N. A. Fufaev [72] is devoted to mechanics of systems withnonretaining constraints. In this treatise the Boltzmann–Hamel equations areused for studying the motion of nonholonomic systems, and the possibility ofrestoring nonholonomic constraints is investigated on the basis of behaviour ofsolution curves in the common space of generalized coordinates and quasive-locities. In this Appendix E the Maggi equations, which make it possible toeasily determine the generalized reaction forces of nonholonomic constraints,are applied; the beginning and stop of wheels sideslip being determined bythese constraint forces.

We now return to examples II. 4 and II. 5 considered in Chapter II. Payattention that the Boltzmann-Hamel and Maggi equations are formed therefor the realized constraints (4.16). In this case the turning of a car is studied,saying figuratively, under the "dynamic control when the turning momentL1(t), resisting moment L2(θ), and restoring moment L3(θ) are applied tothe rotating front axle (see Fig. II. 4). This scheme required introducing fourgeneralized coordinates ϕ, θ, ξC , ηC , that was reasonable from the methodicalpoint of view, for in this case we get an example, in which with two con-straints (4.16) we have to obtain two Boltzmann-Hamel’s equations or twoMaggi’s equations. This mathematical model can be of interest in studyingthe motion of wheeled robot vehicles, development of which is given muchattention at present (see, for ex., works by V. N. Belotelov, V. I. Kalyonova,A. V. Karapetyan, A. I. Kobrin, A. V. Lenskii, Yu. G. Martynenko, V. M.Morozov, D. E. Okhotsimskii, M. A. Salmina [146-148, 423]).

Let us go to the "kinematic control under which the turn of the frontaxle is determined by a driver as a certain time function θ = θ(t). In sucha scheme a turning car has three degrees of freedom. In this case, we shallconsider nonholonomic constraints

ϕ1≡ −ξC sin ϕ + ηC cos ϕ − l2ϕ = 0 , (E.1)

ϕ2≡ −ξC sin(ϕ + θ) + ηC cos(ϕ + θ) + l1ϕ cos θ = 0 , (E.2)

245

246 Appendix E

which should be satisfied by the car motion, as nonretaining. The activeforces F1(t) and F2(t) have the same meaning as in examples II. 4 and II. 5,of Chapter II.

§ 2. The turning movement of a car

with retaining (bilateral) constraints

We shall study the car motion in the horizontal plane with respect to thefixed system of coordinates Oξηζ (see Fig. E. 1). We shall set the position ofa car by generalized coordinates q1 = ϕ (the angle between the longitudinalCx-axis of the car and the Oξ-axis ), q2 = ξC , q3 = ηC (the coordinates ofpoint C). The angle θ is equal to the angle between the front axle and aperpendicular to the Cx-axis . It is a given time function:

θ = θ(t) .

Two nonholonmic constraints (E.1) and (E.2), expressing the absence of sideslipping of the front and rear axles of the car are imposed on the car motion.

The kinetic energy of the system consists of the kinetic energies of the carbody and front axle and is calculated according to the formula:

2T =M∗( ˙ξC

2+ ˙ηC

2)+J∗ϕ2+J2θ2 + 2J2ϕθ + 2M2l1ϕ(−ξC sin ϕ + ηC cos ϕ) ,

M∗ = M1 + M2, J∗ = J1 + J2 + M2l21 .

(E.3)Using the expression for virtual elementary work

δA = Qϕδϕ + QξCδξC + QηC

δηC ,

we shall find the generalized forces acting on the car, as was done in ex-amples II. 4, II. 5, Chapter II. For the rear drive car we obtain the following

Fig. E. 1

Appendix E 247

expressions:Q1 ≡ Qϕ = 0 ,

Q2 ≡ QξC= F1(t) cos ϕ − F2(vC)ξC/vC ,

Q3 ≡ QηC= F1(t) sin ϕ − F2(vC)ηC/vC ,

vC =

√˙ξC

2+ ˙ηC

2 .

(E.4)

In order to form the Maggi equations describing the vehicle motion weintroduce new nonholonomic variables by to the formulas:

v1∗

= ϕ, v2∗

= −l2ϕ − ξC sin ϕ + ηC cos ϕ ,

v3∗

= l1ϕ cos θ − ξC sin(ϕ + θ) + ηC cos(ϕ + θ) ,

and write the reverse transformation

q1≡ ϕ = v1

∗, q2

≡ ξC = β21v1

∗+ β2

2v2∗

+ β23v3

∗,

q3≡ ηC = β3

1v1∗

+ β32v2

∗+ β3

3v3∗,

(E.5)

where

β21 =

l1 cos ϕ cos θ + l2 cos(ϕ + θ)

sin θ,

β22 =

cos(ϕ + θ)

sin θ, β2

3 = −

cos ϕ

sin θ,

β31 =

l1 sin ϕ cos θ + l2 sin(ϕ + θ)

sin θ,

β32 =

sin(ϕ + θ)

sin θ, β3

3 = −

sin ϕ

sin θ.

(E.6)

The first group of the Maggi equations in this case consists of a singleequation

(MW1 − Q1)∂q1

∂v1∗

+ (MW2 − Q2)∂q2

∂v1∗

+ (MW3 − Q3)∂q3

∂v1∗

= 0 . (E.7)

The expressions MWσ may be calculated using kinetic energy by the formulas

MWσ =d

dt

∂T

∂qσ−

∂T

∂qσ, σ = 1, 3 .

Finally, using expressions (E.3), (E.4), (E.5), (E.6), we represent the equationof motion (E.7) in the following expanded form:

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ)+

+β21(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC)+

+β31(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ + F2(vC)ηC/vC) = 0 .

(E.8)The equations of constraints (E.1) and (E.2) should be added to this equation.

248 Appendix E

If the initial conditions and analytic representation of the functions F1(t),F2(vC) are given, then after numerical integrating the nonlinear system ofdifferential equations (E.1), (E.2), (E.8) we shall find the law of the car mo-tion:

ϕ = ϕ(t), ξC = ξC(t), ηC = ηC(t). (E.9)

Now we can determine the generalized reaction forces. The second groupof Maggi’s equations will be written as follows:

Λ1 = (MW1 − Q1)∂q1

∂v2∗

+ (MW2 − Q2)∂q2

∂v2∗

+ (MW3 − Q3)∂q3

∂v2∗

,

Λ2 = (MW1 − Q1)∂q1

∂v3∗

+ (MW2 − Q2)∂q2

∂v3∗

+ (MW3 − Q3)∂q3

∂v3∗

,

or in the extended form for the rear drive vehicle:

Λ1 = β22(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC)+

+β32(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ+ (E.10)

+F2(vC)ηC/vC) ,

Λ2 = β23(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC)+

+β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ+ (E.11)

+F2(vC)ηC/vC) .

After inserting expressions (E.9) into these formulas we find the law of varyingthe generalized reaction forces

Λi = Λi(t), i = 1, 2 .

These functions allow us to investigate the possibility of realizing the non-holonomic constraints (E.1), (E.2). If the reaction forces appear to exceedthe forces provided by Coulomb’s frictional forces, then these constraints willnot be realized and the vehicle will begin to slip along the axles to which thewheels are fastened.

In order to write the conditions of the beginning of side slipping of thewheels in the analytical form, it is necessary to establish the relation betweenthe determined generalized reactions Λ1, Λ2 and reaction forces RB , RA

applied to the wheels from the road (see Fig. E. 1).This is a question of principal importance, so let us consider the relation

between the generalized reaction force Λ of the nonholonomic constraint andthe reaction force R for the following quite general case. Assume that theequation of the nonholonomic constraint sets the condition of the fact thatfor the plane motion the velocity v of a point of mechanical system alongthe direction of the unit vector n is equal to zero, i. e. assume that constraintequation written in vectorial form is as follows:

ϕn = v · n = 0 .

Appendix E 249

This equation in a scalar form appears as

ϕn = xnx + yny = 0 .

If the constraint is ideal, then the reaction force R can be represented as

R = Rxi + Ryj =

= Λ(∂ϕn

∂xi +

∂ϕn

∂yj)

= Λn ,

where i and j are unit vectors in x− and y− directions. Hence, the generalized

reaction force Λ is equal to the projection of the constraint reaction force R

onto the direction of vector n.It is clear that this representation of the vector R in the form Λn can be

extended also to the constraints (E.1), (E.2). Writing these constraints in thevector form

ϕ1 = vB · j = 0 , (E.12)

ϕ2 = vA · j1 = vA · (−i sin θ + j cos θ) = 0 , (E.13)

where j1 is the unit vector of the ordinate axis of the movable frame Ax1y1

of the car front axle, we obtain

RB = Λ1j , (E.14)

RA = Λ2(−i sin θ + j cos θ) . (E.15)

Remark that if the constraints (E.12) and (E.13) are violated, then nonze-ro values of ϕ1 and ϕ2 are equal to projections of velocities of the points B

and A onto the vectors j and j1, correspondingly. In this case the resultingfriction forces applied to the wheels may be represented as

RfrB

= −Λfr1 sign(ϕ1)j ,

RfrA

= −Λfr2 sign(ϕ2)(−i sin θ + j cos θ) .

Finding the positive quantities Λfr1 and Λfr

2 will be reported below.

§ 3. The turning movement of a rear-drive car

with nonretaining constraints

General remarks. Let us return to the question considered in the pre-vioius paragraphs. Note that the Maggi equation (E.8) was obtained for thesatisfied constraints (E.1), (E.2), i. e. when these nonholonomic constraintswere retaining (bilateral).

Let us study the vehicle motion in the case when the constraints (E.1),(E.2) may be nonretaining, i. e. when side slipping of the front or rear wheels

250 Appendix E

(or both front and rear wheels simultaneously) begins. The dynamic condi-tions of realizing the kinematic constraints (E.1), (E.2) is the requirementthat the forces of interaction between the wheels and the road should not ex-ceed the corresponding Coulomb’s friction forces. For the driven front wheelsin accordance with formula (E.15) this is expressed by inequality:

|Λ2| < F fr2 = k2N2 , (E.16)

where F fr2 , k2 are the frictional force and the coefficient of friction between

front wheels and the road, respectively, N2 is the normal pressure of the frontaxle.

