Transformation Groups and Lie Algebras

Transformation Groups and Lie Algebras 变换群和李代数

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NONLINEAR PHYSICAL SCIENCE

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NONLINEAR PHYSICAL SCIENCENonlinear Physical Science focuses on recent advances of fundamental theories andprinciples, analytical and symbolic approaches, as well ascomputational techniquesin nonlinear physical science and nonlinear mathematics with engineering applica-tions.

Topics of interest inNonlinear Physical Science include but are not limited to:

- New findings and discoveries in nonlinear physics and mathematics- Nonlinearity, complexity and mathematical structures innonlinear physics- Nonlinear phenomena and observations in nature and engineering- Computational methods and theories in complex systems- Lie group analysis, new theories and principles in mathematical modeling- Stability, bifurcation, chaos and fractals in physical science and engineering- Nonlinear chemical and biological physics- Discontinuity, synchronization and natural complexity in the physical sciences

SERIES EDITORS

Albert C.J. LuoDepartment of Mechanical and Industrial

Engineering

Southern Illinois University Edwardsville

Edwardsville, IL 62026-1805, USA

Email: [email protected]

Nail H. IbragimovDepartment of Mathematics and Science

Blekinge Institute of Technology

S-371 79 Karlskrona, Sweden

Email: [email protected]

INTERNATIONAL ADVISORY BOARDPing Ao, University of Washington, USA; Email: [email protected]

Jan Awrejcewicz, The Technical University of Lodz, Poland; Email: [email protected] Benilov, University of Limerick, Ireland; Email; [email protected]

Eshel Ben-Jacob, Tel Aviv University, Israel; Email: [email protected] Courbage, Universite Paris 7, France; Email: [email protected] Gidea, Northeastern Illinois University, USA; Email: [email protected]

James A. Glazier, Indiana University, USA; Email: [email protected] Liao, Shanghai Jiaotong University, China; Email: [email protected] Antonio Tenreiro Machado, ISEP-Institute of Engineering of Porto, Portugal; Email: [email protected]

Nikolai A. Magnitskii, Russian Academy of Sciences, Russia; Email: [email protected] J. Masdemont, Universitat Politecnica de Catalunya (UPC), Spain; Email:[email protected] E. Pelinovsky, McMaster University, Canada; Email: [email protected]

Sergey Prants, V.I.Il’ichev Pacific Oceanological Institute of the Russian Academy of Sciences. Russia;Email: [email protected] I. Shrira, Keele University, UK; Email: [email protected]

Jian Qiao Sun, University of California, USA; Email: [email protected] Wazwaz, Saint Xavier University, USA; Email: [email protected]

Pei Yu, The University of Western Ontario, Canada; Email: [email protected]

Nail H. Ibragimov

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Preface

The termtransformation grouprefers to the following properties of a collectionGof invertible transformations ¯x = T(x) of certain objectsx :

1◦. G contains the identity transformationI .2◦. G contains the inverseT−1 of anyT ∈ G.3◦. G contains the productT2T1 of anyT1,T2 ∈ G.

Note that the identity transformationI is defined by the equationI(x) = x. TheproductT2T1 is defined as a successive action ofT1 andT2, i.e.

(T2T1

)(x) = T2

(T1(x)

).

Finally, the inverseT−1 is defined by the equationsT−1T = TT−1 = I .The group property ofG is closely connected with theinvarianceof sets of the

objectsx under the transformationsT ∈ G. We can formulate the statement in thefollowing form.

Proposition. Let S be a set of objectsx andG be the collection of all invertibletransformationsT defined onSand mapping anyx∈ S into T(x) = x∈ S. ThenG isa group.

Proof. Let us verify that the group properties 1◦−3◦ hold. The validity of the prop-erty 1◦ is obvious becausex ∈ S implies I(x) = x ∈ S. Hence,I ∈ G. Furthermore,T(x) = x∈ Simplies thatT−1(x) = x∈ S, and henceT−1 ∈G, i.e. the property 2◦ isalso satisfied. Finally, to verify the property 3◦, we note that ifT1,T2 ∈ G, then theactionT2

(T1(x)

)is defined becauseT1(x) ∈ S, andT2

(T1(x)

)∈ SbecauseT2 maps

any element ofS into an element ofS. Hence,T1,T2 ∈ G. This completes the proof.In particular, ifx denotes a solution of a given differential equationF = 0 and

S is the totality of the solutions ofF = 0, then the above statement shows that thecollection of all transformations mapping any solution ofF = 0 into a solution ofthe same differential equation compose a group. It is calledthegroup admitted bythe differential equation,or thesymmetry groupof the equation in question.

Part I of these notes introduces the reader to the basic concepts of the classicaltheory of local transformation groups and their Lie algebras. It has been designedfor the graduate course onTransformation groups and Lie algebrasthat I have beenteaching at Blekinge Institute of Technology, Karlskrona,Sweden, since 2002. The

v

vi Preface

aim of this course was to augment a preliminary knowledge on symmetries of differ-ential equations obtained by students during the courseDifferential equationsbasedon my book [17],A practical course in differential equations and mathematicalmodelling.

Part II of these notes provides an easy to follow introduction to the new topic.It is based on my talks at various conferences, in particularon the plenary lectureat the International Workshop on “Differential equations and chaos” (University ofWitwatersrand, Johannesburg, South Africa, January 1996). The final form of thepresentation of this material, used in the present book, wasprepared for my lec-tures “Approximate transformation groups” delivered for MSc students at BlekingeInstitute of Technology since 2009.

Each part of the book contains an Assignment provided by detailed solutionsof all problems. I hope that these assignments will be usefulboth for students andteachers.

Nail H. Ibragimov

Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . v

Part I Local Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Changes of frames of reference and point transformations . . . . . . . . . 3

1.1.1 Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 31.1.2 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 31.1.3 Galilean transformation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 4

1.2 Introduction of transformation groups . . . . . . . . . . . . . .. . . . . . . . . . . 51.2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 51.2.2 Different types of groups . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 10

1.3 Some useful groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 131.3.1 Finite continuous groups on the straight line . . . . . . .. . . . . . . 131.3.2 Groups on the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 141.3.3 Groups in IRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Exercises to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 21

2 One-parameter groups and their invariants . . . . . . . . . . . . . . . . . . . . . . . 232.1 Local groups of transformations . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 23

2.1.1 Notation and definition . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 232.1.2 Groups written in a canonical parameter . . . . . . . . . . . .. . . . . 252.1.3 Infinitesimal transformations and generators . . . . . .. . . . . . . . 252.1.4 Lie equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 272.1.5 Exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 292.1.6 Determination of a canonical parameter . . . . . . . . . . . .. . . . . 32

2.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 342.2.1 Definition and infinitesimal test . . . . . . . . . . . . . . . . . .. . . . . . 342.2.2 Canonical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 362.2.3 Construction of groups using canonical variables . . .. . . . . . . 38

vii

viii Contents

2.2.4 Frequently used groups in the plane . . . . . . . . . . . . . . . .. . . . . 402.3 Invariant equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 41

2.3.1 Definition and infinitesimal test . . . . . . . . . . . . . . . . . .. . . . . . 412.3.2 Invariant representation of invariant manifolds . . .. . . . . . . . . 432.3.3 Proof of Theorem 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 442.3.4 Examples on Theorem 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 45


3 Groups admitted by differential equations . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 51

3.1.1 Differential variables and functions . . . . . . . . . . . . .. . . . . . . . 513.1.2 Point transformations . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 533.1.3 Frame of differential equations . . . . . . . . . . . . . . . . . .. . . . . . . 53

3.2 Prolongation of group transformations . . . . . . . . . . . . . .. . . . . . . . . . . 543.2.1 One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 543.2.2 Prolongation with several differential variables . .. . . . . . . . . 553.2.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 56

3.3 Prolongation of group generators . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 563.3.1 One-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 563.3.2 Several differential variables . . . . . . . . . . . . . . . . . .. . . . . . . . . 593.3.3 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 60

3.4 First definition of symmetry groups . . . . . . . . . . . . . . . . . .. . . . . . . . . 623.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 623.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 62

3.5 Second definition of symmetry groups . . . . . . . . . . . . . . . . .. . . . . . . . 673.5.1 Definition and determining equations . . . . . . . . . . . . . .. . . . . 673.5.2 Determining equation for second-order ODEs . . . . . . . .. . . . 683.5.3 Examples on solution of determining equations . . . . . .. . . . . 68


4 Lie algebras of operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 75

4.1.1 Commutator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 754.1.2 Properties of the commutator . . . . . . . . . . . . . . . . . . . . .. . . . . 774.1.3 Properties of determining equations . . . . . . . . . . . . . .. . . . . . . 794.1.4 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 80

4.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 814.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 814.2.2 Subalgebra and ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 814.2.3 Derived algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 824.2.4 Solvable Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 83

4.3 Isomorphism and similarity . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 844.3.1 Isomorphic Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 844.3.2 Similar Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 86

4.4 Low-dimensional Lie algebras . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 88

Contents ix

4.4.1 One-dimensional algebras . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 884.4.2 Two-dimensional algebras in the plane . . . . . . . . . . . . .. . . . . 894.4.3 Three-dimensional algebras in the plane . . . . . . . . . . .. . . . . . 974.4.4 Three-dimensional algebras in IR3 . . . . . . . . . . . . . . . . . . . . . . 99

4.5 Lie algebras and multi-parameter groups . . . . . . . . . . . . .. . . . . . . . . . 1014.5.1 Definition of multi-parameter groups . . . . . . . . . . . . . .. . . . . . 1014.5.2 Construction of multi-parameter groups . . . . . . . . . . .. . . . . . 102


5 Galois groups via symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1075.2 Symmetries of algebraic equations . . . . . . . . . . . . . . . . . .. . . . . . . . . . 108

5.2.1 Determining equation . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1085.2.2 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1095.2.3 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1115.2.4 Third example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 112

5.3 Construction of Galois groups . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 1135.3.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1135.3.2 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1145.3.3 Third example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1155.3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 116

Assignment to Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Part II Approximate Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . 125

6 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 1276.2 A sketch on Lie transformation groups . . . . . . . . . . . . . . . .. . . . . . . . . 129

6.2.1 One-parameter transformation groups . . . . . . . . . . . . .. . . . . . 1296.2.2 Canonical parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1306.2.3 Group generator and Lie equations . . . . . . . . . . . . . . . . .. . . . . 1316.2.4 Exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 133

6.3 Approximate Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 1346.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1346.3.2 Definition of the approximate Cauchy problem . . . . . . . .. . . 136

7 Approximate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.1 Approximate transformations defined . . . . . . . . . . . . . . . .. . . . . . . . . . 1397.2 Approximate one-parameter groups . . . . . . . . . . . . . . . . . .. . . . . . . . . 140

7.2.1 Introductory remark . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1407.2.2 Definition of one-parameter approximate

transformation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1407.2.3 Generator of approximate transformation group . . . . .. . . . . . 141

7.3 Infinitesimal description . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 1427.3.1 Approximate Lie equations . . . . . . . . . . . . . . . . . . . . . . .. . . . . 142

x Contents

7.3.2 Approximate exponential map . . . . . . . . . . . . . . . . . . . . .. . . . 146Exercises to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 150

8 Approximate symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.1 Definition of approximate symmetries . . . . . . . . . . . . . . . .. . . . . . . . . 1518.2 Calculation of approximate symmetries . . . . . . . . . . . . . .. . . . . . . . . . 152

8.2.1 Determining equations . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 1528.2.2 Stable symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1528.2.3 Algorithm for calculation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 153

8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 1548.3.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1548.3.2 Approximate commutator and Lie algebras . . . . . . . . . . .. . . . 1558.3.3 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 1568.3.4 Third example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 157


9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1619.1 Integration of equations with a small parameter using

approximate symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 1619.1.1 Equation having no exact point symmetries . . . . . . . . . .. . . . 1619.1.2 Utilization of stable symmetries . . . . . . . . . . . . . . . . .. . . . . . . 162

9.2 Approximately invariant solutions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 1669.2.1 Nonlinear wave equation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 1669.2.2 Approximate travelling waves of KdV equation . . . . . . .. . . . 170

9.3 Approximate conservation laws . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 172Exercises to Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 174

Assignment to Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 181

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 183

Part ILocal Transformation Groups

Calculations show that groups admitted by differential equations involve one ormore parameters and depend continuously on these parameters. This circumstanceled Lie to the concept ofcontinuous transformation groups.Multi-parameter contin-uous transformation groups are composed byone-parameter groupsdepending ona single continuous parameter. Each one-parameter group isdetermined by itsin-finitesimal transformationor the corresponding first-order linear differential opera-tor termed thegeneratorof the one-parameter group. One-parameter transformationgroups and their generators are connected by means of the so-calledLie equations.Since the existence of solutions of the Lie equations is guaranteed, in general, onlyfor values of the group parameter in a small neighborhood of its initial value, onearrives at what is calledlocal groupsof continuous transformations.

The generators of multi-parameter transformation groups form specific linearspaces known asLie algebras.Description of continuous transformation groups interms of their Lie algebras simplifies the calculation and use of groups admittedby differential equations significantly. Namely, the generators of continuous groupsadmitted by a given differential equation are defined by solving an over-determinedsystem of linear differential equations known asdetermining equations.The charac-teristic property of determining equations is thatthe totality of their solutions spansa Lie algebra.

Due to the fundamental role of one-parameter groups in Lie’stheory of contin-uous groups, it is natural to begin the study of the general theory of transforma-tion groups and symmetries of differential equations by considering one-parametergroups and their generators.

2

Chapter 1Preliminaries

This chapter introduces the reader to a general idea of transformations and exhibitsa variety of transformation groups. The duality between changes of frames of ref-erence and point transformations is useful in group analysis. We discuss the idea ofthe duality in this chapter and will employ it in the next chapter for the prolongationof point transformation groups to derivatives.

1.1 Changes of frames of reference and point transformations

1.1.1 Translations

Consider, in the(x,y) plane, a pointP having the coordinates(x,y) in the rectangularCartesian reference frame with the axesOx,Oy. Let e = (e1,e2) be a fixed unitvector. Consider a new pair of rectangular axesOx,Oy parallel to the former axessuch thatO has the coordinates(−ae1,−be2) with respect to the original frame ofreference, wherea is an arbitrary real parameter. Then the coordinates(x, y) of thepointP in the new frame of reference are given by

x = x+ae1 , y = y+be2. (1.1.1)

An alternative interpretation of Eqs. (1.1.1) is as follows. One ignores the newaxesOx,Oy and regards(x,y) and(x, y) as the coordinates of pointsP andP, re-spectively, each referred to the original frameOx,Oy. Then Eqs. (1.1.1) define atransformation of the pointP(x,y) into the new positionP(x, y) in the (x,y) plane.Accordingly, equations (1.1.1) determine the displacement (translation) of all pointsP of the plane through the distancea in the direction of the vectore.

1.1.2 Rotations

Consider again the rectangular Cartesian reference frame with the axesOx,Oy. LetOx,Oy be the new pair of axes obtained by rotating the original axesround the origin

3

4 1 Preliminaries

O counter-clockwise through an anglea. Let (x,y) and(x, y) be the coordinates of apointP referred to the axesOx,Oy andOx,Oy, respectively. Then we have

x = x cosa+ysina, y = y cosa−xsina. (1.1.2)

Indeed, in the polar coordinates(r,θ ), connected with the Cartesian coordinates bythe equations

x = r cosθ , y = r sinθ , (1.1.3)

the rotation by the anglea about the origin clockwise is written

r = r, θ = θ −a. (1.1.4)

Equations (1.1.3), (1.1.4) yield the following transformation:

x = r cosθ = r cos(θ −a), y = r sinθ = r sin(θ −a).

Expanding cos(θ −a) and sin(θ −a) and substitutingr cosθ = x, r sinθ = y, onearrives at Eqs. (1.1.2).

An alternative interpretation of Eqs. (1.1.2) is as follows. We regard(x,y) and(x, y) as the coordinates of the pointsP andP, respectively, each referred to thesame axesOx,Oy. Then Eqs. (1.1.2) accomplish the rotation of all points of theplane aboutO clockwise through the anglea.

1.1.3 Galilean transformation

Everyone travelling by train can observe the duality between uniform motions ofhis local frame of reference (a train) and outside points (people or other objectson a depot). This remarkable exhibition of the duality, whenone cannot determinewho is actually moving, is known in the classical mechanics as Galileo’s relativityprinciple. It is equivalent to the invariance of equations of motion of mechanicalsystems under the transformation

t = t, x = x+ tV , (1.1.5)

whereV is the constant velocity. Differentiation ofx with respect tot = t yields

v = v +V . (1.1.6)

The transformation (1.1.6) of the velocity is a mathematical expression of Galileo’srelativity principle. The transformation (1.1.5) is knownas the Galilean transforma-tion and lies at the core of theGalilean groupwhich is one of the most importantgroups in non-relativistic physics.

1.2 Introduction of transformation groups 5

1.2 Introduction of transformation groups

1.2.1 Definitions and examples

We will consider invertible transformations in ann-dimensional Euclidean spaceIRn defined, in coordinates, by equations of the form

x i = f i(x), i = 1, . . . ,n, (1.2.1)

where the vector-functionf = ( f 1, . . . , f n) is continuous together with its derivativesinvolved in further discussions. Since the transformation(1.2.1) is invertible, thereexists the inverse transformation

xi = ( f−1)i(x), i = 1, . . . ,n. (1.2.2)

Let us denote the transformation (1.2.1) byT and its inverse (1.2.2) byT−1.Thus,T carries any point

x = (x1, . . . ,xn) ∈ IRn

into a new position

x = (x1, . . . ,xn) ∈ IRn,

andT−1 returnsx into the original positionx. It is assumed that the coordinatesxi

andx i of pointsx andx, respectively, are referred to one and the same coordinatesystem. The identical transformation

x i = xi , i = 1, . . . ,n, (1.2.3)

will be denoted byI .Let T1 andT2 be two transformations of the form (1.2.1) with functionsf i

1 and f i2,

respectively. Theirproduct T2T1 (termed alsocompositionand denoted byT2◦T1) isdefined as the consecutive application of these transformations and is given by

x i = f i2(x) = f i

2( f1(x)), i = 1, . . . ,n. (1.2.4)

The geometric interpretation of the product is as follows. SinceT1 carries the pointx to the pointx= T1(x), whichT2 carries to the new positionx = T2(x), the effect ofthe productT2T1 is to carryx directly to its final locationx, without a stopover atx.Thus, equation (1.2.4) means that

xdef= T2(x) = T2T1(x). (1.2.5)

In this notation, the definition of the inverse transformation (1.2.2) means

TT−1 = T−1T = I . (1.2.6)

6 1 Preliminaries

Definition 1.1. A setG of transformations (1.2.1) in IRn containing the identityI iscalled a transformation group if it contains the inverseT−1 of every transformationT ∈ G and the productT1T2 of any transformationsT1,T2 ∈ G. Thus, the attributesof the groupG are:

I ∈ G, and T−1 ∈ G, T1T2 ∈ G wheneverT,T1,T2 ∈ G. (1.2.7)

Example 1.1. The setG = {I ,T1, . . . ,T5} of the transformations

I : x = x, T1 : x = 1−x, T2 : x =1x

,

T3 : x =1

1−x, T4 : x =

xx−1

, T5 : x =x−1

x

(1.2.8)

on the straight line is a group containing six elements (see,e.g., [6],§9). The groupproperties (1.2.7) can be verified by computing the inversesand products of thetransformations (1.2.8), e.g.