When considering the rear driving wheels it is necessary to take intoaccount that the value of this wheels-road interaction force FB is determinedby the vector sum of the driving force F1 and side reaction force RB given byformula (E.14) (see Fig. E. 2). To provide the absence of side slipping of therear axle, the following condition should be satisfied (the introduced notationis analogous to the notation used for the front axle):

FB =√

(F1)2 + (Λ1)2 < F fr1 = k1N1 . (E.17)

According to Fig. E. 2 this means that the end of the force vector FB shouldnot go beyond the circle of radius F fr

1 . Otherwise the road will not be ableto develop such reaction value |Λ1| that is required for realization of thenonholonomic constraint (E.1). Thus, this constraint becomes nonretaining,the side velocity component of driven wheels appears, and Coulomb’s frictionforce F fr

1 starts acting to them from the road. This Coulomb’s friction forceF fr

1 arises from simultaneous action of the driving force F1 and side frictionforce Λfr

1 , so that

(FB)2 ≡ (F fr1 )2 ≡ (k1N1)

2 = (F1)2 + (Λfr

1 )2 . (E.18)

Fig. E. 2

Appendix E 251

Note that at the beginning of side slipping the driver sets

F1 = 0 .

Possible types of the car motion. We shall explain possible differenttypes of motion of the mechanical model of a car. In Fig. E. 3 in the phasespace of variables qσ, qσ, σ = 1, 3, we see the representation of two hypersur-faces. The first one corresponds to the constraint given by equation (E.12),and the second one corresponds to the constraint given by equation (E.13).In an explicit form these constraints are presented by formulas (E.1), (E.2).

For simultaneous realization of nonholonomic constraints (E.1) and (E.2)the point of the phase space should be located in the line of intersectionof these hypersurfaces. This corresponds to the I-st type of the car motion(bold curve I in Fig. E. 3). If the first constraint is violated (FB = F fr

1 ) andthe second constraint is satisfied, then the representation point is located atthe hypersurface ϕ2 = 0 (II-nd type of motion). If the second constraint isviolated, but the first constraint is fulfilled ϕ1 = 0, then the representationpoint belongs to hypersurface ϕ1 = 0 (III-rd type of motion). In the caseif both constraints are violated, the representation point does not belong tohypersurfaces, as this takes place the vehicle moves in the presence of sidefriction forces acting on the front and rear axles (IV-th type of motion).

From any type of motion the representation point can change to any othertype of motion. For example, in the I-st type of motion, if inequality (E.17)is not satisfied , the vehicle becomes released of the constraint (E.1). If inthis case inequality (E.16) is still satisfied, then constraint (E.2) keeps onworking, thus, the representation point can move only over the hypersurfaceϕ2 = 0 (the car changes to the II-nd type of motion). Here two cases ofpossible restoring the I-st type of motion should be distinguished.

In some area G1 the solution curves go through the curve I, without stop-

Fig. E. 3

252 Appendix E

ping there (see Fig. E. 3). This instantaneous realization of the constraint(E.1) corresponds to the stop of side motion of the rear axle in one directionand change of the same axle to the side motion in the reverse direction. Incontrast to this the behaviour of solution curves within the area G2 charac-terizes restoration of the constraint ϕ1 = 0 and change from the II-nd typeof motion to the I-st one.

Without preliminary studies of behaviour of solution curves in the com-mon space of generalized coordinates and quasivelocities [72], it is possibleto find out in which area G1 or G2 the equation ϕ1 = 0 turned out to besatisfied, in the following manner. By the values of phase variables, such thatthe constraint (E.1) is fulfilled, let us calculate the reaction Λ1 by the for-mula (E.10). If for the obtained value of Λ1 the inequality (E.17) is satisfied,then the constraint ϕ1 = 0 becomes retaining (bilateral) (the solution curveis within the area G2), otherwise this constraint is not restored (the solutioncurve is within the area G1).

In investigating the II-nd type of motion it is necessary also to ensurethat inequality (E.16) is satisfied, for if it is violated the vehicle will changeto the IV-th type of motion. If constraint (E.1) is restored and constraint(E.2) is violated at the same time, then the III-rd type of motion will occur.

Note that for the sake of simplicity, it was assumed in the foregoing thatthe static and dynamic coefficicients of Coulomb’s friction force are equal toeach other. The difference of these quantities could be taken into account in asimilar way as it has been done in § 4 of Chapter I, when studying acceleratedmotion of a car with the possible slipping of its driving wheels.

Let us write out the equations of motion for the turning car four types ofmotion cosidered.

I-st type of motion. For this motion both constraints (E.1) и (E.2) arefulfilled :

ϕ1 = 0 , ϕ2 = 0 .

Maggi’s equation for a rear-wheel drive vehicle takes the form (E.8), whichshould be integrated together with the equations of constraints (E.1) and(E.2). Having obtained the law of motion

ϕ = ϕ(t) , ξC = ξC(t) , ηC = ηC(t) ,

the generalized reactions can be found from (E.6), (E.10), (E.11)

Λ1 = Λ1(t) , Λ2 = Λ2(t) .

By these values, fulfillment of inequelities (E.16) and (E.17) is being checked.When one of them is violated, the vehicle changes to the II-nd or III-rd typeof motion, and when both of them are violated simultaneously it changes tothe IV-th type.

II-nd type of motion. For this type of motion only the second constraintis fulfilled:

ϕ1= 0 , ϕ2 = 0 .

Appendix E 253

The rear axle of the vehicle executes lateral motion, therefore the lateralfrictional force Λfr

1 , calculated by formula (E.18) is applied to it. As thistakes place, if ϕ1 > 0, then according to formula (E.12) the rear wheelssideslip in the positive direction of the y-axis. Therefore, the lateral frictionalforce is opposed to the y-axis, and if ϕ1 < 0, it is aligned with the y-axis (seeFig. E. 1).

Let us obtain Maggi’s equations in the presence of one constraint (E.2).Let us go to quasivelocities by formulas:

v1∗

= ϕ , v2∗

= ξC ,

v3∗

= −ξC sin(ϕ + θ) + ηC cos(ϕ + θ) + l1ϕ cos θ .

Let us find the inverse transformation:

ϕ = v1∗, ξC = v2

∗, ηC = β3

1v1∗

+ β32v2

∗+ β3

3v3∗,

where

β31 = −l1 cos θ/ cos(ϕ + θ) , β3

2 = tg(ϕ + θ) , β33 = 1/ cos(ϕ + θ) . (E.19)

Now we may get two Maggi’s equations for the rear-wheel drive vehicle

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) − Λfr1 sign(ϕ1)l2+

+β31(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)−

−F1(t) sin ϕ + F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) = 0 ,

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC−

−Λfr1 sign(ϕ1) sin ϕ + β3

2(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ+

+F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) = 0 .

(E.20)From the second group of Maggi’s equations there remains one equation

for determination of the generalized reaction Λ2. For the vehicle with drivingrear wheels it is as follows:

Λ2 = β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ+

+F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) .

(E.21)

The equations of motion (E.20) are integrated together with the constraintequation (E.2). If the dynamic condition (E.16) for the constraint (E.2) tobe realized holds for the obtained value of Λ2, then the II-nd type of motioncontinues. If the condition (E.16) is violated, then the vehicle will change toIV-th type of motion.

In the course of checking inequality (E.16) it is necessary to keep watch-ing if the constraint ϕ1 = 0 begins to hold. If this constraint is realizedunder certain obtained values of t, qσ, qσ, σ = 1, 3, and if inequality (E.17)

254 Appendix E

holds for the value Λ1 calculated by formula (E.10), then the constraintϕ1 = 0 is restored, the rear axle ceases to execute lateral motion and thecar changes to the I-st type of motion. If inequality (E.17) is not fulfilledfor the value Λ1 calculated by formula (E.10), then the car keeps the II-ndtype of motion (rear axle begins lateral motion in the oppositedirection).

Theoretically the car may change from the II-nd type of motion to theIII-rd one: for this purpose, at a certain time instant inequality (E.16) mustcease to hold and simultaneously the constraint ϕ1 = 0 must be restored.

III-rd type of motion. This motion is studied in a similar way to theII-nd type. Now the following should be fulfilled:

ϕ1 = 0 , ϕ2= 0 .

Due to side slipping of the front axle of the car this front axle is acted uponby the side friction force

Λfr2 = k2N2 . (E.22)

In order to form Maggi’s equations for this nonholonomic problem withone constraint (E.1) let us change to quasivelocities by using the formulas:

v1∗

= ϕ , v2∗

= ξC ,

v3∗

= −ξC sin ϕ + ηC cos ϕ − l2ϕ .

This corresponds to the reverse transformation:

ϕ = v1∗, ξC = v2

∗, ηC = β3

1v1∗

+ β32v2

∗+ β3

3v3∗,

whereβ3

1 = l2/ cos ϕ , β32 = tg ϕ , β3

3 = 1/ cos ϕ . (E.23)

Two Maggi equations for the car with driving rear wheels have the form:

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) + Λfr2 sign(ϕ2)l1 cos θ+

+β31(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)−

−F1(t) sin ϕ + F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) = 0 , (E.24)

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC−

−Λfr2 sign(ϕ2) sin(ϕ + θ) + β3

2(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)−

−F1(t) sin ϕ + F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) = 0 .

The generalized reaction Λ1 is expressed as

Λ1 = β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ+ (E.25)

+F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) .

Appendix E 255

The equations of motion (E.24) are integrated together with the constraintequation (E.1). If the dynamic condition (E.17) for realizing the constraint(E.1) is satisfied for the value of Λ1 obtained by formula (E.25), then theIII-rd type of motion continues. If the condition (E.17) is violated, then thecar changes to the IV-th type of motion.

In the course of checking inequality (E.17) it is necessary to keep watchingif the constraint ϕ2 = 0 begins to be realized. If this constraint is realizedunder certain calculated values t, qσ, qσ, σ = 1, 3, then these values of thevariables should be substituted in formula (E.11). If for the obtained Λ2 theinequality (E.16) is satisfied, then the constraint ϕ2 = 0 is restored, the frontaxles ceases to execute lateral motion, and the car changes to the I-st typeof motion. If for the calculated value of Λ2 inequality (E.16) is not satisfied,then the car continues the III-rd type of motion (the front axle begins lateralmotion in the opposite direction).

Theoretically the III-rd type of motion can change to the II-nd one: forthis purpose, at a certain instant inequality (E.17) must cease to hold, andat the same time the constraint ϕ2 = 0 must be restored.