T−11 = T1, T−1

2 = T2, T−13 = T5, T−1

4 = T4, T−15 = T3,

T21 = I , T2

2 = I , T23 = T5, T2

4 = I , T25 = T3, (1.2.9)

T2T1 = T3, T1T2 = T5, T3T1 = T2, T1T3 = T4.

Example 1.2. Consider the setG of all translations (displacements)Ta :

x = x+a (1.2.10)

on the straight line. Sincex = x whena = 0, the setG contains the identityI = T0.Furthermore, the combined effect of two translations,Ta andTb, acting in succes-sion, is to displacex through the distancea+b. Hence,

TbTa = Ta+b. (1.2.11)

Equation (1.2.11) shows thatT−1

a = T−a.

Thus, the transformations (1.2.10) obey the group properties (1.2.7), and hence de-fine aone-parameter group G, i.e. a group containing one arbitrary parametera.This group is known as thetranslation groupand provides one of the simplest illus-trations to the following definition.

Definition 1.2. A set G of transformationsTa in IRn depending continuously on aparametera, wherea ranges over all real numbers from a given intervalU ⊂ R, iscalled aone-parameter groupif there is a unique valuea = a0 in U providing theidentical transformation,Ta0 = I , and the following conditions hold for alla,b∈U :

T−1a = Ta−1 ∈ G, TbTa = Tc ∈ G, (1.2.12)


wherea−1,c∈U and c = φ(a,b) is a continuous function.

Definition 1.3. A group containing a finite number of parameters and dependingcontinuously on these parameters was termed by Lie afinite continuous group.A continuous group which depends onr < ∞ essential parameters is called anr-parameter group and is denoted byGr .

Remark 1.1. A continuous group of a different type is provided by invertible trans-formationsx = f (x) of a straight line. The set of all these transformations, wheref (x) ranges over all continuously differentiable functions satisfying the invertibil-ity condition f ′(x) 6= 0, is a group. Continuous groups involving arbitrary functionsare calledinfinite continuous groups.Thus,x = f (x) is an example of an infinitecontinuous group involving one arbitrary functionf (x).

Example 1.3. The setG of the transformations

x = a1 +a2x, a2 6= 0 (1.2.13)

provides an example of atwo-parameter group, i.e. a group involving two parame-ters,a1,a2. This group is known as thegeneral linear group.

Let us verify that the group properties (1.2.7) are satisfied. We will introduce thevector valued parameter

a = (a1,a2)

and denote the transformation (1.2.13) byTa. Thus,

Ta : x = a1 +a2x. (1.2.14)

The identity transformation is obtained by letting

a1 = 0, a2 = 1. (1.2.15)

Furthermore, the transformationTb with the parameterb = (b1,b2) maps the pointx into x defined as follows:

Tb : x = b1 +b2x = b1 +b2(a1 +a2x). (1.2.16)

Hence, the combined effect of the transformationsTa andTb acting in succession is

TbTa : x = b1 +b2a1 +b2a2x = c1 +c2x.

It means thatTbTa = Tc, (1.2.17)

wherec = (c1,c2) is the vector valued parameter with the components

c1 = b1 +b2a1, c2 = b2a2. (1.2.18)

Let us find the inverse transformation to (1.2.14). Comparing Eqs. (1.2.6), (1.2.17)and using Eqs. (1.2.15), (1.2.18), we see that the inverse transformationTb = T−1

a

8 1 Preliminaries

to Ta is found by solving the equations

b1 +b2a1 = 0, b2a2 = 1,

whence

b2 =1a2

, b1 = −a1

a2·

One can verify by substituting these values ofb1,b2 in Eqs. (1.2.15) that one hasindeedx = x. Thus,T−1

a = Ta−1, wherea−1 is the vector valued parameter

a−1 =

(−a1

a2,

1a2

).

Definition 1.4. Let G be a transformation group. Its subsetH ⊂ G is called asubgroup ofG, if H possesses all group properties (1.2.7), i.e.I ∈ H andT−1 ∈H, T1T2 ∈ H wheneverT,T1,T2 ∈ H.

Example 1.4. The setH = {I ,T1}

is a subgroup of the groupG= {I ,T1, . . . ,T5} of the transformations (1.2.8). Indeed,H contains the identity transformationI . Furthermore,

T−11 ,T2

1 ∈ H

becauseT21 = I , and henceT−1

1 = T1.

Example 1.5. The two-parameter group (1.2.13) has two one-parameter subgroups,namely, the translation group (1.2.10) witha = a1,

x = x+a,

and the dilation groupTa : x = ax, a 6= 0, (1.2.19)

with the parametera = a2. The group properties for the dilation (1.2.19) can beexamined as above. Namely, one can readily verify that the multiplication of thedilationsTa andTb yields (compare with Eq. (1.2.11))

TbTa = Tab. (1.2.20)

Equation (1.2.20) shows that the inverse transformation toTa is the dilation with theparametera−1 = 1/a :

T−1a = Ta−1.

The identity transformation is obtained by lettinga = 1.

Definition 1.5. Two transformation groups are said to besimilar if one can be ob-tained from another by an appropriate change of the variablesxi .


Example 1.6. The setG of transformations

x = x+ α1− ln(1+ α2ex)

involving two parameters,α1,α2, where|α2| is sufficiently small, defines a two-parameter group ([14], Section 6.3.1). Indeed, let us rewrite the above transforma-tions in the form

x = − ln[e−α1

(e−x + α2

)](1.2.21)

and denote byTα andTβ two transformations (1.2.21) with the vector parametersα = (α1,α2) andβ = (β1,β2), respectively. Thus,

Tα : x = − ln[e−α1

(e−x + α2

)],

Tβ : x = − ln[e−β1

(e−x + β2

)].

It is clear from (1.2.21) thatG contains the identical transformation and that itobtained by setting

α1 = α2 = 0. (1.2.22)

Furthermore, we have

Tβ Tα : x = − ln[e−β1

(e−x + β2

)]

= − ln[e−β1

(eln[(e−x+α2)e

−α1 ] + β2

)].

The result can be written

x = − ln[e−(α1+β1)

(e−x + α2 + β2e

α1)]

= − ln[e−γ1

(e−x + γ2

)],

whereγ1 = α1 + β1, γ2 = α2 + β2e

α1. (1.2.23)

HenceTβ Tα = Tγ ∈ G, (1.2.24)

whereγ = (γ1,γ2) is defined by Eqs. (1.2.23). Equations (1.2.23) and (1.2.24)showthat the parameterβ of the inverse transformationTβ = T−1

α to Tα is obtained fromEqs. (1.2.23) by lettingγ1 = γ2 = 0. Hence,

T−1α = Tα−1 ∈ G, where α−1 = (−α1,−α2e−α1). (1.2.25)

Thus,G is a group. It is similar to the general linear group (1.2.13). Namely, afterthe substitutiony = e−x the transformation (1.2.21) is written in the form (1.2.13):

y = a1 +a2y,

where

10 1 Preliminaries

a1 = α2e−α1 , a2 = e−α1.

1.2.2 Different types of groups

Continuous, discontinuous and mixed groups

Definition 1.6. A groupG of transformations in IRn is said to becontinuous,if anytwo transformationsT1,T2 ∈ G can be connected via a continuous set of elementswithin the group. In other words,T1 can be continuously deformed intoT2 withinthe groupG.

One-parameter groupsG (Definition 1.2) are manifestly continuous in the abovesense. It means geometrically that any pointx ∈ IRn is carried by the group trans-formations into the pointsx = Ta(x) whose locus is a continuous curve (passingthroughx) and is called apath curveof the groupG. The group property means thatany point of a path curve is carried by G into points of the samecurve.The locus ofthe imagesTa(x) is also termed theG-orbit of the pointx and denoted byG(x).

In the case of continuousr-parameter groupsGr (Definition 1.3), orbitsGr(x)are continuousr-dimensional manifolds.

Definition 1.7. A group of transformations in IRn is said to bediscontinuousif itcontains no transformations whose effects onx∈ IRn differ infinitesimally.

Example 1.7. The group of transformations (1.2.8) from Example 1.1 is discontin-uous.

There are also groups that are neither continuous nor discontinuous (see [14],Section 6.3.4). An example of such amixed groupis offered by symmetry trans-formations of the rectangular coordinate axes on the plane,i.e. by the followingtransformations leaving invariant the equationxy= 0 :

T : x = f (x), y = g(y); R : x = p(y), y = q(x), (1.2.26)

where f (x), g(y), p(y), andq(x) are arbitrary functions such that

f (0) = g(0) = p(0) = q(0) = 0,

f ′(0) = g′(0) = p′(0) = q′(0) = 1.

The setG of all transformations (1.2.26) is a group. The groupG is infinite sinceit involves arbitrary functions. But it is not continuous because transformations oftype T cannot be continuously deformed into transformations of type R, and viceversa. However,G contains an infinite continuous subgroup composed of the trans-formations (1.2.26) of typeT.


Global and local groups

Example 1.8. Consider transformationTa :

x =x

1−ax(1.2.27)

known as aprojective transformation. It depends on the continuous parametera anddefines a one-parameter set of transformationsG.

We haveI ∈G whena= 0. The reckoning shows that the consecutive applicationof two transformations (1.2.27)Ta,Tb yields

x =x

1− (a+b)x· (1.2.28)

It follows from (1.2.28) thatT−1

a = T−a ∈ G

andTbTa = Ta+b ∈ G.

One naturally may conclude thatG is a group. However, this conclusion presumesthat the following expressions do not vanish:

1−ax, 1−bx, 1− (a+b)x.

This simple observation is of fundamental significance for the theory of Lie groupanalysis.

To clarify the situation, let us consider a fixed pointx0 > 0 and move along thepath curve throughx0 when the parametera ranges over all real numbers from acertain interval 0≤ a≤ ε. Two transformations (1.2.27),Ta andTb, are well definedin the interval

[0, 1/x0),

i.e. 1−ax0 and 1−bx0 do not vanish in this interval, but their consecutive applica-tion may give the prohibited value

a+b= 1/x0

when the product (1.2.28) is not determined. This is the case, e.g. fora = 1/(3x0)and b = 2/(3x0). Therefore, let us assume thata and b are taken from a closervicinity of a = 0, e.g. from the interval

[0, 1/(2x0)).

Thena+b< 1/x0

and the product (1.2.28) is well defined. However, a further multiplication may againresult in an unacceptable value of the parameter. For example, when

12 1 Preliminaries

a = b = 1/(3x0)

one hasa+b= 2/(3x0).

HenceTaTb = Ta+b

is determined, butTaTa+b = T2a+b

is not. It can be readily seen that iterated multiplication of the projective transforma-tions (1.2.27) inevitably leads to the prohibited value of the group parameter. Onecannot solve the problem by merely fixing a “tiny” interval 0≤ a≤ ε, the true na-ture of the problem being that the orbit of any pointx (excluding the isolated pointx = 0) has a singularity whena = 1/x.

The projective group (1.2.27) provides an example of what iscalled a localgroup, meaning that the composition is defined only for transformations that aresufficiently close to the identity. The vicinity of the identical transformation, wherethe composition is determined, may depend upon a transformed pointx. An alterna-tive definition of a local groupG is that path curves ofG have singularities.

Example 1.9. The rotation groupis given by Eqs. (1.1.2):

x = x cosa+y sina,

y = y cosa−x sina,(1.2.29)

and provides another simple example of a local group. Here again the identitytransformation corresponds toa = 0. The consecutive application of two rotations(1.2.29) through the anglesa andb yields:

x = x cos(a+b)+ysin(a+b),

y = y cos(a+b)−xsin(a+b).(1.2.30)

Equations (1.2.30) show that the rotation (1.2.29) does notsatisfy Definition 1.2 of aone-parameter group. Namely, it does not satisfy the requirement on existence of anintervalU ⊂ R such that there is a unique valuea = a0 in U providing the identicaltransformationTa0 = I . To illustrate this statement, let us take, e.g. the interval

U =

(−π

2,

32

π)

,

wherea0 providing the identical transformation is unique, namelya0 = 0. But if weperform the consecutive rotations, say by the angles

a = π ∈U, b =54

π ∈U,

1.3 Some useful groups 13

then the parameter

a+b=94

π

of resulting rotation (1.2.30) will be outside of the intervalU. If we extend the initialintervalU in order to include the above value ofa+ b, we lose the uniqueness ofa0 because an extended interval will have at least two values,a0 = 0 anda0 = 2π ,providing the identical transformation (1.2.29).

A transformation group is called aglobal groupif the composition of any trans-formations is defined simultaneously at all generic pointsx. A common represen-tative of a global group is the translation group (1.2.10). Other global groups of aphysical significance are provided by the Galilean transformation (1.1.5) with thevector valued group parameterV and the Lorentz transformation

x = x cosha+y sinha,

y = y cosha+x sinha,(1.2.31)

considered in Section 1.3.2, Example 1.13, using a different notation.

Remark 1.2. In what follows, we will apply the nomenclaturegroupboth to globaland local groups.

1.3 Some useful groups

1.3.1 Finite continuous groups on the straight line

Example 1.10. The linear fractional transformation

x =a1 +a2xa3x+a4

, a1a3−a2a4 6= 0 (1.3.1)

forms a group. It involves four arbitrary constantsai subject to the invertibility con-dition,a1a3−a2a4 6= 0. But one of these constants can be eliminated by dividing theright-hand side of Eq. (1.3.1) by any non-vanishing coefficientai . If we deal withthe transformations (1.3.1) in the vicinity of the identical transformation (i.e. neara1 = a3 = 0,a2 = a4 = 1), thena4 6= 0. Therefore dividing bya4 one arrives at whatis called theprojective groupon the straight line:

x =a1 +a2x1+a3x

, a2 6= a1a3. (1.3.2)

The group of the general linear fractional transformations(1.3.1) contains, as itssubgroup, the general linear group (1.2.13) which itself has the translation (1.2.10)and dilation (1.2.19) as its subgroups. The projective group (1.3.2) is a three-

14 1 Preliminaries

parameter group (Definition 1.3), while Eqs. (1.2.13) and (1.2.19) provide its two-and one-parameter subgroups, respectively.

The following theorem states that the projective group and its subgroups are theonly types of finite continuous groups on the straight line.

Theorem 1.1. Any finite continuous groupGr on the straight line contains at mostthree essential parameters, i.e.r ≤ 3. The groupGr is similar to the three-parameterprojective group (1.3.2) ifr = 3, to the general linear group (1.2.13) ifr = 2, and tothe translation group (1.2.10) ifr = 1.

This important result was first published by Lie in 1874. Its detailed proof isprovided in [20], § 1, pp. 2–6. One can also find in Chapter 1 of the book [20]Lie’s enumeration of continuous groups on the plane which exhibits that the plane,unlike the straight line, contains a large variety of different types of groups. Considerexamples of groups on the plane.

1.3.2 Groups on the plane

Example 1.11. The general translation group on the plane is a two-parameter groupgiven by the transformation

x = x+a1, y = y+a2. (1.3.3)

Two independent translations in the directions of thex axis,

x = x+a1, y = y,

and they axis,x = x, y = y+a2,

provide one-parameter subgroups and compose the two-parameter group (1.3.3).Furthermore, the translations (1.1.1) in the direction of any vectore = (e1,e2) forma one-parameter subgroup of the group (1.3.3).

Example 1.12. The composition of the rotation (1.2.29) and the translation (1.3.3),

x = x cosa+ysina+a1, y = y cosa−xsina+a2, (1.3.4)

provides thegroup of isometric motionson the plane. It is the largest continuousgroup that changes only the location and orientation of geometric figures (trian-gles, circles, etc.), but not their magnitude and shape. In other words, the isomet-ric motions (1.3.4) transpose geometric figures as rigid bodies. Consequently, thetransformation (1.3.4) (and its generalization to three dimensions) is a mainstay ofEuclidean geometry and is known as the group of Euclidean motions, or briefly theEuclidean group.


Example 1.13. ThePoincare group (known also as thenon-homogeneous Lorentzgroup) of the special theory of relativity comprises (in the case of one spatial vari-ablex) translations oft andx and the Lorentz transformation:

t = t cosh(a/c)+ (x/c) sinh(a/c),

x = x cosh(a/c)+ct sinh(a/c),(1.3.5)

wherea is a group parameter andc = 2.99793×1010 cm/s is the velocity of lightin vacuum. Introducing the new parameterV = c tanh(a/c), one can rewrite thetransformation (1.3.5) in the form similar to the Galilean transformation (1.1.5):

t =t +(xV/c2)√1− (V2/c2)

, x =x+Vt√

1− (V2/c2)· (1.3.6)

The physical meaning of the transformation (1.3.6) and of its parameterV can beillustrated by considering the motion of a particle in a direction parallel to thex-axis.It follows from (1.3.6) that if an observer at rest detects the velocityv of a particle,then an observer moving with the velocityV along thex-axis will detect the velocity

v =v+V

1+(vV/c2)· (1.3.7)

Consequently, the transformation (1.3.6) is known in the physical literature as theLorentz boost. In the limit of classical mechanics, whenV/c→ 0, the Lorentz boosttakes the form of the Galilean transformation,

t = t, x = x+Vt,

and the formula (1.3.7) reduces to Galileo’s relativity principle (1.1.6),

v = v+V.

Example 1.14. The generalprojective groupon the plane is given by the linearfractional transformation

x =a11x+a12y+a1

b1x+b2y+b3, y =

a21x+a22y+a2

b1x+b2y+b3· (1.3.8)

The identity transformation is obtained by setting

a11 = a22 = b3 = 1, a12 = a21 = a1 = a2 = b1 = b2 = 0. (1.3.9)

In the vicinity of these values of the parameters the transformation (1.3.8) is invert-ible and, upon dividing byb3 6= 0, defines the eight-parameter group. The maingeometric significance of the projective group (1.3.8) is that it maps any straightline on the plane again into a straight line. Let us verify this property. Since thetransformation (1.3.8) is invertible, it suffices to show that if the transformed points

16 1 Preliminaries

(x,y) are located on a straight line, the original points(x,y) also lie on a straightline. Thus, let us assume that

Ax+By+C = 0.

The substitution of (1.3.8) yields

kx+ ly+m= 0,

where

k = Aa11+Ba21+Cb1,

l = Aa12+Ba22+Cb2,

m= Aa1+Ba2+Cb3.

Hence, the image of any straight line is again a straight line.

Remark 1.3. The well-known subgroups of the general projective group are thesix-parameterlinear group:

x = a11x+a12y+a1, y = a21x+a22y+a2;

the four-parameterlinear homogeneous group:

x = a11x+a12y, y = a21x+a22y;

and the one-parameterspecial projective group:

x =x

1−ax, y =

y1−ax

· (1.3.10)

Example 1.15. The transformationS:

x = R2 xr2 , y = R2 y

r2 , (1.3.11)

whereR 6= 0 is any constant andr2 = x2+y2, is known as theinversionwith respectto a circle of radiusR centered atO. The pointO = (0,0) and the positive numberR are called the center and the radius of inversion (1.3.11). It follows from (1.3.11)that rr = R2, where r2 = x2 + y2. Therefore the inversion is also known as thetransformation of reciprocal radii. We have:

S2 = 1. (1.3.12)

It follows thatS−1 = S, and henceG = {I ,S} is a group containing two elements.Note that one can transpose the center of inversion to any point as well as to chooseany positive number (orR= ∞) as a radius of inversion.