IV-th type of motion. For such motion the following must take place:

ϕ1= 0 , ϕ2

= 0 .

This means that the car moves as a holonomic system when its wheels areacted upon by side frictional forces Λfr

1 and Λfr2 set by formulas (E.18) and

(E.22). The motion of the rear wheel-drive car is determined by the followingLagrange equations of the second kind:

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ)−

−Λfr1 sign(ϕ1)l2 + Λfr

2 sign(ϕ2)l1 cos θ = 0 ,

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos ϕ + F2(vC)ξC/vC−

−Λfr1 sign(ϕ1) sin ϕ − Λfr

2 sign(ϕ2) sin(ϕ + θ) = 0 , (E.26)

M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin ϕ + F2(vC)ηC/vC+

+Λfr1 sign(ϕ1) cos ϕ + Λfr

2 sign(ϕ2) cos(ϕ + θ) = 0 .

In course of calculation of motion by equations (E.26) it is necessary tokeep watching if either function ϕ1 or ϕ2 vanishes, or both functions ϕ1 andϕ2 do so for the current values of

t, qσ , qσ , σ = 1, 3 . (E.27)

If ϕ1 = 0 holds for the values (E.27), then Λ1 should be calculated forthese values of variables by formula (E.25). If for this value of Λ1 inequality(E.17) is satisfied, then the car changes to the III-rd type of motion, otherwiseit keeps the motion of the IV-th type.

256 Appendix E

If it turns out that ϕ2 = 0 for the values (E.27), then for these values ofvariables Λ2 should be calculated by formula (E.21). If the inequality (E.16)is satisfied for this value of Λ2, then the car changes to the II-nd type ofmotion, otherwise it keeps the motion of the IV-th type.

If it turns out that for the values (E.27) the both functions ϕ1 and ϕ2

vanish simultaneously, then Λ1 and Λ2 should be found from formulas (E.10),(E.11). If both inequalities (E.16) and (E.17) are fulfilled for these values,then the car changes to the I-st type of motion. If only inequality (E.16)is satisfied, then the II-nd type of motion begins. If only inequality (E.17)is fulfilled, then from this point on the car will execute the III-rd type ofmotion.

§ 4. Equations of motion of a turning

front-drive car with non-retaining constraints

Consider a motion of a front-drive car. All necessary changes in the equa-tions of motion are caused by the fact that the application point of the forceF1(t) changes. Now the force is applied to the point A and aligned with theaxis Ax1 (see Fig. E. 1). So, for a front-drive car the expressions of generalizedforces appear as

Q1 ≡ Qϕ = l1F1(t) sin θ ,

Q2 ≡ QξC= F1(t) cos(ϕ + θ) − F2(vC)ξC/vC ,

Q3 ≡ QηC= F1(t) sin(ϕ + θ) − F2(vC)ηC/vC ,

vC =

√˙ξC

2+ ˙ηC

2 .

Below we present the equations of motion for the four types of motion.

I-st type of motion. The car motion without slipping.In this case the Maggi equations have the form

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) − l1F1(t) sin θ

+β21(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ)

+F2(vC)ξC/vC) + β31(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin(ϕ + θ)

+F2(vC)ηC/vC) = 0 .

(E.28)As this takes place, the generalized constraint reaction forces are expressed as

Λ1 = β22(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ)

+F2(vC)ξC/vC) + β32(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC) ,

(E.29)

Appendix E 257

Λ2 = β23(M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ)

+F2(vC)ξC/vC) + β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC) .

(E.30)

In equation (E.28) and relations (E.29), (E.30) the quantities β21 , β3

1 , β22 , β3

2 ,

β23 , β3

3 should be calculated according to formulae (E.6).In the case of a front-drive car, inequalities, the fulfillment of which means

the realization of constraints (E.1) and (E.2), appear as

|Λ1| < F fr1 = k1N1 , (E.31)

FA =√

(F1)2 + (Λ2)2 < F fr2 = k2N2 , (E.32)

for the driven rear wheels and driving front wheels, correspondingly. If in-equation (E.31) is violated, it means that side slipping of the rear axle begins(the violation of the constraint ϕ1 = 0, the change to the II-nd type of mo-tion). If inequation (E.32) is violated, it means that side slipping of the frontaxle begins (the violation of the constraint ϕ2 = 0, the change to the III-rdtype of motion).

II-nd type of motion. The car rear axle executes side (lateral) motion.For a front-drive car two Maggi’s equations have the form

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) − l1F1(t) sin θ

+Λfr1 sign(ϕ1)l2 + β3

1(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) = 0 ,

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ) + F2(vC)ξC/vC

−Λfr1 sign(ϕ1) sin ϕ + β3

2(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) = 0 ,

where coefficients β31 , β3

2 , β33 are determined from (E.19), and the quantity

Λfr1 is defined as

Λfr1 = k1N1 . (E.33)

The relation for determining the generalized reaction Λ2 is

Λ2 = β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin(ϕ + θ)+

+F2(vC)ηC/vC + Λfr1 sign(ϕ1) cos ϕ) .

III-rd type of motion. The car front axle executes side (lateral) motion.In this case the, for a front-drive car the two Maggi equations take the

formJ∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) − l1F1(t) sin θ+

+Λfr2 sign(ϕ2)l1 cos θ + β3

1(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)−

258 Appendix E

Fig. E. 4

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) = 0 ,

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ) + F2(vC)ξC/vC−

−Λfr2 sign(ϕ2) sin(ϕ + θ) + β3

2(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ)−

−F1(t) sin(ϕ + θ) + F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) = 0 .

Here the quantities β31 , β3

2 , β33 are defined from (E.23), and the friction force

Λfr2 is expressed as

(k2N2)2 = (F1)

2 + (Λfr2 )2 . (E.34)

In this case the generalized constraint reaction force Λ1 is

Λ1 = β33(M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin(ϕ + θ)+

+F2(vC)ηC/vC + Λfr2 sign(ϕ2) cos(ϕ + θ)) .

IV-th type of motion. Both car axles execute side motion.Equations of motion of a front-drive car slipping on the horizontal surface

(level) are

J∗ϕ + J2θ + M2l1(−ξC sin ϕ + ηC cos ϕ) − l1F1(t) sin θ+

+Λfr1 sign(ϕ1)l2 + Λfr

2 sign(ϕ2)l1 cos θ = 0 ,

M∗ξC − M2l1(ϕ sin ϕ + ϕ2 cos ϕ) − F1(t) cos(ϕ + θ) + F2(vC)ξC/vC−

−Λfr1 sign(ϕ1) sin ϕ − Λfr

2 sign(ϕ2) sin(ϕ + θ) = 0 ,

M∗ηC + M2l1(ϕ cos ϕ − ϕ2 sin ϕ) − F1(t) sin(ϕ + θ) + F2(vC)ηC/vC+

+Λfr1 sign(ϕ1) cos ϕ + Λfr

2 sign(ϕ2) cos(ϕ + θ) = 0 .

The values of side friction forces Λfr1 and Λfr

2 are defined from (E.33) and(E.34).

Taking into account conditions (E.31), (E.32) of constraints realizationand formulae for defining the values of friction forces (E.33), (E.34), the logic

Appendix E 259

of change from one type of motion to another is the same as in the case of arear-drive car (see § 3).

§ 5. Calculation of motion

of a certain car

As an example, let us consider the motion of a hypothetical compactmotor car with M1 = 1000 kg ; M2 = 110 kg ; J1 = 1500 kg·m2; J2 = 30 kg·m2 ;l1 = 0.75 m ; l2 = 1.65 m ; k1fr = 0.4 ; k2fr = 0.4 for the power characteristics:

F2(vC) = k2vC N ; k2 = 100 N·s·m−1 .The following car motion is studied. In the beginning the vehicle moves

rectilinearly (the planes of the front and rear wheels are parallel) duringeight seconds, in this case ϕ = π/6. During this time the function F1(t)changes by the law F1(t) = 200t (F1 is measured in Newtons, t is measuredin seconds), i. e. at the initial time F1(0) = 0, and at the end of rectilinearmotion F1(8) = 1600. Graphs of dependences of coordinates on time arepresented in Fig. E. 4.

After eight seconds of rectilinear motion the driver starts to turn thesteering wheel at a smooth manner at the angle θ = π(t − 8)/8 , that is, intwo seconds the angle θ is equal to π/4 . For this motion F1(t) = 1600 . Bythe computed values of constraint reactions Λ1 and Λ2 we get graphs shownin Fig. E. 5 . It follows from the graphs that inequality (E.17) is satisfied, butcondition (E.16) is violated when t1 = 9.5147 , θ(t1) = 0.5948 . Thus, when8 < t < 9.5147, the car moves by the I-st type, but after t1 = 9.5147 itchanges to the III-rd one.

After occurrence of the III-rd type of motion a driver tries to eliminatethe side slipping of front wheels of the car, by the way of setting F1 = 0and changing a turning angle of the front axle according to the law θ =−10(t − t1) + θ1 . Let us calculate the constraint reaction |Λ1| and check ifdynamic condition (E.17) is satisfied. As we can see from Fig. E. 6, in case thecar keeps moving by the III-rd of motion, then the force FB does not exceedthe friction force at least during the time interval 9.5147 < t < 13, that isthe dynamic condition (E.17) of realizing the constraint (E.1) is satisfied. At

Fig. E. 5

260 Appendix E

the same time we check if the condition ϕ2 = 0 is fulfilled. As follows fromFig. E. 6 , it starts to be satisfied at the moment t2 = 9.8415; in this case aswe can see from calculations, the constraint reaction force |Λ2| becomes closein value to the friction force between wheels and the road, and the front axlestops moving in side directions.

Fig. E. 6

Fig. E. 7

Fig. E. 8

Appendix E 261

Thus, for t1 < t < t2 the car moves according to the III-rd type, but aftert2 = 9.8415 it returns to the I-st type of motion.

Now suppose that for t1 < t < 14 the driving force is varied by thelaw F1 = 200(t − t2)/(2 − t2) . In this case, according to Fig. E. 7 dynamicconditions (E.16) and (E.17) are satisfied, that is, the restored constraintϕ2 = 0 will be realized further. So, the car is in the I-st type of motion.

In Fig. E. 8 graphs of functions during all the time interval of the carmotion are given.

§ 6. Reasonable choice of quasivelocities

Previously, when studying possible types of the car motion, we had touse different forms of the equations of motion (E.8), (E.20), (E.24), (E.26).This makes certain difficulties, especially when numerically integrating thegiven systems of differential equations with the help of computer. For simi-lar problems with nonholonomic nonretaining constraints N. A. Fufayev [72]suggests to use a single form of Boltzmann-Hamel’s equations. Let us see,how this idea may be applied in the case of using Maggi’s equations in anal-ogous problems. (We notice that for solving similar problems the equationsof motion of nonholonomic systems with variable kinematic structure can beeffective [221]).

Quite different forms of the equations of motion (E.8), (E.20), (E.24),(E.26) were obtained due to the fact that for different types of the car motionnew transition formulas for quasivelocities were chosen every time, or thegeneralized coordinates were used directly to get the Lagrange equations ofthe second kind. Now we shall use the form of Maggi’s equations for all thefour types of motion, the generalized velocities being always expressed interms of quasivelocities by the same formulas (E.5). In these formulas thequasivelocities have a certain physical meaning: v1

∗is the angular velocity

of rotation of the car body, v2∗

and v3∗

are, according to formulas (E.12) and(E.13), the side velocities of the rear and front axles, correspondingly, alignedwith the vectors j and j1. If the nonholonomic constraints (E.1), (E.2) arerealized, quasivelocities v2

∗and v3

∗vanish, and if these constraints turn out to

be nonretaining, then these quasivelocities have real nonzero values (exceptfor the instant stops of axles in their side motion).

For the motion of the I-st type we still use the equation of motion (E.8)(or (E.21)) and the formulas for determination of the generalized reactions(E.10), (E.11).

For the II-nd type of motion, if the constraint ϕ2 = 0 holds, then thegeneralized reaction Λ2 calculated by the formula (E.11) arises. In this casethe equation of motion (E.8) should be completed with the differential equa-tion (E.10), where Λ1 is changed for the projection of the side friction force(−Λfr

1 sign(ϕ1)) acting on the rear axle during its side slipping. It is necessaryto add the constraint equation (E.2) to these differential equations.

262 Appendix E

For the III-rd type of motion Λ1 is calculated in the same way by formula(E.10), and equation (E.11), in which the reaction Λ2 is replaced with theprojection of the side friction force (−Λfr

2 sign(ϕ2)), is added to equation (E.8).The constraint equation (E.1) is added to these differential equations.

Maggi’s equations are linear combinations of the Lagrange equations ofthe second kind, therefore, in order to keep the uniformity of differentialequations and for the IV-th type of motion corresponding to the holonomicproblem, it is convenient to use the form of Maggi’s equations. Eventuallythe equations of motion will take the form (E.8), (E.10), (E.11), where Λ1

and Λ2 are replaced with (−Λfr1 sign(ϕ1)) and (−Λfr

2 sign(ϕ2)).The logic of change from one type of motion to another is the same as

in § 3.

Pay attention that the obtained equations of motion have a singularityat θ = 0. Therefore, the difficulties may occur in calculations, when turningbegins with rectilinear motion. In this case, instead of some possible modifica-tions of the system of differential equations, which we used in the calculationsgiven above, we can advise to change initially to the special system of curve-linear coordinates suggested in works [423].

A P P E N D I X F

CONSIDERATION OF REACTION FORCES OF HOLONOMIC

CONSTRAINTS AS GENERALIZED COORDINATES

IN APPROXIMATE DETERMINATION

OF LOWER FREQUENCIES OF ELASTIC SYSTEMS

A new method for determination of lower frequencies of mechanical sys-

tems consisting of elastic bodies connected to each other is offered. The con-

ditions of connection of bodies are written as holonomic constraints, the reac-

tions of which are considered as generalized coordinates. Therefore the number

of degrees of freedom proves to be equal to the number of constraints.

On the possibility of introducing generalized reaction forces as

Lagrangean coordinates. This Appendix presents a development of themethod suggested in the Chapter VI. The equation of frequencies (6.12) ofthis chapter makes it possible, if necessary, to determine any number of thesystem’s natural frequencies for a reasonably great number N of dynamicallyconsidered oscillation modes of the system elements. However, as a rule, it isnecessary to know only several first frequencies and modes. When calculatingthem one can use the following approximate approach to this problem.

The potential energy of the system consisting of elastic bodies connectedto each other can be represented as a positively defined quadratic form of thegeneralized constraint reactions introduced

Π =1

2

n∑

i,j=1

cijΛiΛj , (F.1)

when considering all the natural vibration modes of the system’s elementsquasi-statically.

Recall that the coefficients of this form are calculated by formulas (5.15),(5.13) of Chapter VI.

In quasistatics the deformed state of all system elements is uniquely deter-mined by setting the quantities Λi, i = 1, n. The given elastic system comesto this state as a result of the fact that its points have obtained displacements,which can be found as linear functions of the reactions Λi, i = 1, n. Hencethe position of all points of the system at the time t is uniquely determinedby setting the quantities Λi, i = 1, n. Therefore, they can be considered asthe generalized Lagrange coordinates; and the kinetic energy of the systemcan be represented in the form

T =1

2

n∑

i,j=1

aijΛiΛj . (F.2)

263

264 Appendix F

Here aij , i, j = 1, n, are some constants, the calculation procedure for whichwill be shown below through a number of examples.

Lagrange’s equations of the second kind corresponding to expressions(F.1) and (F.2) are

n∑

j=1

(aijΛj + cijΛj) = 0 , i = 1, n .

By assuming as in § 5 of Chapter VI

Λi = Λi cos(p t + α) , i = 1, n ,

we come to the following equation of frequencies:

det[cij − p2aij ] = 0 . (F.3)

When calculating the factors aij and cij of this determinant, one need notknow the natural frequencies and natural modes of oscillation of the system’selements. It is essential that these factors can be determined rather simplyfor the bars of variable section too.

Let us start analyzing this approach with solving the problem of approxi-mate determining the first natural frequency and mode of bending oscillationsof the cantilever of variable cross-section.

Bending oscillations of the cantilever of variable cross-section.

Let us assume that at the end x = l the bar is rigidly clamped and that thearea of cross-section and the moment of inertia of this section are definedcorrespondingly as follows:

S(x) = A(ξ)S(l) , J(x) = B(ξ)J(l) , ξ =x

l, 0 ξ 1 . (F.4)

Here A(ξ) and B(ξ) are some prescribed functions. Note that they may bestep functions too.

Let us introduce into consideration the deflection of neutral layer of thecantilever y(x, t). As the bar is rigidly clamped at the end x = l, then

y(l, t) = 0 ,∂y

∂x

∣∣∣∣x=l

= 0 . (F.5)

We shall consider these two conditions as holonomic constraints imposedon the motion of a free bar. The constraint reaction forces are the bendingmoment M = Λ1 and the lateral force Q = Λ2 applied to the end x = l ofthe free bar (see Fig. F. 1).

The motion of the free bar under the action of these forces can be repre-sented as, first, translational motion (motion of the center of mass C), sec-ondly, rotation about the center of mass and, third, bending. This bendingdeformation in quasistatics can be found in the following manner.

Appendix F 265

Fig. F. 1

The acceleration of the center of mass Wc and the angular acceleration ϕ

at the time t are

Wc =Λ2(t)

ρ∫

l

0S(x)dx

, ϕ =Λ1(t) + (l − xc)Λ2(t)

ρ∫

l

0S(x)(xc − x)2dx

. (F.6)

Here ρ is the density, and xc is the coordinate of the center of mass.The intensity of inertia forces caused by translational and rotation motion

of the bar appears as

q(x, t) = −ρ(Wc + ϕ(x − xc))S(x) . (F.7)

The bending moment in section x, corresponding to the load q(x, t), is equalto

M(x, t) =

∫x

0

q(x1, t)(x − x1)dx1 . (F.8)

The deflection caused by the action of the bending moment M(x, t) satisfiesthe equation

EJ(x)∂2y

∂x2= M(x, t) .

This equation in dimensionless variables

y =y

l, ξ =

x

l, L(ξ, t) =

M(x, t)l

EJ(l)(F.9)

takes the form:

B(ξ)∂2y

∂ξ2= L(ξ, t) . (F.10)

Formulas (F.4), (F.6)–(F.9) imply that the dimensionless moment L(ξ, t) isequal to

L(ξ, t) = Λ1(t)f1(ξ) + Λ2(t)f2(ξ) . (F.11)

Here

Λ1(t) =Λ1(t)l

EJ(l), Λ2(t) =

Λ2(t)l2

EJ(l),

f1(ξ) =

∫ξ

0

A(η)(c − η)

a(ξ − η) dη ,

266 Appendix F

f2(ξ) =

∫ξ

0

((c − η)A(η)(1 − c)

a−

A(η)

b

)(ξ − η) dη , (F.12)

a =

∫ 1

0

A(ξ)(c − ξ)2 dξ; b =

∫ 1

0

A(ξ) dξ , c =1

b

∫ 1

0

ξA(ξ) dξ .

Integrating (F.10) and taking into account the constraint equation (F.5) pro-duce

y(ξ, t) =

2∑

k=1

Λk(t)hk(ξ) , hk =

∫ 1

ξ

fk(η)(η − ξ)

B(η)dη . (F.13)

The potential energy of the bar

Π =1

2

∫l

0

M2(x, t)

EJ(x)dx

can be represented by using the formulas (F.4), (F.9), (F.11) as

Π =EJ(l)

2l

2∑

i,j=1

cijΛiΛj , (F.14)

where

cij =

∫ 1

0

fi(ξ)fj(ξ)

B(ξ)dξ .

The kinetic energy of the system

T =ρ

2

∫l

0

S(x)

(∂y

∂t

)2

dx ,

as follows from formulas (F.4), (F.9), (F.13), is

T =1

2ρS(l) l3

2∑

i,j=1

aij˙Λi

˙Λj , aij =

∫ 1

0

A(ξ)hi(ξ)hj(ξ) dξ . (F.15)

Equation (F.3) and expressions (F.14), (F.15) imply that the dimension-less frequencies p∗ related to the required frequencies p as

p = p∗1

l2

√EJ(l)

ρS(l)(F.16)

are the roots of the equation

det [cij − p2∗aij ] = 0 , i, j = 1, 2 . (F.17)

For oscillations with the frequencies pk, k = 1, 2, in accordance with ex-pression (F.13) we obtain:

yk(ξ, t) = (˜Λk1h1(ξ) + ˜Λk2h2(ξ)) cos(pkt + α) , k = 1, 2 .