Proposition 1.1. Inversion (1.3.11) (i) leaves unaltered any straight line passingthrough the centerO of inversion, (ii) maps any circle passing throughO into astraight line which does not pass throughO, (iii) maps any straight line that doesnot pass throughO into a circle passing throughO, (iv) maps any circle not passingthroughO again into a circle that does not pass throughO.

Fig. 1.1 Geometry of inversion. Fig. 1.2 Does the oil recovery pump perform aninversion?

The geometric properties (ii) and (iii) of inversion, formulated in Proposition 1.1,explain the theoretical principles of mechanical constructions for transforming recti-linear motions into circular ones (e.g. in steam engines) and vice versa. Let us dwellon this question by considering the construction given in Figure 1.1 (it is discussedin [14], Note 6.6). One takes seven rigid rodsAP,AQ,BM,PM,PN,QM,QN suchthat

AP= AQ,PM = PN = QM = QN.

The rods are joined together by hinges as shown in Figure 1.1.The pointsA andBare fixed so thatAB= BM. Then elementary geometry shows thatAMN is a straightline and that the productAM ·AN is constant, namely

AM ·AN = AP2−PM2.

Hence the pointsM andN are connected by the inversion (1.3.10) with the centerAand

R2 = AP2−PM2.

Thus, whenM moves along the circle with the centerB, the pointN moves alonga straight line. Oil recovery pumps transform circular motions into rectilinear ones.Figure 1.2 invites the reader to discuss if this transformation is inversion.

Remark 1.4. The inversion (1.3.11) is aconformal transformationon the plane.Indeed it satisfies the Cauchy–Riemann equations:

18 1 Preliminaries

∂x∂x

+∂y∂y

= 0,∂x∂y

− ∂y∂x

= 0.

Example 1.16. Using Remark 1.4, we can obtain a one-parameter group of confor-mal transformations as follows. Consider the transformation

Ca = S1TaS1, (1.3.13)

whereS1 is the inversion with respect to the circle of unit radius, and Ta is thetranslation group along thex-axis:

x = x+a, y = y.

The combined transformationS1TaS1 acts as follows:S1 carries(x,y) to the point(x1,y1),

x1 =xr2 , y1 =

yr2 ,

thenTa carries it to the new position

x2 = x1 +a =x+ar2

r2 , y2 = y1 =yr2 ,

after thatS1 brings(x2,y2) to the final location(x,y) of the initial point(x,y):

x =x2

r22

, y =y2

r22

·

Substituting the above values ofx2,y2 into r22 = x2

2 +y22, one obtains

r22 =

(x+ar2)2 +y2

r4 =1+2ax+a2r2

r2 ·

Hence, the transformationCa = S1TaS1 has the form:

x =x+ar2

1+2ax+a2r2 , y =y

1+2ax+a2r2 , (1.3.14)

wherer2 = x2 + y2. Let us demonstrate that the setG of transformations (1.3.14)with varyinga defines a one-parameter group. Writing (1.3.14) in the form (1.3.13)we have

I = C0 ∈ G.

Furthermore, the composition of two transformations is written

CbCa = S1TbS1S1TaS1.

Invoking the equationS1S1 = I and the group propertyTbTa = Ta+b of the translationgroup, we have

CbCa = S1TbTaS1 = S1Ta+bS1 = Ca+b ∈ G.


1.3.3 Groups inIRn

Example 1.17. TheEuclidean groupin IRn comprisesn translations

x i = xi +ai, i = 1, . . . ,n, (1.3.15)

andn(n−1)/2 rotations in all coordinate planes(xi ,x j) :

x i = xi cosa+x j sina, x j = x j cosa−xi sina. (1.3.16)

Hence, the group of isometric motions in then-dimensional Euclidean space IRn

containsr = n(n+1)/2 essential parameters. For example, solid geometry is basedon the six-parameter groupG6 of isometric motions containing three independentrotations and three translations. Continuous and discontinuous subgroups of the six-parameter group of isometric motions in IR3 serve incrystallographyto characterizesymmetries of crystalline substances. Accordingly, grouptheory furnishes a propertool for a rigorous mathematical approach to crystallography.

Example 1.18. The general linear group in IRn has the form

x i = aikx

k +ai, i = 1, . . . ,n,

wherek is the summation index. A compact form of the above group is

x = Ax+a, (1.3.17)

wherex, x, a aren-dimensional vectors, andA= (aik) is ann×nnon-singular matrix

(i.e. detA 6= 0). The general linear group depends onn(n+ 1) essential parametersand contains then(n+1)/2-parameter Euclidean group as a subgroup.

Example 1.19. The linear homogeneous group is a subgroup of the general lineargroup. It containsn2 parameters and has the form:

x = Ax, detA 6= 0. (1.3.18)

The composition ofx = Axandx = Bx yields

x = Bx = BAx,

whereBA is the product of the matricesA andB. Hence, the composition of transfor-mations (1.3.18) is represented by the usual multiplication of matrices. Furthermore,the inverse to the transformation (1.3.18) is

x = A−1x,

whereA−1 is the inverse matrix. Thus, the linear homogeneous group (1.3.18) canbe regarded as thematrix groupconsisting of all non-singular matricesA, where thegroup composition law is the usual multiplication of matrices. The identity element

20 1 Preliminaries

of the group is the unit matrixI = (δ ik), whereδ i

k denote the Kronecker symbols:δ i

k = 1 if i = k, andδ ik = 0 if i 6= k.

Definition 1.8. A conformal mappingin IRn is a transformation preserving the an-gles, but not necessarily the magnitudes. For example, a conformal mapping trans-forms a sphere into a sphere, but the latter may have a radius and a center differentfrom those of the original sphere. The set of all conformal mappings is a groupcalled theconformal groupin IRn. The group of isometric motions is a subgroup ofthe conformal group.

Example 1.20. Inversion in IR3 with respect to a sphere of radiusR centered atO = (0,0,0) is given by

x = R2 xr2 , y = R2 y

r2 , z= R2 zr2 , (1.3.19)

wherer2 = x2 +y2 +z2. The radiir andr of a point(x,y,z) and of its image(x,y,z)are in the relation of reciprocal radii (cf. Example 1.15),rr = R2. The inversion pro-vides an example of a non-trivial (i.e. different from an isometric motion) conformalmapping.

Example 1.21. The continuous conformal group in IR3 is a finite group. It can beobtained as follows. Proceeding as in Example 1.16 and extending it to they- andz-directions, one arrives at the following three one-parameter groups of conformalmappings:

x =x+a1r2

1+2a1x+a21r

2, y =

y

1+2a1x+a21r

2, z=

z

1+2a1x+a21r

2,

x =x

1+2a2y+a22r

2, y =

y+a2r2

1+2a2y+a22r

2, z=

z

1+2a2y+a22r

2,

x =x

1+2a3z+a23r2

, y =y

1+2a3z+a23r

2, z=

z+a3r2

1+2a3z+a23r2

·

J. Liouville obtained in 1847 the following result stating that any non-trivial con-formal mapping in IR3 is based on inversion.

Theorem 1.2. The maximal continuous conformal group in IR3 contains 10 essen-tial parameters. It is composed by the six-parameter group of Euclidean motions(i.e. translations and rotations ofx,y,z), three one-parameter groups given in Exam-ple 1.21 and the one-parameter dilation groupx = ax,y = ay,z= az.

Remark 1.5. W. Thomson (Lord Kelvin) showed in 1847 (he published his paperin the same volume ofJ. Math. Pures et Appl.as Liouville’s paper) that the inversion(1.3.19) leaves invariant the Laplace equation

∆u≡ uxx+uyy+uzz= 0

Exercises to Chapter 1 21

provided that the dependent variableu undergoes the transformation

u = ru.

This combined transformation is known in the literature asKelvin’s transformationand used in mathematical physics, e.g. in the theory of potentials for constructingGreen’s function.

Example 1.22. The Galilean group in IRn is the(n+1)(n+2)/2-parameter groupcomprising the Euclidean group (1.3.15)–(1.3.16), the time translationt = t +b, andthe Galilean transformation (1.1.5) withV = (b1, . . . ,bn) :

t = t, x i = xi + tbi, i = 1, . . . ,n.

Example 1.23. The Poincare group (non-homogeneousLorentz group) is thegroupof isometric motions in the Minkowski space-time of specialrelativity. It is ann-parameter group and comprises the Euclidean group (1.3.15)–(1.3.16), the timetranslationt = t + b and the Lorentz transformation (1.3.5) inn different planes(xi , t), i = 1, . . . ,n, namely:

t = t cosh(a/c)+ (xi/c) sinh(a/c),

xi = xi cosh(a/c)+ct sinh(a/c).(1.3.20)

Each one-parameter group (1.3.20) has its own group parametera.

Exercises to Chapter 1

Exercise 1.1.Complete the multiplication table (1.2.9), namely, compute

T1T4, T4T1, T1T5, T5T1, T2T3, T3T2, . . . , T4T5, T5T4

and verify that the group properties (1.2.7) are satisfied for all transformations(1.2.8).

Exercise 1.2.Compute the inverses and the products of the following transforma-tions on the straight line:

I : x = x, T1 : x = −x, T2 : x =1x

, T2 : x = −1x· (1.3.1)

Demonstrate that the setG= {I ,T1,T2,T3} is a discontinuous group containing fourelements.

Exercise 1.3.Show that each of the setsH1 = {I ,T1}, H2 = {I ,T2} andH3 = {I ,T3}is a subgroup of the groupG from Exercise 1.2.

22 1 Preliminaries

Exercise 1.4.Prove that the groupG from Exercise 1.2 does not have subgroupscontaining three different elements.

Exercise 1.5.Derive the group property (1.2.28) for the projective transformation(1.2.27),

x =x

1−ax·

Exercise 1.6. In the polar coordinates(r,θ ), the rotation by the anglea about theorigin clockwise is naturally written

r = r, θ = θ −a.

Show that this translation group is written in the rectangular Cartesian coordinates

x = r cosθ , y = r sinθ

in the form (1.2.29),

x = x cosa+ysina, y = y cosa−xsina.

Exercise 1.7.Derive the group property (1.2.30) for the rotation (1.2.29).

Exercise 1.8.Demonstrate that the Lorentz transformation (1.2.31),

x = x cosha+y sinha, y = y cosha+x sinha,

satisfies the group properties (1.2.7) of Definition 1.1. Show that it is a global group.

Exercise 1.9.Read [20], pp. 2–6, and present the proof of Theorem 1.1.

Exercise 1.10.Derive the relativistic formula (1.3.7) for the velocity inmovingframe,

v =v+V

1+(vV/c2),

by using the representation (1.3.6) of the Lorentz transformation.

Exercise 1.11.Show that the linear fractional transformations

x =x

1−ax, y =

y1−ay

define a one-parameter group and that this group does not convert an arbitrarystraight line into a straight line.

Exercise 1.12.Prove the property (1.3.12) of the inversion (1.3.11).

Chapter 2One-parameter groups and their invariants

This chapter contains the infinitesimal description of one-parameter local transfor-mation groups, their invariants and invariant equations. The connection betweenone-parameter groups and their infinitesimal generators isprovided by the Lie equa-tions and the exponential map.

2.1 Local groups of transformations

2.1.1 Notation and definition

We will consider invertible transformationsTa in IRn given by the equation

x = f (x,a) (2.1.1)

with certain functionsf depending on a real parametera. Herex∈ IRn andx∈ IRn.We assume that the functionsf satisfy the condition that (2.1.1) is the identicaltransformation for a certain valuea0 of the parameter, i.e.

f (x,a0) = x, (2.1.2)

and that there are no other values ofa in a vicinity of a0 reducingTa to the identity.In coordinatesxi , equations (2.1.1) and (2.1.2) are written:

x i = f i(x,a), f i(x,a0) = xi , i = 1, . . . ,n. (2.1.3)

It is assumed that, given a generic pointx, there exists an open intervalU ⊂ Rcontaininga0 such that the functionsf i(x,a) satisfy the regularity conditions dis-cussed in Section 1.2.1 in a vicinity ofx, whenevera∈ U. Furthermore, it is sup-posed thata0 is the only value ofa ∈ U for which Eq. (2.1.1) reduces to the

23

24 2 One-parameter groups and their invariants

identical transformation. In short, equation (2.1.1) defines single-valued transfor-mationsTa : IRn → IRn for all a from the intervalU.

Transformations occurring in practical applications often satisfy Definition 1.2of a one-parameter group only whena andb in Eqs. (1.2.12) are restricted to suf-ficiently small numerical values of the group parameter. Then we have alocal one-parameter groupdefined as follows.

Definition 2.1. A setG of transformationsTa in IRn given by (2.1.3) is called aone-parameter local groupif there exists a subintervalU ′ ⊂ U containinga0 such thatthe functionsf i(x,a) satisfy thecomposition rule

f i( f (x,a),b) = f i(x,c), i = 1, . . . ,n, (2.1.4)

for all valuesa,b∈U ′. Herec∈U is a smooth function,c= φ(a,b), of two variablesa,b∈U ′ such that the equation

φ(a,b) = a0 (2.1.5)

has a unique solutionb for any a ∈ U ′. Given a, the solutionb of Eq. (2.1.5) isdenoted bya−1. Hence, the inverse transformationT−1

a is given by

f i(x,a−1) = xi , i = 1, . . . ,n. (2.1.6)

For brevity, local groups are also termed groups (see Examples 1.8, 1.9 and Remark1.2).

The function

c = φ(a,b) (2.1.7)

defined by the composition rule (2.1.4) is termed agroup composition law.It followsfrom theinitial conditions fi(x,a0) = xi and (2.1.4) that

φ(a,a0) = a, φ(a0,b) = b. (2.1.8)

According to the above definition, a local groupG contains the (unique) identityI = Ta0. The composition rule (2.1.4) means that any two transformationsTa,Tb ∈ Gwith a,b∈U ′ carried out one after the other result in a transformationTc ∈G, wherec belongs to the intervalU but may not belong to the subintervalU ′. Consequently,one cannot obtain a global group merely by assigningU ′ as a basic interval insteadof U. To draw a parallel between global and local one-parameter groups, equations(2.1.4) and (2.1.6) can be written in the form similar to Eqs.(1.2.12):

T−1a = Ta−1 ∈ G, TbTa = Tc ∈ G, where a,b∈U ′, c∈U.

Note that the symbola−1 for the parameter of the inverse transformation (2.1.6)does not signify, in general, 1/a (see Eq. (2.1.13) and Exercise 2.1).

2.1 Local groups of transformations 25

2.1.2 Groups written in a canonical parameter

The composition law (2.1.7) depends on a choice of the group parameter (see, e.g.Exercise 2.1). The composition law assumes the simplest form in canonical param-eters defined as follows.

Definition 2.2. A transformation group

x i = f i(x,a), i = 1, . . . ,n, (2.1.9)

where the functionsf i(x,a) are defined in a neighborhood ofa = 0 and satisfy theconditions

f i(x,0) = xi , i = 1, . . . ,n, (2.1.10)

is said to be written in acanonicalparametera if the composition rule (2.1.4) hasthe form

f i( f (x,a),b) = f i(x,a+b), i = 1, . . . ,n. (2.1.11)

Here,a andb are numerical values of the group parameter taken from a neighbor-hood ofa = 0.

It is manifest that in a canonical parameter the compositionlaw (2.1.7) is written

φ(a,b) = a+b. (2.1.12)

It will be shown in Section 2.1.6 that a canonical parameter can be introduced inany local one-parameter group by a suitable choice of the group parameter.

It follows from Eqs. (2.1.11) and (2.1.10) that

a−1 = −a. (2.1.13)

In other words, the inverse transformation (2.1.6) is obtained merely by changingthe sign ofa :

xi = f i(x,−a), i = 1, . . . ,n. (2.1.14)

Hence, in a canonical parameter, local groups are defined by the initial conditions(2.1.10) and the composition rule (2.1.11).

2.1.3 Infinitesimal transformations and generators

We will assume that a one-parameter groupG of transformations (2.1.9) is given ina canonical parametera and expand the functionsf i(x,a) into the Taylor series inthe group parametera in a neighborhood ofa= 0. Neglecting the terms of the ordero(a) and using the initial conditions (2.1.10), we arrive at the following infinitesimaltransformationof the groupG :

x i ≈ xi +aξ i(x), i = 1, . . . ,n, (2.1.15)


where

ξ i(x) =∂ f i(x,a)

∂a

∣∣∣a=0

. (2.1.16)

Geometrically, the infinitesimal transformation (2.1.15)–(2.1.16) defines the tan-gent vector

ξ (x) = (ξ 1(x), . . . ,ξ n(x))

at the pointx to theG-orbit of x (see Section 1.2.2). Therefore,ξ is called thetangentvector fieldof the groupG. The tangent vector field is often denoted byξ i and iswritten as a first-order linear differential operator1

X = ξ i(x)∂

∂xi · (2.1.17)

The operatorX is called theinfinitesimal generatorof the one-parameter group(2.1.9), or simply the groupgenerator.

The following statement shows that the differential operator (2.1.17) behaves,unlike thevectorξ i , as ascalarunder any change of variables.

Theorem 2.1. Consider a change of variables ˜x = ϕ(x), or in coordinates

xi = ϕ i(x), i = 1, . . . ,n. (2.1.18)

Let x(x) = ϕ−1(x) be the inverse to Eq. (2.1.18). The following equation holdsforany functionF(x) :

X(F(x)

)= X

(F(x(x))

). (2.1.19)

HereX is the differential operator obtained from the operator (2.1.17) by rewritingit in the new variables as follows:

X = X(ϕ1)∂

∂ x1 + · · ·+X(ϕn)∂

∂ xn ≡ X(ϕk)∂

∂ xk , (2.1.20)

where

X(ϕk) = ξ i ∂ϕk(x)∂xi ·

Proof. Using the chain rule for the partial derivatives,

∂∂xi =

n

∑k=1

∂ϕk

∂xi

∂∂ xk ,

we have:

X(F(x)

)= ξ i ∂F(x)

∂xi = ξ i ∂ϕk(x)∂xi

∂F(x(x))∂ xk

= X(ϕk)∂F(x(x))

∂ xk= X

(F(x(x))

), (2.1.21)

1 We use the summation convention, i.e. hereξ i(x)∂

∂xi ≡n

∑i=1

ξ i(x)∂

∂xi .


whereX is defined by Eq. (2.1.20).

Example 2.1. Consider the rotation group (1.2.29),


Denotingx1 = x, x2 = y, we obtain from Eqs. (2.1.16):

ξ 1 =∂ (x cosa+ysina)

∂a

∣∣∣a=0

= (−x sina+ycosa)∣∣a=0 = y,

ξ 2 =∂ (y cosa−x sina)

∂a

∣∣∣a=0

= −(y sina+x cosa)∣∣a=0 = −x.

Hence the infinitesimal transformation is written

x≈ x+ya, y≈ y−xa

and yields the following generator of the rotation group:

X = y∂∂x

−x∂∂y

· (2.1.22)

2.1.4 Lie equations

The following theorem, due to Lie, asserts that one-parameter local groups are de-termined by their infinitesimal transformations.

Theorem 2.2. Let G be a local group (2.1.9) with functionsf i obeying (2.1.10) andthe composition rule (2.1.11). Let Eq. (2.1.17) be the infinitesimal generator of thegroupG. Then the functionsx i = f i(x,a) solve the system of first-order ordinarydifferential equations (known as theLie equations):

dx i

da= ξ i(x), i = 1, . . . ,n, (2.1.23)

with the initial conditionsx i∣∣a=0 = xi , or in the compact form

dxda

= ξ (x), x∣∣a=0 = x. (2.1.24)

In other words, theG-orbit of a pointx∈ IRn is an integral curve of the Lie equations(2.1.23) passing through the pointx.