Appendix F 267

The quantities ˜Λk1,˜Λk2, k = 1, 2, satisfy the equations

(c21 − p2∗k

a21)˜Λk1 + (c22 − p2

∗ka22)

˜Λk2 = 0 , k = 1, 2 .

This yields that the fist two modes of oscillation of the cantilever can beapproximately represented as

Yk(ξ) =Xk(ξ)

Xk(0.5), Xk(ξ) = h1(ξ) −

c12 − p2∗k

a12

c22 − p2∗k

a22h2(ξ) , k = 1, 2 .

The exact solutions for the cantilever of wedge and cone shape were obtainedby Kirchhoff in 1879. These solutions are given in many books, in particular,in the reference book by E. Kamke [421] (Chapter IV, paragraphs 4.22, 4.24).

For the wedge, where

A(ξ) = ξ , B(ξ) = ξ3 ,

natural frequencies p∗ are the roots of the equation

J1(κ)I0(κ) = I1(κ)J0(κ) , κ = 2√

p∗ .

Here J0(κ) and J1(κ) are Bessel’s functions of the first kind, and I0(κ) andI1(κ) are modified Bessel’s functions of the first kind. Natural modes corre-sponding to the natural frequencies p∗ are as follows:

Y (ξ) =X(ξ)

X(0.5), X(ξ) =

J0(κ)I1(κ√

ξ) − I0(κ)J1(κ√

ξ)√

ξ.

In the case of a cone, where

A(ξ) = ξ2 , B(ξ) = ξ4 ,

the equation of frequencies, and functions X(ξ) take the form:

κ(J0(κ)I1(κ) + I0(κ)J1(κ)) = 4J1(κ)I1(κ) ,

X(ξ) =I1(κ)[J1(κ

√

ξ) − κ√

ξ

2 J0(κ√

ξ)]

ξ√

ξ+

J1(κ)[I1(κ√

ξ) − κ√

ξ

2 I0(κ√

ξ)]

ξ√

ξ.

By using the suggested approximate approach we obtain— for the wedge:

p∗1 = 5.3187 , p∗2 = 17.3006 ,

h1(ξ) = 1 −

5ξ

2+ 2ξ2

−

ξ3

2, h2(ξ) =

1

6−

ξ

2+

ξ2

2−

ξ3

6,

— for the cone:p∗1 = 8.73521 , p∗2 = 25.1813 ,

h1(ξ) =7

6− 3ξ +

5ξ2

2−

2ξ3

3, h2(ξ) =

1

6−

ξ

2+

ξ2

2−

ξ3

6.

268 Appendix F

The exact values of the first two frequencies are as follows:— for the wedge: p∗1 = 5.3151 , p∗2 = 15.2072 ,

— for the cone: p∗1 = 8.71926 , p∗2 = 21.1457 .

The second frequency error for the wedge as well as for the cone is greatenough. Therefore this approximate method can be used only for determina-tion of the first frequency and the first mode for the cantilever of variablecross-section.

The first natural modes for the wedge and the cone are shown in Fig. F. 2.Solid curves correspond to the approximate solution, and dashed curves overthem correspond to the exact solution. For visualization of differences be-tween the depicted curves, the deflection at ξ = 1/2 is taken as a unit ofmeasurement for each of them. The cone is a more flexible bar than thewedge and thus the first natural mode of the cone for ξ < 1/2 is locatedhigher than the corresponding curve for the wedge.

For the cone the mass per unit of length decreases while approaching tothe end by the quadratic law, and for the wedge the linear law is applied.For the bar of constant cross-section the mass per unit of length is constant.Pay attention to the following fact. The first frequency error for the cone isequal to 0.2%, for the wedge it is equal to 0.07%. For the bar of constantcross-section we have:— approximately: p∗1 = 3.516035 , p∗2 = 22.7125 ,

— exactly: p∗1 = 3.516015 , p∗2 = 22.0345 .

Thus, the first frequency error makes up only 5.7 · 10−4% . Upon compar-ison of given above errors for the cone, the wedge and the bar of constantcross-section we can expect that for the bar of constant cross-section with themass localized at the end the approximate solution will become practically

6

Y1

5

4

3

2

1

0.2 0.4 0.6 0.8 1 ξ

Fig. F. 2

Appendix F 269

the exact one. Really, in this case we obtain:

A(ξ) = 1 + γδ(ξ) , B(ξ) = 1 , γ =m2

m1.

Here δ(ξ) is the Dirac delta-function, m2 is the load mass, m1 is the bar mass.For γ = 1 we have:

— approximately: p∗1 = 1.5572990 , p∗2 = 16.6203 ,

— exactly: p∗1 = 1.5572976 , p∗2 = 16.2501 .

We see that the first frequency error decreased six times relative to thecase when γ = 0.

For the cantilever with the disk at its end we obtain quite accurate solutionif we consider the presence of disk at the end as the third and the forthholonomic constraints. This system with four degrees of freedom makes itpossible to determine to a rather high accuracy not only the first frequencybut the second and the third ones. So let’s analyze the following problem.

Determination of the lower natural frequencies of bending oscil-

lations of the cantilever of variable cross-section with a disk at its

end. In the rotor dynamics, the urgent problem is accurate determination offirst two critical critical speeds of the cantilever shaft with a disk at its end.We remind that the values of these critical speeds are proportional to naturalfrequencies of the cantilever with a disk. Actually, as for instance in the caseof marine screw (water propeller) or airscrew, there is not a disk at the shaftend but a body of rather complicated shape. There are methods allowing usto determine the moment of inertia of this body relative to the axis that isperpendicular to the shaft axis. Let us assume that this moment is set in theform

I = m2R2,

where m2 = γρlS(l) is mass of the body, and R = rl is its radius of in-ertia. Notice that with given functions A(ξ) and B(ξ) the required naturalfrequencies p∗ will depend on two parameters γ and r.

In the case of the bar of constant cross-section the exact values of thefrequencies p∗ are found from the equation

det

[V (x) + γxU(x) S(x) + γxV (x)

S(x) − γr2x3T (x) T (x) − γr2x3U(x)

]= 0 , x =

√

p∗ . (F.18)

Here

S(x) =1

2(ch x + cos x) , T (x) =

1

2(sh x + sinx) ,

U(x) =1

2(ch x − cos x) , V (x) =

1

2(sh x − sin x)

are the Krylov functions.In approximate determination of the frequencies p∗ we shall consider the

conditions of rigid fixing (F.5) as two holonomic constraints as before. We

270 Appendix F

shall denote now their reaction forces: the bending moment M(t) and lateralforce Q(t) by Λ1(t) and Λ2(t) correspondingly.

The condition that the deflection y(0, t) is equal to the displacement ofmass m2, and the angle of rotation of the bar’s end

ϕ =∂y

∂x

∣∣∣∣x=0

is equal to the angle of the body rotation will be considered as two holonomicconstraints imposed on motion of the free bar. The reaction forces of theseconstraints are the lateral force Λ3(t) and bending moment Λ4(t). They areapplied to the bar at the cross-section x = 0. Positive directions of reactions,applied to the bar are shown in Fig. F. 3.

Formulas (F.6) in this case will take the form:

Wc =Λ2(t) + Λ3(t)

ρ∫

l

0S(x)dx

, ϕ =Λ1(t) − Λ4(t) + (l − xc)Λ2(t) − xcΛ3(t)

ρ∫

l

0S(x)(xc − x)2dx

.

The intensity of inertial forces q(x, t) will be calculated by formula (F.7)as before; formula (F.8) will take the form:

M(x, t) = Λ4(t) + xΛ3(t) +

∫x

0

q(x1, t)(x − x1) dx1 .

When going to dimensionless variables we obtain:

L(ξ, t) =4∑

k=1

Λk(t)fk(ξ) .

Here

Λ1(t) =Λ1(t)l

EJ(l), Λ2(t) =

Λ2(t)l2

EJ(l),

Λ3(t) =Λ3(t)l

2

EJ(l), Λ4(t) =

Λ4(t)l

EJ(l).

The functions f1(ξ) and f2(ξ) are set by formulas (F.12), and the functionsf3(ξ) and f4(ξ) are as follows:

f3(ξ) = ξ +

∫ξ

0

((η − c)cA(η)

a−

A(η)

b

)(ξ − η) dη ,

Fig. F. 3

Appendix F 271

f4(ξ) = 1 +

∫ξ

0

A(η)(η − c)

a(ξ − η) dη .

Formulas (F.13), (F.14), (F.17) remain valid, but their indices i, j and k runnow from 1 to 4.

When calculating the kinetic energy it is necessary to take into accountthe kinetic energy of the disk, therefore the factors aij of determinant (F.17)in this case are as follows:

aij =

∫ 1

0

A(ξ)hi(ξ)hj(ξ) dξ + γhi(0)hj(0) + γr2ϕi(0)ϕj(0) , i, j = 1, 4 .

(F.19)Here

ϕi(ξ) =dhi(ξ)

dξ, i = 1, 4 .

In the case of the bar of constant cross-section the equation (F.18) allowsus to calculate the natural frequencies exactly and so to estimate an error ofthis approximate method.

The radius of inertia for the thin disk R is equal to R1/2, where R1 is theradius of the disk and therefore R1 = 2lr.

If the shaft of radius r1 and the disk of thickness h are made of the samematerial then for r1 = l/20 and h = R1/20 we obtain

γ = 160 r3 . (F.20)

Assuming that γ and r are related to each other with this expression and r

varies within the range from 0 to 1/2, let us follow the change of error for thefirst, second and third frequencies. Upon calculations we obtain the followingvalues for the error in percentage terms (%):

r = 0.000 1.5 · 10−4 0.56 2.67

r = 0.125 3.7 · 10−5 9/5 · 10−2 0.85

r = 0.250 1.4 · 10−6 3.7 · 10−4 0.40

r = 0.500 − 9.0 · 10−6 3.9 · 10−5 0.35

The first column corresponds to the first frequency, the second column cor-responds to the second frequency and the third one corresponds to the thirdfrequency. We see that the higher frequency, the greater error.