Conversely, given an infinitesimal transformation (2.1.15), or an operator (2.1.17),whereξ i(x) are continuously differentiable, the initial value problem (2.1.24) has aunique solutionx = f (x,a) in a neighborhood ofa = 0. This solution satisfies thegroup property (2.1.11). In other words, the solution of Lie’s equations provides aone-parameter local group with a given infinitesimal generator (2.1.17).


Proof. To prove the direct statement, let us setb = ∆a in Eq. (2.1.11):

f i(x,a+ ∆a) = f i( f (x,a),∆a), i = 1, . . . ,n, (2.1.25)

then expand both sides of Eq. (2.1.25) in powers of∆a and single out the leadingterms:

f i(x,a+ ∆a)≈ f i(x,a)+∂

∂a

[f i(x,a)

]·∆a,

f i( f (x,a),∆a) ≈ f i(x,a)+∂

∂∆a

[f i

(f (x,a),∆a

)]∆a=0

·∆a.

Invoking Eq. (2.1.16), one obtains

∂∂∆a

[f i

(f (x,a),∆a

)]∆a=0

= ξ i( f (x,a)).

Now we substitute the above expressions in Eq. (2.1.25) and obtain the Lie equations(2.1.23):

∂ f i(x,a)

∂a= ξ i( f (x,a)

), i = 1, . . . ,n.

To prove the second part of the theorem, it is necessary to verify the group prop-erty (2.1.11) alone. Indeed, existence and uniqueness of the local solution to theproblem (2.1.24) is guaranteed by the classical theorem on existence and unique-ness of the solution to the Cauchy problem. Thus, let us provethat the solutionx = f (x,a) of Eqs. (2.1.24) satisfies Eq. (2.1.11). Consider, for a fixedvaluea, thefunctionsu = (u1, . . . ,un) andv= (v1, . . . ,vn) of a variableb are defined as follows:

u(b) = f (x,b) ≡ f ( f (x,a),b), v(b) = f (x,a+b).

To prove the group property (2.1.11), it suffices to show thatu(b) = v(b) in a neigh-borhood ofb = 0. Since f (x,a) solves Eqs. (2.1.24), one has:

dudb

≡ d f (x,b)

db= ξ (u), u

∣∣b=0 = f (x,a);

anddvdb

≡ d f (x,a+b)

db= ξ (v), v

∣∣b=0 = f (x,a).

Hence, bothz= u(b) andz= v(b) solve the initial value problem

dzdb

= ξ (z), z∣∣b=0 = f (x,a).

One concludes from the uniqueness theorem thatu(b) = v(b) in a neighborhood ofb = 0, thus completing the proof.

Example 2.2. Let us solve the Lie equations for the operator


X = y∂∂x

−x∂∂y

·

Using the notation of Example 2.1, we write Eqs. (2.1.23) in the form

dxda

= y,dyda

= −x. (2.1.26)

The initial conditions are written

x∣∣a=0 = x, y

∣∣a=0 = y. (2.1.27)

A simple way to solve the system (2.1.26) is to reduce it to a single second-order equation. Namely, differentiating the first equationof (2.1.26) and substitutingdy/da from the second equation of (2.1.26), we obtain

d2xda2 +x = 0.

The general solution of this second-order equation is well-known (see, e.g. [17],Example 3.3.2) and has the form

x = C1cosa+C2sina.

Differentiatingx and invoking the first equation of (2.1.26) we get

y = C2cosa−C1sina.

Substituting the above expressions forx andy in Eqs. (2.1.27) for the initial condi-tions we obtainC1 = x, C2 = y, and hence arrive at the rotation group:


2.1.5 Exponential map

If the coefficientsξ i(x) of the generator (2.1.17) are analytic functions ofx =(x1, . . . ,xn), the corresponding group transformation (2.1.9) can be obtained in theform of an infinite series known as the exponential map. This result is given by thefollowing statement.

Theorem 2.3. If ξ i(x) are analytic functions, the solution to the initial value prob-lem (2.1.24) for the Lie equations (2.1.23) is given by theexponential map:

x i = eaX(xi), i = 1, . . . ,n, (2.1.28)

where eaX is the operator given by the infinite series


eaX = 1+a1!

X +a2

2!X2 + · · ·+ as

s!Xs+ · · · . (2.1.29)

Proof. Let us denoteyi = x i and write Eqs. (2.1.24) in the form

dyi

da= ξ i(y), yi

∣∣t=0 = xi , i = 1, . . . ,n. (2.1.30)

It is known that the solution to a Cauchy problem with analytic data is an analyticfunction. Therefore we can look for the solution of Eqs. (2.1.30) in the form of anconverging Taylor series

yi = xi +a1!

(dyi

da

)0+

a2

2!

(d2yi

da2

)0+ · · ·+ as

s!

(dsyi

das

)0+ · · · , (2.1.31)

where the zero means evaluated ata = 0. For any functionh = h(y), the chain ruleof differentiation and the differential equations (2.1.30) yield

(dhda

)0=

( ∂h∂yk

)0

(dyk

da

)0= ξ k ∂h

∂xk,

or (dhda

)0= X(h),

whereX is the operator (2.1.17),

X = ξ k(x)∂

∂xk.

It is manifest thatX(xi) = ξ i(x).

Therefore the repeated differentiations of Eqs. (2.1.30) yield

(dyi

da

)0= ξ i(x) = X(xi),

(d2yi

da2

)0=

(dξ i

da

)0= X(ξ i) = X2(xi),

and hence(dsyi

das

)0= Xs(xi), s= 1,2,3, . . . .

Accordingly, the development (2.1.31) takes the form

yi = xi +a1!

X(xi)+a2

2!X2(xi)+ · · ·+ as

s!Xs(xi)+ · · · . (2.1.32)

This completes the proof of the theorem.

Example 2.3. Let us construct the exponential map for the operator


X = y∂∂x

−x∂∂y

·

Using the notation of Example 2.1, we write Eqs. (2.1.28) in the form

x = eaX(x) = x+a1!

X(x)+a2

2!X2(x)+ · · ·+ as

s!Xs(x)+ · · · ,

y = eaX(y) = y+a1!

X(y)+a2

2!X2(y)+ · · ·+ as

s!Xs(y)+ · · · .

(2.1.33)

In our case we have:X(x) = y, X(y) = −x.

Using these equations, we obtain:

X2(x) = X(X(x)

)= X(y) = −x,

andX2(y) = X

(X(y)

)= X(−x) = −y.

The iteration gives:

X2s−1(x) = (−1)s−1y, X2s(x) = (−1)sx, s= 1,2, . . . ,

X2s−1(y) = (−1)sx, X2s(y) = (−1)sy, s= 1,2, . . . .

Thus,

X(x) = y, X2(x) = −x, X3(x) = −y, X4(x) = x, . . . ,

X(y) = −x, X2(y) = −y, X3(y) = x, X4(y) = y, . . . .(2.1.34)

Substituting Eq. (2.1.34) in Eq. (2.1.33) we obtain

x = x(

1− a2

2!+

a4

4!−·· ·

)+y

(a− a3

3!+

a5

5!−·· ·

),

y = y(

1− a2

2!+

a4

4!−·· ·

)−x

(a− a3

3!+

a5

5!−·· ·

),

and hence arrive again at the rotation group:


Remark 2.1. The exponential map (2.1.28) can be extended to any analyticfunc-tionsF(x) as follows:

F(x) = eaXF(x) ≡(1+

a1!

X +a2

2!X2 + · · ·+ as

s!Xs+ · · ·

)F(x). (2.1.35)


For the proof, see, e.g. [14], Problem 7.6∗.

2.1.6 Determination of a canonical parameter

Recall the definition of a canonical parameter introduced inSection 2.1.2.

Definition 2.3. The group parameter is said to becanonicalif the composition law(2.1.7) reduces to the addition,φ(a,b) = a+ b, i.e. if the group property (2.1.4) iswritten in the form (2.1.11),TbTa = Ta+b. Furthermore, equation (2.1.10) holds, i.e.I = T0.

Theorem 2.4. In any one-parameter group with an arbitrary composition law (2.1.7),c = φ(a,b), there exists a canonical parameter ˜a. It is defined by

a =

∫ a

a0

dsw(s)

, where w(s) =∂φ(s,b)

∂b

∣∣∣∣b=a0

. (2.1.36)

Proof. Let G be a local one-parameter group with an arbitrary composition law(2.1.7). Let us rewrite Eqs. (2.1.4) in the form

f (x,b) = f (x,c), c = φ(a,b),

and differentiate with respect tob to obtain:

∂ f (x,b)

∂b=

∂ f (x,c)∂c

∂φ(a,b)

∂b·

Letting hereb = a0 and invoking Eq. (2.1.8), we get

∂ f (x,b)

∂b

∣∣∣∣b=a0

=∂ f (x,a)

∂a

(∂φ(a,b)

∂b

)

b=a0

. (2.1.37)

Let us denote the second factor in the right-hand side of Eq. (2.1.37) byw(a). It fol-lows from the second equation of (2.1.8) thatw(a0) = 1 and, by continuity,w(a) 6= 0in a neighborhood ofa = a0.

If we denote (cf. (2.1.16))

ξ (x) =∂ f (x,a)

∂a

∣∣∣∣a=a0

,

Equation (2.1.37) will be written

ξ (x) = w(a)dxda

.

Introducing a new parameter ˜a = a(a) we rewrite this equation in the form


ξ (x) = w(a)dada

dxda

. (2.1.38)

Equation. (2.1.38) coincides with the Lie equation (2.1.23),

dxda

= ξ (x),

if we letdada

=1

w(a)· (2.1.39)

Integrating Eq. (2.1.39) and requiring that ã = 0 when a = a0, we obtain Eq.(2.1.36). Since now Eq. (2.1.38) coincides with the Lie equation (2.1.23), it definesby Theorem 2.2 a one-parameter group with the composition rule (2.1.11) with re-spect to the parameter ã. Hence, the composition law (2.1.12) holds:

φ(a, b) = a+ b.

Furthermore, the identical transformation corresponds toa = 0. Thus,a defined byEq. (2.1.36) is a canonical parameter.

Example 2.4. The transformation

x = x+ax, |a| < 1, (2.1.40)

defines a one-parameter group with the composition law

φ(a,b) = a+b+ab

and witha0 = 0. In this case, the formula (2.1.36) provides the functionw(s) = 1+s,and hence the canonical parameter is ã =

∫ a0 (1+s)−1ds, i.e.

a = ln(1+a).

Remark 2.2. Note that the Lie equations written in the form (2.1.23) establish acorrespondence between a one-parameter group and its infinitesimal transformation(2.1.15) for groups written in a canonical parameter but cannot be used, in general,if a group parameter is not canonical. Consider, e.g., the one-parameter group fromExample 2.4. Here the group transformation (2.1.40) is identical with its infinitesi-mal transformation, so thatξ (x) = x. Therefore Eqs. (2.1.24) are written

dxda

= x, x∣∣a=0 = x.

The solution of this initial value problem is given byx = xea. It does not coincidewith the initial group transformation (2.1.40).

Remark 2.3. In what follows we shall adopt the canonical parameter when refer-ring to one-parameter groups unless otherwise stipulated.


2.2 Invariants

2.2.1 Definition and infinitesimal test

Definition 2.4. A function F(x) is called an invariant of a groupG of transforma-tions (2.1.9) ifF(x) = F(x), i.e. if the equation

F( f (x,a)) = F(x)

is satisfied identically inx anda in a neighborhood ofa = 0.

Theorem 2.5. A function F(x) is an invariant of the groupG with the generatorX(2.1.17) if and only if it solves the homogeneous linear partial differential equation

X(F) ≡ ξ i(x)∂F(x)

∂xi = 0. (2.2.1)

Proof. Let F(x) be an invariant. Since

F( f (x,a)) ≈ F(x+aξ (x))≈ F(x)+aξ i(x)∂F(x)

∂xi ,

the invariance conditionF( f (x,a)) = F(x) leads to Eq. (2.2.1).Conversely, letF(x) be any solution of the differential equation (2.2.1). Since

Eq. (2.2.1) holds at any point one can consider it atx = f (x,a) :

ξ i(x)∂F(x)

∂x i = 0.

Hence, invoking the Lie equations (2.1.23) we have

dF( f (x,a))

da=

∂F(x)∂x i

d f i(x,a)

da= ξ i(x)

∂F(x)∂x i = 0.

SinceF( f (x,0)) = F(x) (see (2.1.10)), we conclude that

u(a) = F( f (x,a))

solves the initial value problem

duda

= 0, u∣∣a=0 = F(x).

The solution to the latter problem is unique and is obviouslygiven byu = F(x).Hence, the two solutions,u = F( f (x,a)) andu = F(x), are identical for anyx. Thisis precisely the invariance condition,F( f (x,a)) = F(x).

Alternative proof. If F(x) is an analytic function, one can prove the second partof Theorem 2.5 by using the extension (2.1.35) of the exponential map. Indeed, the

2.2 Invariants 35

equationX(F) = 0 implies

X2(F) = X(X(F)) = 0, . . . , Xs(F) = 0.

Consequently, equation (2.1.35) yieldsF(x) = F(x).

Corollary 2.1. If F(x) is an invariant then any functionΦ(F(x)) is also an invariant.

Proof. SinceF(x) is an invariant, we haveX(F) = 0. Consequently,

X(Φ(F)

)=

dΦdF

X(F) = 0.

Combining the above Theorem 2.5 with Theorem 4.2.1 from [17]we get thefollowing result.

Theorem 2.6. A one-parameter groupG of transformations in IRn has preciselyn−1 functionally independent invariants. Any set of independent invariants,ψ1(x), . . . ,ψn−1(x), is termed abasis of invariantsof G. Basis is not unique. One can take, asbasic invariants, the left-hand sides ofn−1 first integrals

ψ1(x) = C1, . . . ,ψn−1(x) = Cn−1 (2.2.2)

of thecharacteristic systemfor Eq. (2.2.1):

dx1

ξ 1(x)= · · · = dxn

ξ n(x)· (2.2.3)

An arbitrary invariantF(x) of G is given by the formula

F = Φ(ψ1(x), . . . ,ψn−1(x)) . (2.2.4)

Example 2.5. Consider the group of non-uniform dilations in IR3:

x = xea, y = ye2a, z= ze−2a ,

with the generator

X = x∂∂x

+2y∂∂y

−2z∂∂z

·

The system (2.2.3) has the form

dxx

=dy2y

= −dz2z

·

Integrating, e.g. the equationsdxx

=dy2y

anddxx

= −dz2z

,


one obtains two first integrals,yx−2 = C1 andzx2 = C2. Hence,

ψ1 = yx−2, ψ2 = zx2 (2.2.5)

provide a basis of invariants so that an arbitrary invariantis

F = Φ(yx−2,zx2).

On the other hand, if one integrates the equations

dyy

= −dzz

anddxx

= −dz2z

,

one obtains another basis of invariants,ψ ′1 = yz andψ ′

2 = zx2. Then an arbitraryinvariant is represented in the form

F = Φ(yz,zx2).

2.2.2 Canonical variables

Introduction of canonical variables is one of the basic methods for solving ordinarydifferential equations with known symmetries.

Recall that in new variablesx′i given by (2.1.18),

x′i = ϕ i(x), i = 1, . . . ,n, (2.2.6)

the operatorX (2.1.17) is written in the form (2.1.20):

X = X(ϕ i)∂

∂x′i, (2.2.7)

where the coefficients

X(ϕ i) = ξ j(x)∂ϕ i(x)

∂x j

of X should be expressed in terms of the new variablesx′ = (x′1, . . . ,x′n) by solvingEqs. (2.2.6) with respect toxi .

Definition 2.5. Canonical variables x′i for groupG of transformations (2.1.9) aredefined by the condition that the transformations (2.1.9) are written in these vari-ables as a translation, e.g. in the direction ofx′1 :

x′1 = x′1 +a, x′2 = x′2, . . . , x′n = x′n. (2.2.8)

2.2 Invariants 37

Theorem 2.7. Canonical variables exist for any one-parameter group.

Proof. Given a groupG with the generatorX (2.1.17), canonical variablesx′i for Gare furnished by the change of variables:

x′1 = ϕ(x), x′2 = ψ1(x), . . . , x′n = ψn−1(x), (2.2.9)

whereϕ(x) is any solution to the non-homogeneous linear equation

X(ϕ) ≡ ξ i(x)∂ϕ(x)

∂xi = 1, (2.2.10)

andψ1(x), . . . , ψn−1(x) is a basis of invariants of the groupG. Indeed, by definitionof basic invariants, the functionsψ1(x), . . . , ψn−1(x) are functionally independentand solve the equationX(ψs) = 0. Furthermore,ϕ(x) is functionally independentof ψs(x), otherwise it would solve the equationX(ϕ) = 0 instead of Eq. (2.2.10).Hence, the change of variables (2.2.9) is well defined and, according to Eq. (2.2.7),reduces the operatorX to the form

X =∂

∂x′1·

For the above generatorX, the Lie equations (2.1.23) are written

dx′1

da= 1,

dx′s

da= 0 (s= 2, . . . ,n−1). (2.2.11)

Solving these equations and using the initial conditions ¯x′i∣∣a=0 = x′i , one arrives at

the translation group (2.2.8).

Example 2.6. Consider the dilation group with the generator

X = x∂∂x

+2y∂∂y

−2z∂∂z

· (2.2.12)

Let us take a basis of invariants (2.2.5) from Example 2.5, i.e.

ψ1 = yx−2, ψ2 = zx2.

One can easily solve Eq. (2.2.10), e.g. by following the procedure given in [17],Section 4.3. Namely, lettingϕ = ϕ(x), one reduces the equationX(ϕ) = 1 to

xdϕdx

= 1

and ignoring the constant of integration, one obtainsϕ = ln |x|. Hence, the canonicalvariables (2.2.9) are

u = yx−2, v = zx2, w = ln |x|. (2.2.13)


Example 2.7. Let us find canonical variables for the projective group withthe gen-erator

X = x2 ∂∂x

+xy∂∂y

· (2.2.14)

The characteristic equationdxx

=dyy

provides the first integralxy

= C,

and hence the invariantu =

xy·

We easily solve the equationX(t)= 1 as in the previous example, i.e. lettingt = t(x),and obtain

t = −1x·

Thus, we have the canonical variables

u =xy, t = −1

x· (2.2.15)

In these variables the operator (2.2.14) is written

X =∂∂ t

and yields the group transformation

t = t +a, u = u. (2.2.16)

2.2.3 Construction of groups using canonical variables

Group transformations (2.1.9) with given infinitesimal generators are usually con-structed either by solving the Lie equations (2.1.24) or by using the exponential map(2.1.28). In certain complicated cases, the calculations can be simplified by addingthe third method based on introducing canonical variables (see [14], Section 7.1.9).Consider the following examples.

Example 2.8. Let us construct the group with the generator

X = x2 ∂∂x

+xy∂∂y

· (2.2.17)

2.2 Invariants 39

We know from Example 2.7 the canonical variables (2.2.15) for the operatorX andthe group transformation (2.2.16) in the canonical variables. Now we substitute inEq. (2.2.16) the expressions (2.2.15) fort,u and obtain:

−1x

= −1x

+a,xy

=xy·

Solving these equations forx, y we obtain the following transformation group gen-erated by the operator (2.2.17):

x =x

1−ax, y =

y1−ax

· (2.2.18)

Example 2.9. Let us find the group generated by the following operator:

X = (1+ t2)∂∂ t

+ tr∂∂ r

+(r − tvr)∂

∂vr− tvθ

∂∂vθ

−2tρ∂

∂ρ−4t p

∂∂ p

· (2.2.19)

This group describes a specific symmetry of the so-called shallow-water equations(see [14], Section 7.1.9). Heret is time, r andθ are the polar coordinates on theplane,vr andvθ are the components of velocity in the polar coordinates,ρ andp arethe density and pressure, respectively.

Solving the partial differential equations (see also Assignment to Part I, Problem9)

X(s) = 1, X(λ ) = 0, X(U) = 0, X(V) = 0,

X(R) = 0, X(P) = 0

we obtain the following canonical variables:

s= arctant, λ =r√

1+ t2, U = rvr −

tr2

1+ t2 ,

V = rvθ , R= (1+ t2)ρ , P = (1+ t2)2p.

(2.2.20)

In these variables the operatorX becomesX = ∂/∂sand yields the transformations

s= s+a, λ = λ , U = U, V = V, R= R, P = P.

Substituting here the expressions fors, . . . ,P and the similar expressions for thetransformed variables,

s= arctant, . . . , P = (1+ t2)2p,

we obtain the equations

arctant = arctant +a,r√

1+ t 2=

r√1+ t2

,


r vr −t r2

1+ t 2 = rvr −tr2

1+ t2 , rvθ = rvθ ,

(1+ t 2)ρ = (1+ t2)ρ , (1+ t 2)2p = (1+ t2)2p.

Solving them fort, . . . , p, we arrive at the following group transformations:

t =sina+ t cosacosa− t sina

, r =r

cosa− t sina,

vr = (cosa− t sina)vr + r sina, vθ = (cosa− t sina)vθ ,

ρ = (cosa− t sina)2ρ , p = (cosa− t sina)4p.

2.2.4 Frequently used groups in the plane

InvariantsCanonicalt,u

Transformations Generatorsψ(x,y)

so that

X = ∂/∂ tTranslations

alongx : x = x+a, y = y X =∂∂x

ψ = y t = x, u = ψ

alongy : x = x, y = y+a X =∂∂y

ψ = x t = y, u = ψ

alongkx+ ly = 0 :

x = x+ la, y = y−ka X = l∂∂x

−k∂∂y

ψ = kx+ ly t = x/l ,u= ψ

Rotation

x = xcosa+ysina, X = y∂∂x

−x∂∂y

ψ = x2 +y2 t = arctan(x/y),

y = ycosa−xsina u = ψLorentz transformation

x = xcosha+ysinha, X = y∂∂x

+x∂∂y

ψ = y2−x2 t = ln |y+x|,y = ycosha+xsinha u = ψGalilean transformation

x = x+ay, y = y X = y∂∂x

ψ = y t = x/y, u = ψ

Uniform dilation

x = xea, y = yea X = x∂∂x

+y∂∂y

ψ = y/x t = ln |x|, u = ψ

Non-uniform dilation

x = xea, y = yeka X = x∂∂x

+ky∂∂y

ψ = y/xk t = ln |x|, u = ψ

2.3 Invariant equations 41

Continued

InvariantsCanonicalt,u

Transformations Generatorsψ(x,y)

so thatX = ∂/∂ t

Projective transformations

x =x

1−ax, y =

y1−ax

, X = x2 ∂∂x

+xy∂∂y

ψ = y/x t = −1/x, u = ψ

x =x

1−ay, y =

y1−ay

X = xy∂∂x

+y2 ∂∂y

ψ = y/x t = −1/y, u = ψ

Linear fractional

x =x

1−ax, y =

y1−ay

X = x2 ∂∂x

+y2 ∂∂y

ψ =1y− 1

xt = −1/x, u = ψ

Conformal transformation

x =x+ar2

1+2ax+a2r2 , X = (y2−x2)∂∂x

ψ = y/r2 t = x/r2, u= ψ

y =y

1+2ax+a2r2 −2xy∂∂y

2.3 Invariant equations

2.3.1 Definition and infinitesimal test

Consider a system ofs equations in IRn :

Fσ (x) = 0, σ = 1, . . . ,s, (2.3.1)

wherex ∈ IRn ands< n. We impose the condition that the Jacobian matrix of thefunctionsFσ (x) has the ranks,

rank

(∂Fσ (x)

∂xi

)= s, (2.3.2)

at all pointsx satisfying Eqs. (2.3.1). The locus of solutionsx of the system of Eqs.(2.3.1) is an(n−s)-dimensional manifold (surface)M ⊂ IRn.

Definition 2.6. We say that Eqs. (2.3.1) areinvariant with respect to a groupG oftransformationsx = f (x,a), or that Eqs. (2.3.1)admit G, if x satisfies Eqs. (2.3.1)wheneverx solves Eqs. (2.3.1). In other words:

Fσ (x)∣∣∣(2.3.1)

= 0, σ = 1, . . . ,s. (2.3.3)

Geometrically, it means that transformations of the groupG carry any point of the(n−s)-dimensional surfaceM along this surface. In other words, the path curve ofthe groupG passing through any pointx∈ M lies in M. ThereforeM is also termedan invariant manifoldfor G.


We will employ further the generalization of the following simple observation.Using the Maclaurin expansion of sinx,

sinx = x− x3

3!+

x5

5!− x7

7!+ · · ·

we can write the function sinx nearx = 0 in the form

sinx = xh(x),

where

h(x) = 1− x2

3!+

x4

5!− x6

7!+ · · ·

is a bounded function nearx = 0. It is obvious that this example can be extended toany analytic function. Namely, iff (x) is an analytic function nearx= 0 and vanishesat x = 0 then its Maclaurin expansion has the form

f (x) = f ′(0)x+f ′′(0)

2!x2 +

f ′′′(0)

3!x3 + · · ·

and hencef (x) = xh(x), (2.3.4)

where

h(x) = f ′(0)+f ′′(0)

2!x+

f ′′′(0)

3!x2 + · · · .

Lemma 2.1. Let g(x) be an analytic function such thatg′(x) 6= 0 at pointsx satis-fying the equationg(x) = 0. If f (y) is an analytic function satisfying the conditionf (0) = 0, then there exists a regular functionh(x) such that

f (g(x)) = h(x)g(x). (2.3.5)

Proof. Using the conditions for the functiong(x) we introduce the new variabley = g(x) and write Eq. (2.3.4),f (y) = yh(y), or

f (g(x)) = g(x)h(g(x)).

Denoting hereh(g(x)) = h(x) we arrive at Eq. (2.3.5).We now return to invariant equations. LetG be a one-parameter group with a gen-

eratorX (2.1.17). Theinfinitesimal testfor a system of Eqs. (2.3.1) to be invariantunder the groupG is given by the following theorem.

Theorem 2.8. The system of Eqs. (2.3.1) is invariant under the groupG with theinfinitesimal generatorX if and only if

XFσ (x)∣∣∣(2.3.1)

= 0, σ = 1, . . . ,s, (2.3.6)

where the symbol∣∣(2.3.1)

means evaluated on the manifoldM (2.3.1).


Proof. Let the system of Eqs. (2.3.1) be invariant. One readily arrives at Eqs. (2.3.6)by substituting in Eq. (2.3.3) the infinitesimal expressionfor Fσ (x) :

Fσ (x) ≈ Fσ (x)+aXFσ(x).

Conversely, let us suppose that Eqs. (2.3.6) hold and derivethe invariance condi-tions (2.3.3). We will assume thatFσ (x) andXFσ (x) are analytic in a neighborhoodof the manifoldM (2.3.1). Using a multidimensional version of Lemma 2.1, we willwrite Eqs. (2.3.6) in the form

XFσ (x) = λ νσ (x)Fν(x), σ = 1, . . . ,s, (2.3.7)

where the coefficientsλ νσ (x) are bounded in a neighborhood of the varietyM. It

follows from Eqs. (2.3.7) that

X2Fσ = X(λ νσ )Fν + λ ν

σ X(Fν) =(X(λ µ

σ )+ λ νσ λ µ

ν)

Fµ .

Iteration and substitution into the exponential map (2.1.35) yields

Fσ (x) = Λ νσ (x)Fν (x). (2.3.8)

Equations (2.3.8) represent an equivalent form of Eqs. (2.3.3).

Remark 2.4. If one rewrites Eqs. (2.3.6) in the form

(ξ ·∇Fσ )∣∣M = 0,

it becomes evident that Eqs. (2.3.6) is the condition for thevector field

ξ = (ξ 1, . . . ,ξ n)

to be tangent to the manifoldM.

2.3.2 Invariant representation of invariant manifolds

Given a groupG, invariant manifolds forG can be equivalently represented by manydifferent systems of Eqs. (2.3.1). To develop a general procedure for constructinginvariant manifolds, it is advantageous to distinguishsingular and regular (non-singular) invariant manifolds.

Definition 2.7. Let M be an invariant manifold for a groupG. SinceG transportspointsx∈ M alongM, the action of the groupG confined toM is also a group. It istermed a group induced onM by G, or briefly aninduced group.The induced group,its generator (termed aninduced generator) and invariants will be denotedG, Xandψ , respectively.

Definition 2.8. Let G be a one-parameter group with the generator (2.1.17),


X = ξ i(x)∂

∂xi ·

An invariant manifoldM for G is said to beregular(with respect toG) if the inducedgeneratorX does not vanish, andsingular if X = 0. In other words,M is a regularinvariant manifold if at least one of the coefficientsξ i(x) does not vanish onM,and it is singular if all coefficientsξ i(x) of the generatorX vanish identically onM.Since manifolds are considered locally near generic pointsand since all functionsunder consideration are supposed to be continuous, these two possibilities cover allcases.

Example 2.10. Consider the groupG of transformationsx = xea,y = y with thegeneratorX = x∂/∂x. The equationx= 0 is invariant underG, and hence they-axisis an invariant manifold forG. It is singular sinceX = 0 whenx = 0.

Theorem 2.9. Let a system of Eqs. (2.3.1) admit a groupG. Let the invariant man-ifold M defined by the system of Eqs. (2.3.1) be regular with respect to G. ThenM can be represented by a system whose equations are given by invariant functionswith respect toG, i.e. in the form (cf. formula (2.2.4)):

Φσ (ψ1(x), . . . ,ψn−1(x)) = 0, σ = 1, . . . ,s, (2.3.9)

whereψ1(x), . . . , ψn−1(x) (2.3.10)

is a basis of invariants ofG. Hence, equations (2.3.9) with arbitrary functionsΦσ ofn−1 variables furnish the general form of regular invariant manifolds forG.

2.3.3 Proof of Theorem 2.9

The proof given here is due to L.V. Ovsyannikov [22], Section8.7 (see also [14],Section 7.2.2).

Let M be a regular invariant manifold given by Eqs. (2.3.1). Due tothe condition(2.3.2), the following change of variables is well defined:

x′1 = F1(x), . . . ,x′s = Fs(x), x′s+1 = ϕs+1(x), . . . ,x

′n = ϕn(x). (2.3.11)

In the new variables the manifoldM is given by the equations

x′σ = 0, σ = 1, . . . ,s. (2.3.12)

Then the induced groupG acts in the(n− s)-dimensional space IRn−s of variablesx′ = (x′s+1, . . . ,x′n). SinceM is regular, the induced generatorX 6= 0. Hence, theinduced group has preciselyn−s−1 independent invariants. In particular,

ψ1(x′), . . . , ψn−1(x

′) (2.3.13)


obtained from (2.3.10) by restricting them toM, are invariants ofG. Some of thesen− 1 invariants will be functionally dependent, since the number of independentinvariants is not greater thann− s− 1. Let n− s′− 1 of the invariants (2.3.13) beindependent. Sincen−s′−1≤ n−s−1, there exists′ ≥ s relations:

Φσ(ψ1(x

′), . . . , ψn−1(x′))

= 0, σ = 1, . . . ,s′. (2.3.14)

Assuming that the relations (2.3.14) (i.e. the functionsΦσ therein) are known, definean(n−s′)-dimensional manifoldM′ ⊂ IRn by the equations:

Φσ (ψ1(x), . . . ,ψn−1(x)) = 0, σ = 1, . . . ,s′. (2.3.15)

If x ∈ M, equations (2.3.15) reduce to relations (2.3.14). Hence,M ⊂ M′. Conse-quently, dimM′ ≥ dimM, i.e. n− s′ ≥ n− s. It follows thats′ ≤ s. However,s′ ≥ sby definition of relations (2.3.14). Hence,s′ = s. It follows from dimM′ = dimMand M ⊂ M′ that, locally,M′ = M. Thus, equations (2.3.15) furnish an invariantrepresentation (2.3.9) of the manifoldM. This completes the proof of the theorem.

Remark 2.5. Whenn = 2, the relation (2.3.14) reduces to

ψ1(x′) = C

with an arbitrary constantC.

2.3.4 Examples on Theorem 2.9

Example 2.11. It is shown in Example 2.10 that they-axis given by the equationx = 0 is a singular invariant manifold for the groupG of dilationsx = xea,y = y.Accordingly, they-axis cannot be represented via invariants of the groupG. Indeed,the general form of invariants ofG is ψ(y), so that the most general equation of theform (2.3.9) providesy = const., i.e. the straight lines parallel to thex-axis.

Example 2.12. Consider, in the(x,y,z) space, the paraboloid given by

F(x,y,z) ≡ x2 +y2−z= 0. (2.3.16)

Let G be the group of dilations

x = xea, y = yea, z= ze2a

with the generator

X = x∂∂x

+y∂∂y

+2z∂∂z

.

We haveF(x,y,z) = e2aF(x,y,z),


or in infinitesimals,XF = 2F.

It follows that Eq. (2.3.16),F = 0, is invariant under the groupG, but the functionF is not an invariant of this group.

The generatorX vanishes at the pointO = (0,0,0). Let us exclude this singularpoint from the paraboloid (2.3.16), denote byM the resulting regular invariant sur-face and find its invariant representation (2.3.9). In our example, a basis of invariants(2.3.10) is provided, e.g. by

ψ1 =x2

z, ψ2 =

y2

z·

Let us take the change of variables (2.3.11) in the form

z′ = z−x2−y2, x′ = x, y′ = y.

Then the manifoldM is given byz′ = 0, and the induced invariants are written

ψ1 =x′2

r2 , ψ2 =y′2

r2 ,

wherer2 = x′2 +y′2. It is manifest that

ψ1 + ψ2 = 1.

This is the relation (2.3.14). Hence, the invariant representation (2.3.9) of theparaboloid (2.3.16) with the originO excluded has the form

ψ1 + ψ2 = 1,

i.e.x2 +y2

z−1 = 0. (2.3.17)

Example 2.13. Consider Newton’s equations for Kepler’s problem in two dimen-sions (see Eqs. (2.2.8) in [17]):

wi + αxi

r3 = 0, i = 1,2, (2.3.18)

whereα is a positive constant and

wi =d2xi

dt2 ·

Let G be the group of simultaneous rotations of the vectorsx = (x1,x2) andw =(w1,w2), i.e. the group with the generator


X = x2 ∂∂x1 −x1 ∂

∂x2 +w2 ∂∂w1 −w1 ∂

∂w2 · (2.3.19)

Geometry provides the following four invariants:

ψ1 = |x| ≡√

(x1)2 +(x2)2, ψ2 = |w| ≡√

(w1)2 +(w2)2,

ψ3 = |x×w| ≡ |x1w2−x2w1|, ψ4 = x ·w ≡ x1w1 +x2w2.

But only three of them are independent. Indeed, they are connected by the relation

ψ23 + ψ2

4 = ψ21 ψ2

2 .

This relation follows from the definition of scalar and vector products,

x ·w = |x||w|cosω , |x×w|= |x||w|sinω ,

whereω is the angle betweenx andw. A basis of invariants is given by

ψ1 = |x|, ψ2 = |w|, ψ3 = |x×w|. (2.3.20)

The induced invariants are

ψ1 = |x|, ψ2 = α|x|−2, ψ3 = 0.

Hence, two relations of the form (2.3.14) are

ψ21 ψ2 = α, ψ3 = 0.

Therefore, the invariant representation (2.3.9) of Newton’s equations (2.3.18) canbe written in the form

ψ21 ψ2 = α, ψ3 = 0,

or invoking Eqs. (2.3.20), in the form

|x|2|w| = α, |x×w|= 0. (2.3.21)


Exercise 2.1.Find the group composition law (2.1.7) and the parametera−1 of theinverse transformation (2.1.6) for each of following two representations of the dila-tion group: (i)x = ax, (ii) x = eax.

Exercise 2.2.Find the group composition law (2.1.7) and the infinitesimalgenera-tor (2.1.17) for the Lorentz transformation group (1.2.31),

x = x cosha+y sinha, y = y cosha+x sinha.


Exercise 2.3.Find the group composition law (2.1.7) and the infinitesimalgenera-tor (2.1.17) for thespiral transformation group

x = ea(xcosa−ysina), y = ea(ycosa+xsina).

Exercise 2.4.Find the generator (2.1.17) of the conformal group (1.3.14),

x =x+ar2

1+2ax+a2r2 , y =y

1+2ax+a2r2 , (r2 = x2 +y2).

Exercise 2.5.Find the generator of the conformal group in the three-dimensionalspace (see Example 1.21),

x =x+ar2

1+2ax+a2r2 , y =y

1+2ax+a2r2 ,

z=z

1+2ax+a2r2 ,

wherer2 = x2 +y2+z2.

Exercise 2.6.Explain the major difference between global and local groups andprovide your own example of a local group.

Exercise 2.7.Construct the transformation group with the generator (2.2.12),

X = x∂∂x

+2y∂∂y

−2z∂∂z

,

by two methods: solving the Lie equations and using the exponential map. Thenverify thatu,v,w given by Eqs. (2.2.13),

u = yx−2, v = zx2, w = ln |x|,

are canonical variables for this group.

Exercise 2.8.Construct the transformation group with the generatorX from Exer-cise 2.7 by using the canonical variablesu,v,w given there.

Exercise 2.9. In the(t,x,u)-space, find the one-parameter group with the generator

X = 2t∂∂x

−xu∂∂u

,

by using all three methods: Lie equations, exponential map and canonical variables.Is this group local or global?

Exercise 2.10.Using the generator of the spiral transformation group found in Ex-ercise 2.3, check if the equation


√x2 +y2 = earctan(y/x)

is invariant with respect to the spiral group.

Exercise 2.11.Recall that the polar coordinatesr, θ and the rectangular Cartesiancoordinatesx,y in the plane are connected by the equations

x = r cosθ , y = r sinθ

orr =

√x2 +y2 , θ = arctan(y/x).

Rewrite the generator of the spiral transformation group found in Exercise 2.3 inthe polar coordinatesr, θ . Using this generator, construct the spiral transformationgroup in the polar coordinates.

Exercise 2.12.Solve Exercise 2.10 using the polar coordinates.

Exercise 2.13.Provide the calculations leading to the canonical variables (2.2.20)for the operator (2.2.19).

Chapter 3Groups admitted by differential equations

This chapter provides a simple introduction to definition and calculation of groupsadmitted by differential equations. An admitted group is called also a symmetrygroup. The generators of symmetry groups are often termed infinitesimal symme-tries or brieflysymmetriesof differential equations.

3.1 Preliminaries

The concept of a symmetry group of a differential equation can be specified eitherin terms of its solutions or its frame. Both definitions are useful in group analysis ofdifferential equations.