For r 0.125 the error for the first frequency is close to the limits of ac-curacy which is provided by the software package "Mathematica 5.2". In thisregard one can say that this method permits to determine the first frequencyexactly. Therefore it may be used both for the rotor dynamics and for testingthe programs for analysis of complicated mechanical systems.

In rotor engineering it is important to have the analytical dependence ofthe first natural frequency on the system’s parameters. This method basedon consideration of four holonomic constraints does not allow us to do that

272 Appendix F

as it leads to the solution of algebraic equation of the fourth order. But ifwe limit ourselves to consideration of only two constraints at the end wherethe disk is located, then the required first frequency will be determined inanalytical form as a root of biquadratic equation.

Let us prove, that this simple solution also makes it possible to find thefirst frequency accurately enough. When getting this solution it is reasonableto measure the coordinate of the bar cross-section not from the free end butfrom the end that is rigidly clamped. Formulas (F.4) and (F.17) remain valid,but now S(l) and J(l) will correspond not to the rigidly clamped end, but tothe place of disk fixation.

The bending moment Λ1(t) and lateral force Λ2(t), applied to the end x =l, are constraint reactions and considered in this problem as the generalizedcoordinates. Their positive directions, as well as the positive direction of themoment M(x, t) applied to the cross-section x, are shown in Fig. F. 4.

The dimensionless bending moment L(ξ, t) introduced by formula (F.9)is equal in this case to

L(ξ, t) = Λ1(t)f1(ξ) + Λ2(t)f2(ξ) ,

f1(ξ) = 1 , f2(ξ) = 1 − ξ , Λ1(t) =Λ1(t)l

EJ(l), Λ2(t) =

Λ2(t)l2

EJ(l).

(F.21)

Expression (F.11), as seen, survives and therefore the potential energy willbe written in the form (F.14).

Integrating equation (F.10) and taking into account that

y(0, t) =∂y

∂ξ

∣∣∣∣ξ=0

= 0 ,

imply expression (F.13), where now

hk(ξ) =

∫ξ

0

fk(η)(η − ξ)

B(η)dη , k = 1, 2 . (F.22)

As the deflection is represented in the same form (F.13), the kinetic energywill be written in the same form (F.15) too. The factors aij in this case should

Fig. F. 4

Appendix F 273

be calculated by formulas (F.19), but now hi(0) should be replaced with hi(1),and ϕi(0) should be replaced with ϕi(1).

When calculating for the bar of constant cross-section the error of thefirst and second frequencies in percentage terms (%) for the same relation(F.20) between γ and r, we obtain:

r = 0.000 0.47 58r = 0.125 8.4 · 10−2 15.6r = 0.250 2.6 · 10−3 0.21r = 0.500 1.1 · 10−5 1.1 · 10−3

For r 0.25 we can say that for the first frequency we obtain the exactvalue. Notice, however that for r = 0.25 the disk diameter is equal to theshaft length, and for r = 0.5 it is two times greater. For such relation betweenthese quantities for the assumed values r1 = l/20 and h = R1/20 this disk cannot be regarded as a perfectly rigid body. It is necessary to take into accountthe influence of its compliance on the natural frequencies of the system. Itis feasible but it will require additional calculations, the basic framework ofwhich will be shown through the example of the cantilever with a flexible barat its end. This example will require no new mathematical apparatus. It isreduced to the same calculations as above.

Determination of the first three frequencies of the cantilever

with a flexible bar at its end. Let us analyze the problem, when thebar executing longitudinal oscillations in the mechanical system depicted inFig. VI. 2 is absent (see Fig. F. 5). Within the frames of such problem wehave three constraints and three reaction forces correspondingly. The bendingmoment Λ1(t) and the lateral force Λ2(t) are applied to the cantilever as isshown in Fig. F. 4. The third reaction force is the lateral force Λ3(t) appliedto the bar which is perpendicular to the cantilever.

Both kinetic and potential energy of the cantilever are determined by theformulas given above. Therefore it is necessary to take into account only thesecond bar. When released from the constraints it becomes free and similarto the bar shown in Fig. F. 1, but now the bending moment M(t) = Λ1(t)and lateral force Q(t) = Λ3(t) are applied not to the end of the bar but tothe cross-section x∗ = zl. Therefore, the constraint equations will be writtenin the form

y(x∗, t) = 0 ,∂y

∂x

∣∣∣∣x=x∗

= 0 . (F.23)

274 Appendix F

We shall not provide the parameters of the second bar with indices whenconsidering the question how the deflection curve will change depending onthe place of application of the reactions. We shall do that upon obtainingexpressions for the potential energy of its deflection and for the deflectioncurve.

Formulas (F.6) in this case will appear as

Wc =Λ3(t)

ρ∫

l

0S(x) dx

, ϕ =Λ1(t) + (x∗ − xc)Λ3(t)

ρ∫

l

0S(x)(xc − x)2dx

,

and formula (F.7) remains valid.The bending moment M(x, t) applied to the left of the cross-section x =

x∗ is set by expression (F.8), and the bending moment applied to the rightof cross-section takes the form

M(x, t) =

∫l

x

q(x1, t)(x1 − x) dx1 , x∗ < x < l .

Hence the bar is divided into two sections and the deflections of its leftand right parts have to be calculated independently. Denoting the bendingmoment M(x, t) for 0 < x < x∗ by M1(x, t), and for x∗ < x < l by M2(x, t),and going to dimensionless variables (F.9), we obtain:

Ln(ξ, t) = Λ(2)1 (t)f1n(ξ) + Λ

(2)3 (t)f3n(ξ) , n = 1, 2 .

Fig. F. 5

Appendix F 275

Here

Λ(2)1 (t) =

Λ1(t)l

EJ(l), Λ

(2)3 (t) =

Λ3(t)l2

EJ(l),

f11(ξ) =

∫ξ

0

(c − η)A(η)

a(ξ − η) dη , 0 ξ z ,

f31(ξ) =

∫ξ

0

((c − η)(z − c)A(η)

a−

A(η)

b

)(ξ − η) dη , 0 ξ z ,

f12(ξ) =

∫ 1

ξ

A(η)(η − c)

a(ξ − η) dη , z ξ 1 ,

f32(ξ) =

∫ 1

ξ

((η − c)(z − c)A(η)

a+

A(η)

b

)(ξ − η) dη , z ξ 1 .

(F.24)We remind that the values a, b, c, included in these expressions are calculated

by formulas (F.12). The index "2"of the quantities Λ(2)1 (t) and Λ

(2)3 (t) means

that transition to the dimensionless variables corresponds to the parametersl, E and J(l) of the second bar (see Fig. 5). The functions A(ξ), B(ξ) andthe values a, b, c, should be also provided with index "2"hereinafter, but forthe sake of simplicity they are omitted.

Integrating equation (F.10) for L(ξ, t) = L1(ξ, t), and then for L(ξ, t) =L2(ξ, t), and taking into account the constraint equations (F.23) imply

y(ξ, t) = Λ(2)1 (t)h11(ξ) + Λ

(2)3 (t)h31(ξ) , 0 ξ z ,

y(ξ, t) = Λ(2)1 (t)h12(ξ) + Λ

(2)3 (t)h32(ξ) , z ξ 1 ,

(F.25)

where

fk1(ξ) =

∫z

ξ

fk1(η)(η − ξ)

B(η)dη , 0 ξ z ,

fk2(ξ) =

∫ξ

z

fk2(η)(ξ − η)

B(η)dη , z ξ 1 , k = 1, 3 .

By using the unit function

U(x) =

1 , x 0 ,

0 , x < 0 ,

we represent expressions (F.25) as

y(ξ, t) = Λ(2)1 (t)h1(ξ) + Λ

(2)3 (t)h3(ξ) , 0 ξ 1 . (F.26)

Herehk(ξ) = hk1(ξ)U(z − ξ) + hk2(ξ)U(ξ − z) . (F.27)

276 Appendix F

The potential energy of deformation of the second bar has to be calculatedindependently for its right and left sections. Calculating and summing theseenergies produce

Π =EJ(l)

2l(c

(2)11 (Λ

(2)1 (t))2 + 2c

(2)13 Λ

(2)1 (t)Λ

(2)3 (t) + c

(2)33 (Λ

(2)3 (t))2) , (F.28)

where

c(2)kk

=

∫z

0

f2k1(ξ)

B(ξ)dξ +

∫ 1

z

f2k2(ξ)

B(ξ)dξ , k = 1, 3 ,

c(2)13 =

∫z

0

f11(ξ)f31(ξ)

B(ξ)dξ +

∫ 1

z

f12(ξ)f32(ξ)

B(ξ)dξ .

Adding the potential energy of bending of the first bar to potential energy(F.28), we represent their sum in the form

Π =E1J1(l1)

2l1

3∑

i,j=1

cijΛ(1)i

Λ(1)j

. (F.29)

Here index "1" means that this quantity corresponds to the first bar. The

dimensionless variables Λ(1)i

, i = 1, 3, are introduced by the formulas:

Λ(1)1 (t) =

Λ1(t)l1E1J1(l1)

, Λ(1)k

(t) =Λk(t)l21E1J1(l1)

, k = 2, 3 .

Note that in these formulas J1(l1) corresponds not to the place of rigid fixing,as it was in the beginning of this Appendix, but to the point where the firstbar is connected to the second one (see Fig. F 5).

In formulas (F.24) and (F.28) all quantities refer to the second bar. In-troduction of the parameters

α =E1J1(l1)l

32

E2J2(l2)l31, β =

l2

l1

allows us to represent the potential energy (F.28) of the second bar as

Π2 = αE1J1(l)

2l1(c

(2)11 (Λ

(1)1 (t))2β−2 + 2c

(2)13 Λ

(1)1 (t)Λ

(1)3 (t)β−1 + c

(2)33 (Λ

(1)3 (t))2) .

This implies that the factors cij in expression (F.29) are as follows:

c11 = c(1)11 + αβ−2c

(2)11 , c12 = c

(1)12 ,

c13 = αβ−1c(2)13 , c22 = c

(1)22 , c23 = 0 , c33 = αc

(2)33 .