The first definition treats a symmetry group of a differentialequation as a groupof transformations mapping every solution of the equation into its solution. It doesnot need the theory of prolongations of transformations. This definition dealing withthe totality of solutions of a given differential equation is natural, but does not givea practical method for findingall symmetriesof an equation. See appropriate exam-ples in Section 3.4.

The second definition treats a symmetry group of a differential equation as agroup of transformations whoseprolongation, i.e. extension to the derivatives in-volved in a differential equation in question, leaves invariant theframeof the differ-ential equation. This differential algebraic definition does not assume knowledge ofsolutions. Moreover, it is crucial for the infinitesimal formulation of the invarianceof differential equations and provide a practical method for calculating symmetriesby solving so-calleddetermining equations.

3.1.1 Differential variables and functions

In the differential algebraic approach to symmetries we usethe following infinitenumber of variables (see [14], Chapter 8, or [17], Section 1.4):

51

52 3 Groups admitted by differential equations

x = {xi}, u = {uα}, u(1) = {uαi },

u(2) = {uαi1 i2

}, u(3) = {uαi1 i2 i3

}, . . . ,(3.1.1)

whereα = 1, . . . ,m; i, i1, . . . = 1, . . . ,n. The variablesuαi1 i2

, uαi1 i2 i3

, . . . are symmetricin the subscriptsi1 i2, i1i2 i3, . . . .

The main operation in the calculus of differential algebra is thetotal differentia-tion given by the following formal infinite sums:

Di =∂

∂xi +uαi

∂∂uα +uα

ii1

∂∂uα

i1

+uαii1 i2

∂∂uα

i1 i2

+ · · · , i = 1, . . . ,n. (3.1.2)

We avoid the question on convergence by using the action of total derivativesDi onfunctions involving only finite number of variables (3.1.1), e.g.

Di(

f (x,u,u(1)))

=∂ f∂xi +uα

i∂ f

∂uα +uαii1

∂ f∂uα

i1

·

In particular, lettingf = uα , f = uαj , . . . , one obtains:

uαi = Di(u

α), uαi j = Di(u

αj ) = DiD j(u

α), . . . . (3.1.3)

In view of Eqs. (3.1.3), the variablesuα are calleddifferential variables,andu(1),u(2), . . . are called their successivederivativeswith respect to the independentvariablesxi .

Definition 3.1. A differential function f= f (x, u, u(1), . . .) is a locally analyticfunction (i.e. locally expandable in a Taylor series with respect to all arguments)of a finite number of variables (3.1.1). The highest order of derivatives appearingin f is called the order of the differential function and is denoted by ord( f ), e.g.if f = f (x, u, u(1), . . . ,u(s)) then ord( f ) = s. The set of all differential functions offinite order is denoted byA .

It follows from this definition that the setA is a vector space endowed with theusual multiplication of functions. In other words, iff ,g ∈ A andα andβ are anyconstants then

α f + β g∈ A , ord(α f + β g) ≤ max{ord( f ),ord(g)},

f g∈ A , ord( f g) = max{ord( f ),ord(g)}.Furthermore, the spaceA is closed under the total differentiation: iff ∈ A , then

Di( f ) ∈ A , ord(Di( f )) = ord( f )+1.

Thus, if f ∈ A is a differential function of orders, its successive derivatives

Di( f ), D jDi( f ) = D j(Di( f )), . . .

3.1 Preliminaries 53

are differential functions of respective orderss+1,s+2, . . . . Furthermore, one canprove using the symmetry propertyuα

i j = uαji , . . . , that the successive action of total

differentiationsDi andD j does not depend on the order of differentiation:

D jDi( f ) = DiD j( f ).

3.1.2 Point transformations

Let G be a one-parameter group of transformations (see (2.1.9)) of independentvariablesx = (x1, . . . ,xn) and differential variablesu = (u1, . . . ,um) :

xi = f i(x,u,a), f i∣∣a=0 = xi , (3.1.4)

uα = ϕα(x,u,a), ϕα ∣∣a=0 = uα . (3.1.5)

The generator of the groupG is written in the form

X = ξ i(x,u)∂

∂xi + ηα(x,u)∂

∂uα , (3.1.6)

where

ξ i(x,u) =∂ f i(x,u,a)

∂a

∣∣∣∣a=0

, ηα(x,u) =∂ϕα(x,u,a)

∂a

∣∣∣∣a=0

. (3.1.7)

3.1.3 Frame of differential equations

Consider the equations

Fσ (x,u,u(1), . . . ,u(k)) = 0, σ = 1, . . . ,s, (3.1.8)

whereFσ ∈ A , σ = 1, . . . ,s, are any differential functions.In mathematical analysis, the differential variablesuα are treated as functions

uα(x) and consequently the variablesuαi , uα

i j , . . . are identified with the classicalderivatives

uαi =

∂uα(x)∂xi , uα

i j =∂ 2uα(x)∂xi∂x j , . . . .

In this way we obtain the classical definition of a system ofkth-order differentialequations

Fσ

(x,uα(x),

∂uα(x)∂xi , . . . ,

∂ kuα(x)∂xi1 · · ·∂xik

)= 0, σ = 1, . . . ,s. (3.1.9)


Definition 3.2. The system of Eqs. (3.1.8), wherex,u,u(1), . . . , u(k) are identifiedwith the variables (3.1.1), is called theframeof the system of the differential equa-tions (3.1.9).

3.2 Prolongation of group transformations

3.2.1 One-dimensional case

Let us begin with the case of one independent variablex and one differentialvariable which will be denoted byy. The successive derivatives are denoted byy′,y′′, . . . ,y(s), . . . . Then the total differentiation (3.1.2) is written:

Dx =∂∂x

+y′∂∂y

+y′′∂

∂y′+ · · ·+y(s+1) ∂

∂y(s)+ · · · . (3.2.1)

We will write the transformation (3.1.4)–(3.1.5) of a one-parameter groupG onthe(x,y) in the following form:

x = f (x,y,a), (3.2.2)

y = ϕ(x,y,a). (3.2.3)

Let us denote byDx the total differentiation with respect to the new independentvariablex defined by Eq. (3.2.2),x= f (x,y,a). The chain rule provides the followingrelation between the total differentiations in the old and new independent variables:

Dx = Dx( f )Dx, (3.2.4)

where

Dx( f ) ≡ Dx( f (x,y,a)) =∂ f∂x

+y′∂ f∂y

·

Now we differentiate Eq. (3.2.3) using the respective sidesof Eq. (3.2.4):

Dx( f )Dx(y) = Dx(ϕ(x,y,a)

).

SinceDx(y) = y′ by definition of the total differentiation, the above equation iswritten:

Dx( f )y ′ = Dx(ϕ(x,y,a)

). (3.2.5)

Dividing Eq. (3.2.5) byDx( f ) , we obtain the following transformation ofy′ :

y′ ≡ Dx(y) =Dx(ϕ)

Dx( f )· (3.2.6)

3.2 Prolongation of group transformations 55

Thus, starting from a point transformation (3.2.2)–(3.2.3) and then adding the for-mula (3.2.6), one obtains the transformation in the space ofthree independent vari-ables,x,y,y′. This is what is known as anextendedpoint transformation, or theprolongationof a point transformation. Specifically, the prolongation of point trans-formations to the first derivative is called thefirst prolongation.

Theorem 3.1. The first prolongation of the one-parameter groupG is again a one-parameter group. It is called aprolonged group.

Proof. We deal with a canonical parametera. Therefore the group property is givenby Eq. (2.1.11), or our case

x≡ f (x,y,b) = f (x,y,a+b),

y≡ ϕ(x,y,b) = ϕ(x,y,a+b).(3.2.7)

Let us rewrite transformation (3.2.6) in the form

y′ = ψ(x,y,y′,a) ≡ Dx(ϕ(x,y,a)

)

Dx(

f (x,y,a)) ·

To prove the theorem, we have to show that

y′ ≡ ψ(x,y,y′,b) = ψ(x,y,y′,a+b). (3.2.8)

We have

y′ = ψ(x,y,y ′,b) =Dx

(ϕ(x,y,b)

)

Dx(

f (x,y,b)) ·

Multiplying the numerator and denominator of the right-hand side of this equationby Dx( f (x,y,a)) and using Eqs. (3.2.4) and (3.2.7) we obtain:

Dx( f (x,y,a))Dx(ϕ(x,y,b)

)

Dx( f (x,y,a))Dx(

f (x,y,b)) =

Dx(ϕ(x,y,a+b)

)

Dx(

f (x,y,a+b)) = ψ(x,y,y′,a+b).

Hence, we have verified Eq. (3.2.8) thus completing the proof.Applying the prolongation procedure to the transformations (3.2.2), (3.2.3) and

(3.2.6), one obtains thesecond prolongationof the point transformation groupGacting in the space of the variablesx,y,y′,y′′. It is manifest that Theorem 3.1 appliesalso to the second prolongation. Continuing this procedureone can obtain prolon-gations of higher order.

3.2.2 Prolongation with several differential variables

Consider point transformations (3.1.4)–(3.1.5) with one independent variablex andm differential variablesu = (u1, . . . ,um) :


x = f (x,u,a),

uα = ϕα(x,u,a), α = 1, . . . ,m.(3.2.9)

In this case Eq. (3.2.6) for the prolongation to the first derivatives is replaced by

uα1 ≡ duα

dx=

Dx(ϕα)

Dx( f ), α = 1, . . . ,m. (3.2.10)

Equation (3.2.10) can also be rewritten in the form (3.2.5):

uα1 Dx( f ) = Dx(ϕα). (3.2.11)

3.2.3 General case

In the case of several independent and several differentialvariables the chain rule(3.2.4) is written

Di = Di( f j )D j . (3.2.12)

Acting by Eq. (3.2.12) on the first equation from Eq. (3.1.5),uα = ϕα(x,u,a),we obtain:

Di( f j )D j (uα) = Di (ϕα(x,u,a)) .

Invoking thatDi(u

α) = uαi

we arrive at the followingrule for the change of derivatives:

uαj Di( f j ) = Di(ϕα). (3.2.13)

Upon solving Eq. (3.2.13) with respect touαi , one obtains the transformation of the

first derivatives,uα

i = ψαi (x,u,u(1),a).

The prolongations to higher-order derivatives are obtained by further differenti-ating equation (3.2.13) by means of Eq. (3.2.12). The prolongation of a group oftransformations (3.1.4)–(3.1.5) to derivativesu(1), . . . ,u(s) of any order is again aone-parameter group (see Theorem 3.1) and is called theprolonged group.

3.3 Prolongation of group generators

3.3.1 One-dimensional case

The generator of the prolonged group can be obtained by applying Eqs. (2.1.16) and(2.1.17) to the extended transformation given by Eqs. (3.2.2), (3.2.3) and (3.2.6).

3.3 Prolongation of group generators 57

But the procedure can be simplified significantly by using theinfinitesimal transfor-mation (2.1.15),

x≈ x+aξ (x,y), (3.3.1)

y≈ y+aη(x,y), (3.3.2)

of the groupG instead of the transformation (3.2.2) and (3.2.3), or the generator

X = ξ (x,y)∂∂x

+ η(x,y)∂∂y

· (3.3.3)

Substituting the transformations (3.3.1) and (3.3.2) in Eq. (3.2.6), one obtains theinfinitesimal transformation ofy′ :

y′ ≈ y′ +aζ1(x,y,y′). (3.3.4)

Namely,

y′ ≈ Dx(y+aη)

Dx(x+aξ )=

y′ +aDx(η)

1+aDx(ξ )≈ (y′ +aDx(η))(1−aDx(ξ )),

ory ′ ≈ y′ +a(Dx(η)−y′Dx(ξ )). (3.3.5)

Comparing Eqs. (3.3.4) and (3.3.5) we obtain:

ζ1 = Dx(η)−y′Dx(ξ ). (3.3.6)

Thus, given a one-parameter group with the generator (3.3.3), theprolongation X(1)

of the operator toy′ has the form

X(1) = ξ∂∂x

+ η∂∂y

+ ζ1∂

∂y′· (3.3.7)

Equation (3.3.6) is called thefirst prolongation formula.

Example 3.1. Using the prolongation formula (3.3.6), one obtains the followingfirst prolongation of the generator (2.1.22) of the rotationgroup:

X(1) = y∂∂x

−x∂∂y

− (1+y′2)∂

∂y′· (3.3.8)

The (s+ 1)-times extended infinitesimal transformations, and hence the corre-sponding generatorsX(s+1) are obtained recursively:

y(s+1) ≈ (y(s+1) +aDx(ζs))(1−aDx(ξ )) (3.3.9)

≈ y(s+1) +a(Dx(ζs)−y(s+1)Dx(ξ )), (3.3.10)


whence, by settingy(s+1) ≈ y(s+1) +aζs+1,

we obtain thegeneral prolongation formula:

ζs+1 = Dx(ζs)−y(s+1)Dx(ξ ), s= 1,2, . . . . (3.3.11)

In practical applications, one often needs the coordinatesζ1,ζ2 and ζ3 of thesecond and third prolongations,X(2) andX(3). Namely,

ζ1 = D(η)−y′D(ξ ) = ηx +(ηy− ξx)y′− ξyy′2,

ζ2 = D(ζ1)−y′′D(ξ )

= ηxx+(2ηxy− ξxx)y′ +(ηyy−2ξxy)y

′2

− ξyyy′3 +(ηy−2ξx−3ξyy′)y′′, (3.3.12)

ζ3 = D(ζ2)−y′′′D(ξ )

= ηxxx+(3ηxxy− ξxxx)y′ +3(ηxyy− ξxxy)y

′2

+(ηyyy−3ξxyy)y′3− ξyyyy′4 +3[ηxy− ξxx+(ηyy−3ξxy)y

′

−2ξyyy′2]y′′−3ξyy′′2 +(ηy−3ξx−4ξyy′)y′′′.

Thus, the first, second, and third prolongations of the generator X (3.3.3) aregiven, respectively, by (3.3.7) and by

X(2) = ξ∂∂x

+ η∂∂y

+ ζ1∂

∂y′+ ζ2

∂∂y′′

,

X(3) = ξ∂∂x

+ η∂∂y

+ ζ1∂

∂y′+ ζ2

∂∂y′′

+ ζ3∂

∂y′′′·

(3.3.13)

Remark 3.1. The notationX(p) for p-times prolonged infinitesimal generators istaken from Eisenhart [9]. Lie [19] used for the infinitesimalgenerator the termsym-bol of an infinitesimal transformation and denoted it byU ; its first, second, etc.prolongations he denoted byU ′,U ′′, etc. Cohen [7] and Dickson [8] adopt Lie’snotation. In examples we will employ Eisenhart’s notation.In differential algebra,however, it is advantageous to consider generatorsX extended to all derivatives.ThenX acts in the universal spaceA , and there is no ambiguity if theprolongationof a generator X to all derivativesis denoted by the same symbolX. The operatorX understood in this way truncates when it acts on differential functions f ∈ A .Namely,X acts asX(1) if ord( f ) = 1, asX(2) if ord( f ) = 2, etc.


Remark 3.2. Theorem 3.1 guarantees that one can construct the first prolongationof a point transformation group by solving the Lie equations(2.1.24) for the pro-longed generator (3.3.7), i.e. the equations

dxda

= ξ (x,y), x∣∣a=0 = x,

dyda

= η(x,y), y∣∣a=0 = y,

dy′

da= ζ1(x,y,y

′), y′∣∣a=0 = y′.

(3.3.14)

This construction can be extended to all higher-order prolongations, e.g.

dxda

= ξ (x,y), x∣∣a=0 = x,

dyda

= η(x,y), y∣∣a=0 = y,

dy′

da= ζ1(x,y,y

′), y′∣∣a=0 = y′,

dy′′

da= ζ2(x,y,y

′,y′′), y′′∣∣a=0 = y′′

(3.3.15)

for the second prolongation.

3.3.2 Several differential variables

We substitute in Eq. (3.2.10) the infinitesimal transformation (3.2.9),

x≈ x+aξ (x,u),

uα ≈ uα +aηα(x,u),(3.3.16)

and obtain the following first prolongation formula (cf. (3.3.6)):

ζ α1 = Dx(ηα )−uα

1 Dx(ξ ), α = 1, . . . ,m. (3.3.17)

Furthermore, the general prolongation formula (3.3.11) isreplaced by

ζ αs+1 = Dx(ζ α

s )−uαs+1Dx(ξ ), s= 1,2, . . . . (3.3.18)

Hence, given a one-parameter group with the generator


X = ξ (x,u)∂∂x

+ ηα(x,u)∂

∂uα ,

we obtain, e.g., the twice-prolonged generator by the formula

X(2) = ξ (x,u)∂∂x

+ ηα(x,u)∂

∂uα + ζ α1

∂∂uα

1+ ζ α

2∂

∂uα2

(3.3.19)

whereζ α1 is given by Eq. (3.3.17) andζ α

2 has the form

ζ α2 = Dx(ζ α

1 )−uα2 Dx(ξ ). (3.3.20)

3.3.3 General case

Taking the infinitesimal transformation (3.1.4) and (3.1.5):

f i = xi +aξ i, ϕα = uα +aηα ,

and setting

uαi = uα

i +aζ αi ,

one obtains from Eq. (3.2.13) thefirst prolongation formula:

ζ αi = Di(ηα)−uα

j Di(ξ j). (3.3.21)

Hence, the prolongation of the generator (3.1.6) has the form:

X(1) = ξ i ∂∂xi + ηα ∂

∂uα + ζ αi

∂∂uα

i· (3.3.22)

Continuing this procedure, one obtains the twice-prolonged generator

X(2) = ξ i ∂∂xi + ηα ∂

∂uα + ζ αi

∂∂uα

i+ ζ α

i1i2

∂∂uα

i1i2

, (3.3.23)

wherei2 ≥ i1 andζ αi1i2

are defined by thesecond prolongation formula:

ζ αi1i2 = Di2(ζ

αi1 )−uα

ji1Di2(ξj) (3.3.24)

≡ Di2Di1(ηα )−uα

j Di2Di1(ξj)−uα

ji1Di2(ξj)−uα

ji2Di1(ξj).

The higher-order prolongations are defined recursively:

ζ αi1...is = Dis(ζ

αi1...is−1

)−uαji1...is−1

Dis(ξj). (3.3.25)


Example 3.2. Let us consider the second-order prolongation in the case oftwo in-dependent variablesx,y and one differential variableu. In the present case, the totalderivatives (3.1.2) are written

Dx =∂∂x

+ux∂

∂u+uxx

∂∂ux

+uxy∂

∂uy+ · · · ,

Dy =∂∂y

+uy∂

∂u+uxy

∂∂ux

+uyy∂

∂uy+ · · · ·

The generator (3.1.6) has the form

X = ξ 1(x,y,u)∂∂x

+ ξ 2(x,y,u)∂∂y

+ η(x,y,u)∂

∂u(3.3.26)

and its second prolongation is

X = ξ 1 ∂∂x

+ ξ 2 ∂∂y

+ η∂∂u

+ ζ1∂

∂ux+ ζ2

∂∂uy

(3.3.27)

+ζ11∂

∂uxx+ ζ12

∂∂uxy

+ ζ22∂

∂uyy·

The coefficientsζi are given by formula (3.3.21):

ζ1 = Dx(η)−uxDx(ξ 1)−uyDx(ξ 2)

= ηx +uxηu−uxξ 1x − (ux)

2ξ 1u −uyξ 2

x −uxuyξ 2u , (3.3.28)

ζ2 = Dy(η)−uxDy(ξ 1)−uyDy(ξ 2)

= ηy +uyηu−uxξ 1y −uxuyξ 1

u −uyξ 2y − (uy)

2ξ 2u . (3.3.29)

The coefficientsζi j are given by formula (3.3.24):

ζ11 = Dx(ζ1)−uxxDx(ξ 1)−uxyDx(ξ 2) (3.3.30)

= D2x(η)−uxD

2x(ξ

1)−uyD2x(ξ

2)−2uxxDx(ξ 1)−2uxyDx(ξ 2),

ζ12 = Dy(ζ1)−uxxDy(ξ 1)−uxyDy(ξ 2)

= DxDy(η)−uxDxDy(ξ 1)−uyDxDy(ξ 2) (3.3.31)

−uxxDy(ξ 1)−uxyDy(ξ 2)−uxyDx(ξ 1)−uyyDx(ξ 2),

ζ22 = Dy(ζ2)−uxyDy(ξ 1)−uyyDy(ξ 2) (3.3.32)

= D2y(η)−uxD

2y(ξ

1)−uyD2y(ξ

2)−2uxyDy(ξ 1)−2uyyDy(ξ 2).


3.4 First definition of symmetry groups

3.4.1 Definition

Definition 3.3. A groupG of transformations (3.1.4) and (3.1.5) is termed asymme-try group for a system of the differential equations (3.1.9) if the setof all solutionsof the system of Eqs. (3.1.9) is invariant under the groupG. It means that the so-lutions of the system are merely permuted among themselves (or are individuallyunaltered) by every transformation of the groupG. We also say that the system ofEqs. (3.1.9) isinvariant under the groupG, or thatG is admittedby Eqs. (3.1.9).The generatorX of a symmetry groupG is called aninfinitesimal symmetry, or anadmitted operatorfor Eqs. (3.1.9).

3.4.2 Examples

Example 3.3. Consider the second-order equation

y′′ = 0. (3.4.1)

Its general solution describes the straight lines

y = C1x+C2. (3.4.2)

Furthermore, equation (3.4.1) remains the same if we replace the independent anddependent variables,

x↔ y. (3.4.3)

In other words, if we letx = x(y), then Eq. (3.4.1) becomes

x′′ = 0. (3.4.4)

Indeed, the change of variables (3.4.3) yields (see [17], Example 1.4.1):

x′′ = − y′′

y′3·

Equation (3.4.4) describes the straight lines

x = C3y+C4. (3.4.5)

Equation (3.4.2) and (3.4.5) define the variety of all straight lines

kx+ ly+m= 0 (3.4.6)

with arbitrary constant coefficientsk, l ,m.

3.4 First definition of symmetry groups 63

The obvious symmetries of Eq. (3.4.1) are provided by the translations

x = x+a1, y = y,

x = x, y = y+a2(3.4.7)

and the dilations

x = xea3, y = y,

x = x, y = yea4(3.4.8)

with four arbitrary parametersa1, . . . ,a4. These transformations satisfy Definition3.3. Indeed, they map the solutions (3.4.2) of Eq. (3.4.1) among themselves accord-ing to the following equations:

y = C1 x+C2, C2 = C2−a1C1,

y = C1 x+C2, C2 = C2−a2

(3.4.9)

and

y = C1x+C2, C1 = C1e−a3,

y = C1x+C2, C1 = C1ea4, C2 = C2ea4.(3.4.10)

The transformations (3.4.7) and (3.4.8) provide four one-parameter groups with thefollowing infinitesimal generators:

X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

, X4 = y∂∂y

· (3.4.11)

It is also obvious that we can add toy a linear function inx, i.e. that the set of thestraight lines (3.4.2) is invariant under the transformation

x = x, y = y+a5x. (3.4.12)

Indeed, this transformation yields:

y = C1 x+C2, C1 = C1 +a5. (3.4.13)

The transformation (3.4.12) provides a one-parameter symmetry group with the gen-erator

X5 = x∂∂y

· (3.4.14)

We also know from Example 1.14 that the set of the straight lines (3.4.6) is in-variant under the projective group of transformation (1.3.8). Let us specify this in-variance property by considering the behaviour of the solutions (3.4.2) of Eq. (3.4.1)under the special projective group (1.3.10),


x =x

1−a6x, y =

y1−a6x

· (3.4.15)

The inverse transformation to Eqs. (3.4.15) has the form

x =x

1+a6x, y =

y1+a6x

,

whence1

1−a6x= 1+a6x. (3.4.16)

Dividing both sides of Eq. (3.4.2) by 1−a6x and using Eqs. (3.4.15) and (3.4.16)we obtain

y = C1 x+C2 (1+a6x)

ory = C1 x+C2, C1 = C1 +a6C2. (3.4.17)

We see that the set of the straight lines (3.4.2) is invariantunder the transformation(1.3.10). Hence the generator

X6 = x2 ∂∂x

+xy∂∂y

(3.4.18)

of the special projective one-parameter group (1.3.10) is an infinitesimal symmetryfor Eq. (3.4.1).

One can obtain more symmetries using the invariance of Eq. (3.4.1) under thereplacement (3.4.4) of the variables. Namely, after this replacement one obtains,instead of Eqs. (3.4.12) and (3.4.15), the transformations

x = x+a7y, y = y (3.4.19)

andx =

y1−a8y

, y =y

1−a8y, (3.4.20)

respectively. The transformations (3.4.19) and (3.4.20) map the straight lines (3.4.2)into the similar straight lines:

y = C1x+C2, C1 =C1

1+a7C1, C2 =

C2

1+a7C1, (3.4.21)

and

y = C1x+C2, C1 =C1

1−a8C2, C2 =

C2

1−a8C2, (3.4.22)

respectively. The generators of the one-parameter transformation groups (3.4.19)and (3.4.20) have the form

X7 = y∂∂x

, X8 = xy∂∂x

+y2 ∂∂y

· (3.4.23)

3.4 First definition of symmetry groups 65

The operatorsX7 andX8 can be obtained fromX5 andX6, respectively, by the re-placementx↔ y.

Thus, we have obtained the following eight linearly independent infinitesimalsymmetries of Eq. (3.4.1):

X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

, X4 = y∂∂y

, X5 = x∂∂y

,

X6 = x2 ∂∂x

+xy∂∂y

, X7 = y∂∂x

, X8 = xy∂∂x

+y2 ∂∂y

·(3.4.24)

A general theorem due to Lie states (see, e.g. [14], Theorem 2in Section 9.3.1and the references therein) that second-order ordinary differential equations mayadmit maximum eight linearly independent infinitesimal symmetries and that thismaximum is reached precisely by linear and linearizable equations. According tothis theorem, we conclude that we have found all infinitesimal symmetries of Eq.(3.4.1).

Example 3.4. Consider the second-order partial differential equation

uxuxx+uyy = 0 (3.4.25)

describing a potential steady-state transonic planar flow of a gas. Let

u = h(x,y) (3.4.26)

be any solution of Eq. (3.4.25), i.e.

hxhxx+hyy = 0. (3.4.27)

The obvious symmetries of Eq. (3.4.25) are provided by the translations

x = x+a1, y = y, u = u,

x = x, y = y+a2, u = u,

x = x, y = y, u = u+a3.

(3.4.28)

Consider, e.g. the first translation (3.4.28). It maps the solution (3.4.26) into thefollowing function:

u = h(x,y) ≡ h(x−a1,y).

Consequently,ux = hx, uxx = hxx, uyy = hyy,

and hence Eq. (3.4.27) guarantees thatu solves Eq. (3.4.25):

ux uxx +uyy = hxhxx+hyy = 0.


For the second and third translations from (3.4.28) the calculations are similar. Forexample, in the case of the third translation (3.4.28) we have:

u = a3 +h(x,y) ≡ a3 +h(x,y).

Consequently,

ux = hx, uxx = hxx, uyy = hyy,

and hence Eq. (3.4.27) guarantees thatu solves Eq. (3.4.25).It is also obvious that we can add tou a linear function iny because Eq. (3.4.25)

contains only the second-order derivative iny. Hence, equation (3.4.25) is invariantunder the transformation

x = x, y = y, u = u+a4y. (3.4.29)

It maps the solution (3.4.26) into the following function:

u = a4y+h(x,y)≡ a4y+h(x,y).

Consequently,

ux = hx, uxx = hxx, uyy = hyy,

and hence Eq. (3.4.27) guarantees thatu solves Eq. (3.4.25).Furthermore, the inspection shows that Eq. (3.4.25) is invariant under the follow-

ing dilations:

x = xea5, y = y, u = ue3a5,

x = x, y = yea6, u = ue−2a6.(3.4.30)

Indeed, the first dilation from of (3.4.30) maps the solution(3.4.26) into the function

u = e3a5 h(x,y) ≡ e3a5 h(xe−a5,y).

Consequently,

ux = e3a5 hx e−a5 = e2a5 hx, uxx = ea5 hxx, uyy = e3a5 hyy,

and hence Eq. (3.4.27) guarantees thatu solves Eq. (3.4.25):

ux uxx +uyy = e3a5 (hxhxx+hyy) = 0.

For the second transformation from of (3.4.30) the calculations are similar. See Ex-ercise 3.9.

The generators of the transformations (3.4.28), (3.4.29) and (3.4.30) provide thefollowing six linearly independent infinitesimal symmetries for Eq. (3.4.25):

3.5 Second definition of symmetry groups 67

X1 =∂∂x

, X2 =∂∂y

, X3 =∂

∂u, X4 = y

∂∂u

,

X5 = x∂∂x

+3u∂∂u

, X6 = y∂∂y

−2u∂∂u

·(3.4.31)

The above calculations do not guarantee, however, that (3.4.31) contains all in-finitesimal symmetries of Eq. (3.4.25). This can be done by solving the determiningequations introduced in the next section.

3.5 Second definition of symmetry groups

3.5.1 Definition and determining equations

Definition 3.4. A groupG of transformations (3.1.4) and (3.1.5) is termed asym-metry groupfor a system of thekth-order differential equations (3.1.9) if the frame(3.1.8) of the system of Eqs. (3.1.9) is invariant under thekth-order prolongation ofthe groupG.

Theorem 3.2. The system of the differential equations (3.1.9) is invariant under thegroup with an infinitesimal generator (3.1.6),

X = ξ i(x,u)∂

∂xi + ηα(x,u)∂

∂uα ,

if and only if

XFσ(x,u,u(1), . . . ,u(k)

)∣∣∣(3.1.8)

= 0, σ = 1, . . . ,s. (3.5.1)

In Eqs. (3.5.1),X denotes thekth-order prolongation of the generator (3.1.6), andthe symbol|(3.1.8) means evaluated on the frame (3.1.8).

Proof. The statement is a consequence of Definition 3.4 and Theorem 2.8 fromSection 2.3.1.

Definition 3.5. Equations (3.5.1) determine all infinitesimal symmetries of the dif-ferential equations (3.1.9) and therefore they are known asdetermining equations.

Remark 3.3. The determining equations (3.5.1) can also be written in theform(2.3.7),

XFσ = φνσ Fν , σ = 1, . . . ,s, (3.5.2)

where the undeterminate coefficientsφνσ (x,u,u(1), . . .) ∈ A are bounded on the

frame (3.1.8).

Determining equations (3.5.1) seem, at first glance, to be more complicated thanthe differential equations (3.1.9) in question. However, in many practical applica-tions, the determining equations can be integrated. This isdue to the fact that the


left-hand sides of Eqs. (3.5.1) involve not only the variablesx,u and the functionsξ i(x,u) andηα (x,u), but also the derivativesu(1),u(2), . . . . Accordingly, each equa-tion of (3.5.1) splits into several equations because Eqs. (3.5.1) should be satisfiedidentically with respect to all variables involved in the determining equations. Theresulting equations provide an overdetermined system for the unknown functionsξ i(x,u) andηα (x,u). The examples considered below clarify the situation.

3.5.2 Determining equation for second-order ODEs

Let us restrict consideration to second-order equations written in the form

y′′ = f (x,y,y′). (3.5.3)

We look for the infinitesimal symmetries written in the form

X = ξ (x,y)∂∂x

+ η(x,y)∂∂y

(3.5.4)

and obtain from Eqs. (3.5.1) the followingdetermining equationfor the unknowncoefficientsξ andη :

X(

y′′− f (x,y,y′))∣∣∣

(3.5.3)≡

(ζ2− ζ1 fy′ − ξ fx−η fy

)∣∣y′′= f = 0. (3.5.5)

After substitutingζ1 andζ2 from (3.3.12), equation (3.5.5) is written in the follow-ing form:

ηxx+(2ηxy− ξxx)y′ +(ηyy−2ξxy)y′2−y′3ξyy− ξ fx−η fy

+(ηy−2ξx−3y′ξy) f − [ηx +(ηy− ξx)y′−y′2ξy] fy′ = 0.(3.5.6)

Here f (x,y,y′) is a known function when one deals with a given differential equation(3.5.3). Since Eq. (3.5.6) involves all three variablesx,y andy′, buty′ does not occurin ξ and η , the determining equation (3.5.6) decomposes into several equations,thus becoming an over-determined system of differential equations forξ and η .After solving this system, one finds all generators of point transformations admittedby Eq. (3.5.3).

3.5.3 Examples on solution of determining equations

Example 3.5. Let us find all symmetries of the second-order equation

y′′ = 0 (3.5.7)


by solving the determining equation (3.5.5). This equationwas considered in Ex-ample 3.3 for illustrating the first definition of symmetry groups.

The determining equation (3.5.5) for Eq. (3.5.7) is written

ζ2∣∣y′′=0 = 0.

Substituting here the expression (3.3.12) forζ2 we obtain the equation

ηxx+(2ηxy− ξxx)y′ +(ηyy−2ξxy)y

′2−y′3ξyy = 0. (3.5.8)

Since Eq. (3.5.8) should hold identically inx,y,y′, it splits into the following equa-tions obtained by nullifying the coefficients for differentdegrees ofy′ :

ηxx = 0, 2ηxy− ξxx = 0, ηyy−2ξxy = 0, ξyy = 0. (3.5.9)

Integrating the first and the last equations of (3.5.9) we get:

ξ = φ1(x)y+ φ2(x), η = ψ1(y)x+ ψ2(y).

Differentiating the second equation of (3.5.9) with respect to x and the third equationwith respect toy, we obtain

ξxxx = 0, ηyyy = 0.

It follows thatφi(x) andψi(y) are quadratic functions ofxandy, respectively. Hence,

ξ = C1 +C3x+C4y+C7x2 +C8xy+Ax2y,

η = C2 +C5x+C6y+C9y2 +C10xy+Bxy2.

Substitution of the above expressions forξ , η into the second and third equationsof (3.5.9) yieldsAy+C7 = 2By+C10, 2Ax+C8 = Bx+C9, whenceA = B =0, C10 = C7, C9 = C8. We conclude that the following general solution to Eqs.(3.5.9) is given by the following functions depending on eight arbitrary constantsCi :

ξ = C1 +C3x+C4y+C7x2 +C8xy,

η = C2 +C5x+C6y+C7xy+C8y2.(3.5.10)

Substituting (3.5.10) in (3.5.4) we obtain the general infinitesimal symmetry

X = C1X1 +C2X2 +C3X3 +C4X4 +C5X5 +C6X6 +C7X7 +C8X8

for Eq. (3.5.7), where


X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

, X4 = y∂∂x

, X5 = x∂∂y

,

X6 = y∂∂y

, X7 = x2 ∂∂x

+xy∂∂y

, X8 = xy∂∂x

+y2 ∂∂y

·(3.5.11)

The operators (3.5.11) provide eight linearly independentinfinitesimal symme-tries for Eq. (3.5.7) and coincide with the symmetries (3.4.24).

Example 3.6. Let us find all symmetries of the equation

y′′ +e3yy′4 +y′2 = 0. (3.5.12)

Substitutingf = −(e3yy′4 +y′2) in the determining equation (3.5.6) we have

ηxx+(2ηxy− ξxx)y′ +(ηyy−2ξxy)y

′2−y′3ξyy

+3e3yy′4η − (ηy−2ξx−3y′ξy)(e3yy′4 +y′2)

+[ηx +(ηy− ξx)y′−y′2ξy](4e3yy′3 +2y′) = 0.

The left-hand side of this equation is a polynomial of fifth degree iny′. We proceedas in the previous example, i.e. equate to zero the coefficients of y′5,y′4, . . . andobtain the following four independent equations:

(y′

)5: ξy = 0,

(y′

)4: 3(ηy + η)−2ξx = 0,

(y′

)3: ηx = 0,

(y′

)1 : ξxx = 0.

The coefficients for(y′

)2and

(y′

)0vanish together with the coefficients of

(y′

)4

and(y′

)1, respectively.

Let us solve the above four differential equations forξ (x,y) andη(x,y). Thefirst equation of this system,ξy = 0, shows thatξ = ξ (x). Then the fourth equation,ξxx = 0, is writtenξ ′′(x) = 0 and yields

ξ = K1 +K2x, K1,K2 = const.

Likewise, we obtain from the third equation,ηx = 0, thatη = η(y). Finally, substi-tuting the expressions forξ andη into the second equation,

3(ηy + η)−2ξx = 0,

we obtain the first-order linear ordinary differential equation

η ′ + η =23

K2.


Its integration yields:

η =23

K2 +K3e−y.

DenotingK3 = C1, K1 = C2 and 3C3 = K2, we have:

ξ = C2 +3C3x, η = 2C3 +C1e−y,

whereC1,C2,C3 are arbitrary constants. Hence, the operator (3.5.4) admitted by Eq.(3.5.12) has the form

X = C1X1 +C2X2 +C3X3,

where

X1 = e−y ∂∂y

, X2 =∂∂x

, X3 = 3x∂∂x

+2∂∂y

· (3.5.13)

The operators (3.5.13) provide three linearly independentinfinitesimal symmetriesfor Eq. (3.5.12).

Example 3.7. It is shown in [17], Section 6.5.1, that the equation

y′′ =y′

y2 − 1xy

(3.5.14)

has two independent symmetries

X1 = x2 ∂∂x

+xy∂∂y

, X2 = 2x∂∂x

+y∂∂y

· (3.5.15)

Solving the Lie equations for the above generatorsX1 andX2, we obtain the admittedone-parameter groups of transformations

Ta : x =x

1−ax, y =

y1−ax

(3.5.16)

andTb : x = xe2b, y = yeb, (3.5.17)

respectively, wherea andb are arbitrary parameters. The reckoning shows that Def-inition 3.4 for the invariance of Eq. (3.5.14) under the transformations (3.5.16) and(3.5.17) is satisfied in the forms (cf. Eq. (2.3.3))

y′′− y′

y2 +1xy

= (1−ax)

(y′′− y′

y2 +1xy

),

and

y′′− y′

y2 +1xy

= e−3b(

y′′− y′

y2 +1xy

),

respectively. Besides, equation (3.5.14) has the discretesymmetry provided by thereflection with respect to thex axis:


T∗ : x = x, y = −y. (3.5.18)

For this transformation the invariance condition is written

y′′− y′

y2 +1xy

= −(

y′′− y′

y2 +1xy

).

The composition of the transformations (3.5.16), (3.5.17)and (3.5.18) provides thegeneralmixedgroup of symmetries for Eq. (3.5.14).