Here in accordance with formulas (F.14), (F.21)

c(1)11 =

∫ 1

0

dξ

B1(ξ), c

(1)12 =

∫ 1

0

(1 − ξ) dξ

B1(ξ), c

(1)22 =

∫ 1

0

(1 − ξ)2dξ

B1(ξ).

Appendix F 277

The kinetic energy of the first bar will be represented by using expressions(F.15), (F.21), (F.22) in the form

T1 =1

2ρ1S1(l1) l31

2∑

i,j=1

a(1)ij

˙Λ(1)i

˙Λ(1)j

,

a(1)ij

=

∫ 1

0

A1(ξ)h(1)i

(ξ)h(1)j

(ξ) dξ ,

h(1)1 (ξ) =

∫ξ

0

(ξ − η) dη

B1(η), h

(1)2 (ξ) =

∫ξ

0

(1 − η)(ξ − η) dη

B1(η).

Let us calculate the kinetic energy of the second bar now. The assump-tion that amplitude of oscillations of the bars under consideration is smallallows us, as was noted in § 2 of Chapter VI, to calculate the kinetic energyof translational motion of the second bar along the axis independently fromthe kinetic energy of its motion in the direction that is perpendicular to itsaxis.

The kinetic energy of translational motion of the second bar is

T21 =m2l

21

2(h

(1)1 (1) ˙Λ

(1)1 + h

(1)2 (1) ˙Λ

(1)2 )2 ,

m2 = ρ2S2(l2) l2

∫ 1

0

A2(ξ) dξ .

Displacements of the cross-sections of the second bar in the directionperpendicular to the bar axis are caused, first, by rotation of the bar about thecross-section x∗ = zl2, and, secondly, by the deflection defined by expression(F.26). Therefore we have:

y2(ξ, t) = l2(ψ(t)(z − ξ) + Λ(2)1 (t)h

(2)1 (ξ) + Λ

(2)3 (t)h

(2)3 (ξ)) .

Here

ψ(t) = ϕ1(1)Λ(1)1 (t) + ϕ2(1)Λ

(1)2 (t) , ϕk(1) =

dh(1)k

dξ

∣∣∣∣ξ=1

, k = 1, 2 .

Index "2" of the functions h(2)1 (ξ) and h

(2)3 (ξ) means that these functions

defined by expressions (F.27) are calculated for the parameters of the secondbar.

Taking into account that

Λ(2)1 = αβ−2Λ

(1)1 , Λ

(2)3 = αβ−1Λ

(1)3 ,

the kinetic energy

T22 =1

2ρ2

∫l2

0

S2(x)

(∂y2

∂t

)2

dx ,

278 Appendix F

will be represented as

T22 =1

2ρ2S2(l2)l

32

∫ 1

0

A2(ξ)((ϕ1(1) ˙Λ(1)1 + ϕ2(1) ˙Λ

(1)2 )(z − ξ)+

+αβ−2 ˙Λ(1)1 h

(2)1 (ξ) + αβ−1 ˙Λ

(1)3 h

(2)3 (ξ))2dξ .

Introducing into consideration the third parameter

γ =ρ2S2(l2)l2ρ1S1(l1)l1

,

the total kinetic energy of the second bar appears as follows:

T2 =γ

2ρ1S1(l1) l31

3∑

i,j=1

a(2)ij

˙Λ(1)i

˙Λ(1)j

.

Analytic expressions for the factors a(2)ij

, dependent on the functions A2(ξ)and parameters α and β, are rather intricate and thus not given here. Notethat they are easily found with the software package "Mathematica 5.2".

The kinetic energy of both bars is

T =1

2ρ1S1(l1) l31

3∑

i,j=1

aij˙Λ(1)i

˙Λ(1)j

, aij = a(1)ij

+ γa(2)ij

.

We’ll find the required natural frequencies p∗ by solving equation (F.17).Notice that in formula (F.16) of transition to dimensional frequencies allquantities correspond to the first bar at the point of its connection to thesecond bar.

Comparison with the bars of constant cross-section. The problemof bars of constant cross-section has been solved exactly by the methodsof mathematical physics. As this takes place the equation of frequencies isobtained by equating the determinant of sixth order to zero. Its elements arethe Krylov functions, the arguments of which depend on the parameters α, γ

and z. This intricate transcendental equation, the computational solution ofwhich was a matter of some difficulty even for modern computers, was usedfor testing the method suggested in § 3 of Chapter VI. Note that calculationof first three frequencies by using this suggested method does not create anydifficulties.

As stated above, the problem under investigation is a particular case ofthe problem discussed in §§ 5 and 6 of Chapter VI. When the bar executinglongitudinal vibration is absent, the frequency determinant is a determinantof third order. Comparing the roots of the transcendental equation with theroots of the frequency equation shows that the first frequency is determinedwith four valid significant digits in the second approximation, the second fre-quency is determined with the same accuracy in the fourth approximation,

Appendix F 279

and the third one is obtained with the same accuracy in the sixth approxi-mation.

If the first and the second bars are made of the same material and have thesame cross-sections, then at z = 1/2 the solution depends on a single parame-ter β = l2/l1, for in this case γ = β and α = β3. The calculations show that forthe first frequency the error decreases as β rises, and for β = 1/8, 1/4, 1/2, 2it is equal to 0.22, 0.12, 0.056, 0.0022 percent (%) correspondingly. Note thatin this example in case β 0.25, it is reasonable to consider the second baras a concentrated mass located at the end of the cantilever beam and to usethe method presented in the beginning of this Appendix.

Let us discuss briefly the errors of the method under consideration forthe second and third frequencies. Let us examine this problem through theexample of the bars, differing only in length.

For α = β = γ = 1 and z = 1/2 the exact and approximate values of firstthree dimensionless frequencies p∗ are as follows:

1.44851 , 6.20782 , 14.0641 ,

1.44876 , 6.24235 , 14.1204 .

The errors in percentage terms (%) are equal correspondingly to

0.017 , 0.56 , 0.40 .

If the second bar is symmetrically positioned in relation to the first one,there exists a mode of oscillations such that the first bar does not oscillate,and both halves of the second bar oscillate like a cantilever of length l = βl1/2.The first frequency of the cantilever oscillation in the dimensionless variablesis

p∗ = 3.5164

β2. (F.30)

This frequency in the series of frequencies of the system consisting of twobars has the number n. This number increases as β decreases. For example,for β = 1/4 it will be the ninth frequency, and the third root of equation(F.17) will correspond to it. Let us find this root in the explicit form.

When the second bar does not oscillate, then the bending moment Λ1 andthe lateral force Λ2 applied to the end of the first bar vanish. Therefore inthis oscillation mode only the lateral force Λ3 applied to the middle of thesecond bar is not equal to zero. Under the action of this force the second barmoves translationally and bends so that the application point of the forceΛ3 is immovable. In quasistatics the intensity of inertial forces is constant inthis case, therefore either the second or the third root of equation (F.17) atz = 1/2 is equal to

p∗ =

√c

a

4

β2, c =

∫ 1

0

f2(ξ) dξ , f(ξ) =

∫ξ

0

(ξ − η) dη ,

a =

∫ 1

0

h2(ξ) dξ , h(ξ) =

∫ 1

ξ

f(η) (ξ − η) dη .

280 Appendix F

When calculating we obtain

p∗ = 3.5304

β2. (F.31)

This frequency exceeds its exact value obtained by formula (F.30) by 0.40%.For β = 1/2 the frequency approximately defined by expression (F.31)

corresponds to the exact value of the fourth frequency, for β = 1 and β = 2it corresponds to the third frequency, and for β = 4 it corresponds to thesecond frequency.

Hence this approximate method makes it possible to determine the firstfrequency for any values of the system parameters with a rather high degreeof accuracy, and for some values of the parameters it allows us to define thesecond and the third frequencies as well.

A P P E N D I X G

THE DUFFING EQUATION AND

STRANGE ATTRACTOR

The nonhomogeneous Duffing equation with a linear resistance is studied.

In this case the possibility of arising of strange attractors and periodical so-

lutions with a period multiple to the period of excitation, depending on the

excitation level, is investigated by a numerical method. The Appendix presents

the first part of the paper by P. E.Tovstik and T.M.Tovstik [425]. The prob-

ability properties of a strange attractor considered in the second part of this

paper are not covered in the Appendix. A more completed table of solution

properties is given. The relationship of strange attractors with the classical

theory of motion stability is presented in the monograph by G.A. Leonov [426].

In § 4 of Chapter VI the nonhomogeneous Duffing equation (4.9) has beenobtained for describing the lateral vibration of a beam with the supports fixedin the logitudinal directiion. Taking into consideration a linear resistance andsome changes in notation, in dimensionless variables it appears as

d2x

dt2+ c

dx

dt+ x + x3 = b cos ωt , (G.1)

where

ω =(π

L

)2

√EJ

ρSΩ, b =

2L4

π4E√

JSf0 .

Here Ω and f0 are the frequency and the amlitude of the disturbed force,respectively, c is a coefficient characterizing damping, E and ρ are the coeffi-cient of elasticity and density of beam material, respectively, L is the lengthof beam, S is the cross-section area of the beam, J is the moment of inertiaof cross section of the beam with respect to zero line.

Equation (G.1) includes three parameters — c, b, and ω. Let us fix two ofthem (ω = 1, c = 0.25) and vary parameter b in a wide range 0 b 100.

Equation (G.1) has been integrated numerically under the arbitrary giv-en initial conditions x(0), x(0) belonging to the domain of ([−5.0, 5.0] ×[−5.0, 5.0]). It has been determined, how the existance of strange attrac-tors and limiting solutions with a period multiple to the period of exci-tation depends on the value of b and the initial conditions. To this end,the range 0 b 100 has been partitioned with a step 0.1, and for eachbi = 0.1i the qualitative character of a limiting trajectory has been defined.The neighbour values of bi with the same qualitative characteristics havebeen united in intervals. When subsequently varying i from 0 up to 1000,55 intervals with different qualitative behaviour of solutions have been foundin total.

281

282 Appendix G

Таблица G. 1.

b n k b n k

0.0 − 2.9 1 1 52.1 3 A, 2, 23.0 − 9.6 2 1, 1 52.2 − 52.7 2 2, 29.7 − 11.9 1 1 52.8 − 53.1 2 4, 412.0 − 14.8 2 1, 1 53.2 2 A,A

14.9 − 22.9 1 1 53.3 − 54.3 1 A

23.0 − 35.5 2 1, 1 54.3 − 54.7 2 A, 135.6 − 38.6 2 2, 2 54.8 − 54.9 3 A, 5, 138.7 − 38.9 2 4, 4 55.0 8 A, 15, 15, 15, 10, 10, 5, 1

39.0 4 4, 4, 3, 3 55.1 − 57.9 2 A, 139.1 − 39.2 2 4, 4 58.0 1 139.3 − 39.39 2 8, 8 58.1 − 58.2 2 A, 1

39.4 4 A,A, 8, 8 58.3 3 A,A, 139.5 2 A,A 58.4 3 4, 4, 139.6 4 A,A, 10, 10 58.5 − 58.7 3 2, 2, 1

39.7 − 40.6 2 A,A 58.8 4 3, 3, 2, 140.7 − 41.2 1 A 58.9 − 59.0 3 2, 2, 1

41.3 3 A, 5, 5 59.1 − 60.8 3 1, 1, 141.4 − 41.6 1 A 60.9 − 62.0 2 1, 141.7 − 44.3 2 A, 3 62.1 3 A, 1, 144.4 − 48.6 1 3 62.2 − 62.3 4 3, 3, 1, 148.7 − 49.1 2 3, 3 62.4 − 62.7 3 3, 1, 1

49.2 2 6, 6 62.8 − 67.4 2 1, 149.3 3 A, 6, 6 67.5 − 77.3 1 149.4 2 A,A 77.4 − 91.5 2 1, 1

49.5 − 50.6 1 A 91.6 4 1, 1, 1, 150.7 2 4, 4 91.7 − 92.7 3 1, 1, 150.8 2 8, 8 92.8 − 100.0 2 1, 1

50.9 − 52.0 1 A

Table G. 1 contains the results. In it, for corresponding values of b thereare:— the number n of different limiting solutions, which can be obtained whenchanging the initial conditions,— the multiplicity k of a period kT of the limiting solution, the number ofvalues of k given through a comma being equal to n,— in the case, when a periodical solution is not kT -periodical, but a strangeattractor, the number k is substituted in Table G. 1 for the letter A.

For example, for b = 54.9 there are three (n = 3) different stable limitingsolutions: a strange attractor, a 5T -periodical solution, and a T -periodicalsolution. If we compare this table with the results obtained in the paper[425], then we see that this table only complements them. A greater number

Appendix G 283

of intervals b has been got as a result of decreasing a partition step. Thisdemonstrates that a more detailed study of the variation interval of b can leadto arising of new intervals that are qualitatively different from the consideredones.

It follows from the table, that strange attractors occur in the range 39.4

b 62.1. Note that for b > 100 strange attractors also occur, but in this casethe vibration amplitude is so great that this vibration can hardly be modelledby the Duffing equation.

We illustrate the dependence of limiting solutions on the initial conditionsthrough the example b = 4.0. As follows from Table G. 1, for this value ofb there are two stable T -periodical solutions. In Fig. G. 1 on the part of theplane −5 x(0), x(0) 5 the domains of initial conditions that lead to thefirst or second solutions are given. These solutions are presented in Fig. G. 2.The part of the plane −50 x(0), x(0) 50, which is a hundred timesgreater, has been also regarded, in this case it turned out that the structureof the domain is more complicated than in Fig. G. 1.

It is impossible to describe here all the considered variation range of b.We limit ourselves to four consecutive intervals in the range 40.0 b 45.0and regard four successive values of b. The results are given in the form ofthe Poincare diagrams (see Fig. G. 3), on which points with the coordinatesx(mT ), x(mT ) for integer m are dotted. The Poincare diagrams (or cross-sections) are the powerful means, which make it possible to determine thequalitative character of solution behaviour and find out bifurcations, i. e. thetransfers from one qualitative state to another.

Fig.G. 1

284 Appendix G

2 2

x

11

tπ 2π

–1

–2

Fig.G. 2

Fig.G. 3

Appendix G 285

If a limiting solution is kT -periodical, then a diagram has k different dots.The number of dots for a strange attractor depends on duration of integration.In Fig. G. 3 each of the strange attractors contains 800 dots.

Consider consecutively the diagrams shown in Fig. G. 3. On the first ofthem (for b = 40.0) two strange attractors 1 and 2 are depicted, which can beobtained under certain initial conditions. As b increases, attractors approachto each other, and for b = 41.0 there is only one attractor under any initialconditions. For b = 43.0 we have two stable limiting solutions: a strangeattractor 1 and a 3T -periodical solution 2 depicted by three dots on thePoincare diagram. As b grows further, a strange attractor vanishes, and thereremains only one 3T -periodical solution depicted by three dots in Fig. G. 3for b = 45.0.

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I N D E X

Abstract constraints, 151, 152Acceleration vector for arbitrary

mechanical system, 107Amplitude-frequency

characteristics, 164Andronov–Hopf bifurcation, 229Appell function, 200Appell’s equations, 101, 123Appell’s form equations with

third-order constraints, 123Approximate periodic

solutions, 236Approximate solution of Lagrange’s

equations, 237Axes of curvilinear coordinates, 214

Basic metric form, 217Basic metric tensor, 217Basis of the Lie algebra, 200Basis of s-dimensional Lie algebra

with the commutator, 199Bending oscillations of the cantilever

of variable cross-section, 264Bubnov–Galerkin method, 237

Chaplygin’s equations, 30, 39in quasicoordinates, 43

Chaplygin’s type equations, 32,42, 197

Chetaev’s postulate, 76, 92–93Chetaev’s type constraints, 76Christoffel symbols

of first kind, 219of second kind, 219

Coefficients of influence, 156Complementary metric form, 217Complementary metric tensor, 217Condition of a free motion of the

Chaplygin sledge, 242Condition of the ideality of

constraints, 110Conditions of N. G. Chetaev, 202,

203, 207Configuration space of the system,

230Constraints completely defined by their

analytic representations, 9

Contravariant components, 215, 216of tangent vector, 105of velocity vector, 218

Coordinateline, 213plane, 215surface, 213

Covariant components, 216of velocity vector, 218

Curves of static bend (deflection), 158Curvilinear coordinates, 213Cylindrical system of coordinates, 222

D’Alembert–Lagrange principle, 13, 207Dynamic compliances, 156Dynamic control of the motion of a car,

245Dynamic Euler equations, 200

Effective potential, 230Elastic constraints, 165, 167, 168Equations of motion, 214

of nonholonomic systems in quasico-ordinates, 200

in quasicoordinates, 211Equations of noncomplete program

of motion, 96Equations of nonholonomic systems in

the Poincare–Chetaevvariables, 200

Equations, represented in Maggi’sform, for third-orderconstraints, 121

Euclidean structure of the tangentspace, 106

Formula for computing the Christoffelcoefficients of the firstkind, 219

Free (unconstrained) motion ofnonholonomic system, 239

Gaussian function, 116Gaussian principle, 103, 109

generalized, 119, 183General (fundamental) equation of

dynamics, 110–111

327

328 Index

Generalized control force, 127Generalized D’Alembert–Lagrange

principle, 202Generalized forces, 6

corresponding toquasivelocities, 94

Generalized impulses, 7Generalized operator

Appell, 114Lagrange, 114

Generalized problem ofP.L. Chebyshev, 126

Generalized reactions, 82, 150Gradient of the function, 215

Hamel–Boltzmann equations, 31, 44Hamel–Novoselov equations, 32,

43, 197Hamiltonian nabla operator, 216Harmonic coefficients of

influence, 156High-order program

constraints, 135

Ideal constraintsholonomic, 3, 4, 81nonholonomic, 29, 81

Ideal control, 134Ideality condition of

control, 135Introducing generalized reaction

forces as Lagrangeancoordinates, 263

Kinematical characteristics, 194Kinematic control of the motion

of a car, 245Kronecker symbols, 215

Lagrange multipliers, 82, 150Lagrange operator, 6, 222Lagrange’s equations

of first kind, 4, 7of first kind in generalized

coordinates, 11for nonholonomic systems, 33

of second kind with multipliers, 11undetermined, 33

Lame factors, 214Linear transformation of forces, 94

Maggi’s equations, 30, 201second group, 30, 101

Mangeron–Deleanu principle, 135Manifold of positions of the mechanical

system, 105Maximum principle of

Pontryagin, 185Metric tensor, 106Mixed problem of dynamics, 128Motion of dynamically symmetric ball

on absolutely roughened plane,233

Natural (fundamental) basis, 215Necessary and sufficient conditions for

the existence of free motion ofnonholonomicsystems, 240

New class of control problems, 128Newton’s determinacy principle, 136Noncomplete program of motion, 100Nonholonomic bases, 28Nonlinear second-order nonholonomic

constraints, 116Normal (natural) forms (modes) of

oscillations, 153Normal (natural) frequency, 153

Objects of nonholonomicity, 31

Parametrization of constraints, 195Permutable relations, 206, 207Poincare–Chetaev equations, 34Poincare–Chetaev parameters, 202Poincare–Chetaev–Rumyantsev

equations, 44, 201, 206Poincare diagrams, 283Poincare equations, 44, 200Poincare parameters, 194Possible types of the car motion, 251Principle of virtual accelerations, 110Principle of virtual displacements, 111Principle of virtual velocities, 110Program constraints, 116, 119, 135

Quasicoordinates, 41Quasivelocities, 41

Reciprocal (dual) basis, 107, 215Representation point, 1, 2

Index 329

Residual, 235Rule of dummy index, 216Rules of raising and missing an

index, 217

Series in resonancefrequencies, 186

Set, bifurcational by Smale,231

Steady motions, 231of conservative nonholonomic

systems, 229Steady rolling of disk on horizontal

plane, 233Strange attractors, 281Structural constants of Lie

algebra, 199Subspace

of motions, 8of reactions, 8

Suslov–Jourdain principle, 68

Tangent space, 106

Udwadia–Kalaba equations, 34

Variation of the generalizedvelocity, 67

Variations of coordinates, 12, 106Vector of generalized impulse, 113Velocity vector of mechanical

system, 113Virtual displacements, 12,

106, 112Virtual elementary work, 7,

84, 106Virtual velocity, 112Voronets equations, 40Voronets–Hamel coefficients of first

kind, 198Voronets–Hamel equations, 44Voronets–Hamel type

equations, 198