Example 3.8. Consider again Eq. (3.4.25) from Example 3.4:

uxuxx+uyy = 0. (3.5.19)

We seek the infinitesimal symmetries in the form (3.3.26),

X = ξ 1(x,y,u)∂∂x

+ ξ 2(x,y,u)∂∂y

+ η(x,y,u)∂∂u

,

with unknown coefficientsξ 1, ξ 2 andη to be found from Eq. (3.5.1):

(uxxζ1 +uxζ11+ ζ22)∣∣(3.5.19) = 0, (3.5.20)

where we substituteζ1,ζ11 andζ22 given by Eqs. (3.3.28), (3.3.30) and (3.3.32),respectively. We work out the expressions (3.3.30) and (3.3.32) for ζ11, ζ22 andobtain:

ζ11 = ηxx+2uxηxu+uxxηu +(ux)2ηuu−2uxxξ 1

x −uxξ 1xx (3.5.21)

−2(ux)2ξ 1

xu−3uxuxxξ 1u − (ux)

3ξ 1uu−2uxyξ 2

x −uyξ 2xx

−2uxuyξ 2xu− (uyuxx+2uxuxy)ξ 2

u − (ux)2uyξ 2

uu,

ζ22 = ηyy+2uyηyu+uyyηu +(uy)2ηuu−2uyyξ 2

y −uyξ 2yy (3.5.22)

−2(uy)2ξ 2

yu−3uyuyyξ 2u − (uy)

3ξ 2uu−2uxyξ 1

y −uxξ 1yy

−2uxuyξ 1yu− (uxuyy+2uyuxy)ξ 1

u −ux(uy)2ξ 1

uu.

After substituting the expanded expressions (3.3.28), (3.5.21), and (3.5.22) forζ1, ζ11, ζ22 and eliminatinguyy by settinguyy = −uxuxx, the determining equa-tion (3.5.20) contains the variablesx, y, u, ux, uy, uxx, uxy, whereasξ 1, ξ 2, η dependonly onx, y, u. Accordingly, we isolate the terms containinguxy, uxx, ux, uy, and theterms free of these variables, and set each term equal to zero. So, the terms contain-ing uxy lead to the equation

ξ 1y +uxξ 2

x +uyξ 1u +(ux)

2ξ 2u = 0,


whenceξ 1

y = 0, ξ 1u = 0, ξ 2

x = 0, ξ 2u = 0. (3.5.23)

The same argument applied to the terms containinguxx yields

ηx = 0, ηu−3ξ 1x +2ξ 2

y = 0. (3.5.24)

Then Eq. (3.5.20) reduces to

ηyy = 0, 2ηyu− ξ 2yy = 0. (3.5.25)

Thus, we have decomposed Eq. (3.5.20) into the overdetermined system of linearpartial differential equations (3.5.23)–(3.5.25).The reckoning shows that the generalsolution of this system is given by

ξ 1 = C5x+C1, ξ 2 = C6y+C2, η = (3C5−2C6)u+C4y+C3

and contains six arbitrary constantsCi . Proceeding as in Example 3.5 we obtain thefollowing six linearly independent infinitesimal symmetries:

X1 =∂∂x

, X2 =∂∂y

, X3 =∂

∂u, X4 = y

∂∂u

,

X5 = x∂∂x

+3u∂∂u

, X6 = y∂∂y

−2u∂∂u

·(3.5.26)

They coincide with the symmetries (3.4.31) discussed in Example 3.4.


Exercise 3.1.Verify that the transformation

x =x

1−ax, y =

y(1−ax)2

satisfies the group property, find the group composition law (2.1.7) and constructthe group generator.

Exercise 3.2.Verify that the transformation

x = x, y = y+ax2

satisfies the group property and find the group generator.

Exercise 3.3.Construct the first prolongation of the rotation group (1.2.29),

x = x cosa+ysina, y = y cosa−xsina,


by using the formula (3.2.6).

Exercise 3.4.Construct the first prolongation of the rotation group (1.2.29) by solv-ing the Lie equations (3.3.14) for the prolonged generator (3.3.8).

Exercise 3.5.Give the detailed derivation of the first prolongation formula (3.3.17).

Exercise 3.6.Derive the inverse transformation to the special projective group(3.4.15).

Exercise 3.7.Construct the composition (successive application) of thetransforma-tions (3.4.7), (3.4.8), (3.4.12), (3.4.15), (3.4.19), and(3.4.20). Compare the resultwith the general projective group (1.3.8).

Exercise 3.8.Derive the transformations (3.4.21) and (3.4.22).

Exercise 3.9.Show that the second transformation from (3.4.30),

x = x, y = yea6, u = ue−2a6,

maps the solution (3.4.26) into a solution of Eq. (3.4.25).

Exercise 3.10.Expand the expression (3.3.31) forζ12. See Eq. (3.5.21).

Chapter 4Lie algebras of operators

Examples on calculation of infinitesimal symmetries of differential equations ex-hibit the specific property of the determining equations. This property states that theset of all solutions of the determining equations for a givendifferential equation ora system of differential equations is avector space closed with respect to the com-mutator of infinitesimal generators.Such vector spaces are calledLie algebras.Inthis chapter we outline basic concepts and properties from the theory of Lie algebrasof operators. Some of results are proved if the proofs serve practical needs of Liegroup analysis of differential equations.

4.1 Basic definitions

4.1.1 Commutator

Consider two first-order linear differential operators of the form (2.1.17):

X1 = ξ i1(x)

∂∂xi , X2 = ξ i

2(x)∂

∂xi · (4.1.1)

Definition 4.1. The commutator[X1,X2] of the operators (4.1.1) is defined by theformula

[X1,X2] = X1X2−X2X1. (4.1.2)

Lemma 4.1. The commutator (4.1.2) is equal to the following linear partial differ-ential operator:

[X1,X2] =(

X1(ξ i2)−X2(ξ i

1)) ∂

∂xi · (4.1.3)

Proof. Let φ(x) be any differentiable function ofx = (x1, . . . ,xn). We have:

X1(φ) = ξ i1

∂φ∂xi , X2(φ) = ξ i

2∂φ∂xi ·

75

76 4 Lie algebras of operators

Therefore

X1 (X2(φ)) = X1(ξ i

2

) ∂φ∂xi + ξ j

1ξ i2

∂ 2φ∂xi∂x j

and

X2 (X1(φ)) = X2(ξ i

1

) ∂φ∂xi + ξ j

2ξ i1

∂ 2φ∂xi∂x j ·

Exchanging the summation indicesi and j in the last term forX2 (X1(φ)) , we obtain:

X1 (X2(φ))−X2(X1(φ)) =(

X1(ξ i

2

)−X2

(ξ i

1

)) ∂φ∂xi ,

or

[X1,X2](φ) =(

X1(ξ i

2

)−X2

(ξ i

1

)) ∂φ∂xi ·

Sinceφ(x) is an arbitrary function, the latter equation means that theoperator equa-tion (4.1.3) holds. This completes the proof.

Example 4.1. Let us find the commutator of the operators (3.5.15),

X1 = x2 ∂∂x

+xy∂∂y

, X2 = 2x∂∂x

+y∂∂y

·

Equation (4.1.3) yields

[X1,X2] =(

X1(2x)−X2(x2)

) ∂∂x

+(

X1(y)−X2(xy)) ∂

∂y·

We have:

X1(2x) = 2x2, X2(x2) = −4x2, X1(y) = xy, X2(xy) = 3xy.

Consequently, we obtain

[X1,X2] = −2x2 ∂∂x

−2xy∂∂y

,

and hence we have the following commutator of the operators (3.5.15):

[X1,X2] = −2X1. (4.1.4)

Example 4.2. Let us find the commutators for the operators (3.5.13),

X1 = e−y ∂∂y

, X2 =∂∂x

, X3 = 3x∂∂x

+2∂∂y

·

It is convenient to dispose the commutators in acommutator tablewhose entry at theintersection of theXi row with theXj column is[Xi ,Xj ]. It is manifest that the com-mutator (4.1.2) is antisymmetric. Therefore the commutator table is antisymmetric

4.1 Basic definitions 77

as well, with zeros on the main diagonal. Proceeding as in theprevious example,one obtains the following commutator table for the operators (3.5.13).

X1 X2 X3

X1 0 0 2X1

X2 0 0 3X2

X3 −2X1 −3X2 0

(4.1.5)

Example 4.3. Computing the commutators of the operators (3.5.26),

X1 =∂∂x

, X2 =∂∂y

, X3 =∂

∂u, X4 = y

∂∂u

,

X5 = x∂∂x

+3u∂∂u

, X6 = y∂∂y

−2u∂∂u

,

one obtains the following commutator table:

X1 X2 X3 X4 X5 X6

X1 0 0 0 0 X1 0

X2 0 0 0 X3 0 X2

X3 0 0 0 0 3X3 −2X3

X4 0 −X3 0 0 3X4 −3X4

X5 −X1 0 −3X3 −3X4 0 0

X6 0 −X2 2X3 3X4 0 0

(4.1.6)

4.1.2 Properties of the commutator

It follows from the definition (4.1.2) of the commutator thatit is bilinear:


[c1X1 +c2X2,X] = c1[X1,X]+c2[X2,X],

[X,c1X1 +c2X2] = c1[X,X1]+c2[X,X2],(4.1.7)

wherec1,c2 are arbitrary constants, and skew-symmetric:

[X1,X2] = −[X2,X1]. (4.1.8)

Furthermore, the following equation (termed theJacobi identity)

[X, [Y,Z]]+ [Y, [Z,X]]+ [Z, [X,Y]] = 0 (4.1.9)

is satisfied for any three operatorsX,Y,Z of the form (2.1.17).

Lemma 4.2. The commutator (4.1.2) is invariant under changes of variables. Namely,let Xα denote the operatorsXα written in the variables ˜xi given by Eqs. (2.1.18).Then

[Xα ,Xβ ] = [Xα ,Xβ ]. (4.1.10)

The proof of Lemma 4.2 in the two-dimensional case is given further, in Section4.4.2, Lemma 4.10. For the general case, see, e.g. [14], Section 4.5.2.

Lemma 4.3. The commutator (4.1.2) is compatible with the prolongation(3.3.22).Namely, letX andY be two operators of the form (3.1.6),

X = ξ i1(x,u)

∂∂xi + ηα

1 (x,u)∂

∂uα , Y = ξ i2(x,u)

∂∂xi + ηα

2 (x,u)∂

∂uα ,

and letX(1), Y(1) and [X,Y](1) be the first prolongations (3.3.22) of the operatorsX, Y and of their commutator[X,Y], respectively. Then

[X,Y](1) = [X(1),Y(1)]. (4.1.11)

In words:The prolongation of the commutator is equal to the commutator of pro-longations.The iteration of Eq. (4.1.11) yields the similar equation for higher-orderprolongations:

[X,Y](k) = [X(k),Y(k)]. (4.1.12)

Lemma 4.3 is an infinitesimal consequence of the Propositiongiven in Preface.Let us illustrate Lemma 4.3 by means of the operators (3.5.15) from Example 4.1.We take the first prolongation of the operators (3.5.15):

X1(1) = x2 ∂∂x

+xy∂∂y

+(y−xy′)∂

∂y′, X2(1) = 2x

∂∂x

+y∂∂y

−y′∂∂y

,

and calculate their commutator to obtain:

[X1(1),X2(1)] = −2x2 ∂∂x

−2xy∂∂y

−2(y−xy′)∂

∂y′·

Hence,

4.1 Basic definitions 79

[X1(1),X2(1)] = −2X1(1).

On the other hand, equation (4.1.4) yields

[X1,X2](1) = −2X1(1).

Thus,[X1,X2](1) = [X1(1),X2(1)], i.e. equation (4.1.11) is satisfied.

4.1.3 Properties of determining equations

We will discuss now the main properties of the determining equations (3.5.1). Notethat Eqs. (3.5.1) provide a system of linear homogeneous partial differential equa-tions for unknown coefficientsξ i(x,u), η i(x,u) of the operatorX. Therefore wehave the following property of the determining equations for any system of the dif-ferential equations (3.1.9),

Fσ

(x,uα(x),

∂uα(x)∂xi , . . . ,

∂ kuα(x)∂xi1 · · ·∂xik

)= 0, σ = 1, . . . ,s.

Lemma 4.4. The set of all solutionsX of the determining equations (3.5.1),

XFσ (x,u,u(1), . . . ,u(k))∣∣∣(3.1.8)

= 0, σ = 1, . . . ,s,

is a vector space.

This vector space has the following specific property.

Lemma 4.5. If the operatorsX1 andX2 of the form (3.1.6) satisfy the determiningequations (3.5.1) then the commutator

X = [X1,X2] (4.1.13)

also solves Eqs. (3.5.1).

Proof. We will use the determining equations written in the form (3.5.2). Namely,let

X1Fσ = φνσ Fν , X2Fσ = ϕν

σ Fν , (4.1.14)

whereX1, X2 denote thekth-order prolongations of the operatorsX1, X2. Equations(4.1.13), (4.1.12), and (4.1.14) yield:

XFσ = X1 (ϕνσ Fν)−X2(φν

σ Fν) . (4.1.15)

WritingX1 (ϕν

σ Fν) = X1(ϕνσ )Fν + ϕν

σ X1 (Fν) ,

X2(φνσ Fν) = X2 (φν

σ )Fν + φνσ X2 (Fν)


and invoking Eqs. (4.1.14) we conclude from Eq. (4.1.15) that the determining equa-tion (3.5.2) is satisfied forX :

XFσ = ψνσ Fν , (4.1.16)

whereψν

σ = X1(ϕνσ )−X2(φν

σ )+ ϕµσ φν

µ −φ µσ ϕν

µ .

4.1.4 Lie algebras

Definition 4.2. A vector spaceL of operators of the form (2.1.17),

X = ξ i(x)∂

∂xi

is called aLie algebraif it is closed under the commutator (4.1.2), i.e. ifX,Y ∈ Limplies [X,Y] ∈ L. The Lie algebra is denoted by the same letterL. ThedimensiondimL of the Lie algebra is identified with the dimension of the vector spaceL. Weshall use the symbolLr to denote anr-dimensional Lie algebra.

Consider a Lie algebraLr of a finite dimensionr. Let

Xα = ξ iα(x)

∂∂xi , α = 1, . . . , r (4.1.17)

be a basis of the vector spaceLr . Then anyX ∈ Lr is a linear combination of theoperators (4.1.17),

X = cαXα ≡ c1X1 + · · ·+crXr ,

with constant coefficientscα . We also say thatLr is spannedby the operators(4.1.17) or thatLr is thelinear spanof the operators (4.1.17).

In particular,[Xα ,Xβ ] ∈ Lr , and hence

[Xα ,Xβ ] = cγαβ Xγ , α,β = 1, . . . , r. (4.1.18)

Conversely, givenr linearly independent operators (4.1.17), their linear span is anr-dimensional Lie algebra if the relations (4.1.18) hold. This is an immediate con-sequence of the bilinearity of the commutator.

Thus, thecommutator relations(4.1.18) furnish a simple test for the linear spanof operatorsXα to be a Lie algebra. The constant coefficientscγ

αβ are called thestructure constantsof the algebraLr .

Lemma 4.4 and Lemma 4.5 from Section 4.1.3 prove the following basic propertyof the determining equations.

Theorem 4.1. The set of all solutionsX of the determining equations (3.5.1) is aLie algebra. It is called aLie algebra admittedby the differential equations (3.1.9),or asymmetry Lie algebrafor Eqs. (3.1.9).

4.2 Basic properties 81

Example 4.4. According to Example 3.7, the second-order ODE (3.5.14) hasthetwo-dimensional symmetry Lie algebraL2 spanned by the operators (3.5.15). Thecommutator relations (4.1.18) are given by Eq. (4.1.4),[X1,X2] = −2X1. Hence, thestructure constants ofL2 are

c112 = −2, c1

21 = 2, c212 = c2

21 = 0.

4.2 Basic properties

4.2.1 Notation

The linear span of any set of operatorsX1, . . . ,Xs will be denoted by〈X1, . . . ,Xs〉.For example, a Lie algebraLr with the basis (4.1.17) is written

Lr = 〈X1, . . . ,Xr〉. (4.2.1)

LetL be a vector space, and letK andN be subspaces ofL. The set of all operatorsX +Y with X ∈ K andY ∈ N is denoted by

K +N. (4.2.2)

Likewise, the linear span of all commutators[X,Y] with X ∈ K andY ∈ N will bedenoted by

[K,N]. (4.2.3)

For example, the condition forL to be closed under commutation, i.e. to be a Liealgebra, is written

[L,L] ⊂ L. (4.2.4)

4.2.2 Subalgebra and ideal

Definition 4.3. Let L be a Lie algebra. A subspaceK ⊂ L of the vector spaceL iscalled asubalgebraof the Lie algebraL if K is closed under commutation, i.e.

[K,K] ⊂ K. (4.2.5)

According to the notation (4.2.3), equation (4.2.5) means that

[X,Y] ∈ K for all X,Y ∈ K. (4.2.6)

One-dimensional subalgebras exist in any Lie algebraL. Indeed, anyX ∈ L spansa one-dimensional subalgebra with elementscX, c = const.


Existence of two-dimensional subalgebras in Lie algebras is guaranteed, in gen-eral, only in the complex domain.

Example 4.5. (See [14], Section 7.3.3). The Lie algebraL3 spanned by

X1 =(1+x2

) ∂∂x

+xy∂∂y

, X2 = xy∂∂x

+(1+y2

) ∂∂y

,

X3 = y∂∂x

−x∂∂y

has the subalgebraL2 = 〈X,Y〉, where

X = X1, Y = X2 + iX3

and the commutator relation has the form

[X,Y] = −iY.

Definition 4.4. A subalgebraK is called anidealof L if

[K,L] ⊂ K. (4.2.7)

According to the notation (4.2.3), equation (4.2.7) means that

[X,Y] ∈ K for all X ∈ K, Y ∈ L. (4.2.8)

Example 4.6. Consider the Lie algebraL6 = 〈X1, . . . ,X6〉 spanned by the operators(3.5.26). The commutator table (4.1.6) shows that the subalgebraL3 = 〈X1,X2,X3〉is an ideal ofL6.

4.2.3 Derived algebras

Lemma 4.6. Let L be a Lie algebra. The linear span of the commutators[X,Y] ofall operatorsX,Y ∈ L is a Lie algebra.

Definition 4.5. The Lie algebra spanned by the commutators[X,Y] of all operatorsX,Y ∈ L is called thederived algebraof the Lie algebraL. The derived algebra isdenoted byL′ or L(1). Thus,

L′ = [L,L].

Higher derivatives are defined by induction:

L(s+1) = (L(s))′ ≡ [L(s),L(s)], s= 1,2, . . . .

Remark 4.1. Let Lr be a Lie algebra with a basis (4.1.17). Then the derived algebraL′

r is the linear span of the commutators[Xα ,Xβ ] of the basis. In other words,L′r is

4.2 Basic properties 83

spanned by the linearly independent operators that appear in the table of commuta-tors forLr .

Example 4.7. Let L6 be the Lie algebra spanned by the operators (3.5.26). Ta-ble (4.1.6) and Remark 4.1 show that the derived algebra is the following four-dimensional Lie algebra:

L′6 = 〈X1,X2,X3,X4〉.

Remark 4.2. A Lie algebraL is said to beAbelianif L(1) = 0, i.e. if [X,Y] = 0 forall X,Y ∈ L. It is clear that a Lie algebra with a basis (4.1.17) is Abelianif and onlyif the basis operators commute,[Xα ,Xβ ] = 0.

4.2.4 Solvable Lie algebras

Definition 4.6. A Lie algebraLr , r < ∞, is said to be solvable if there is a sequence

Lr ⊃ Lr−1 ⊃ ·· · ⊃ L1 (4.2.9)

of subalgebras of respective dimensionsr, r − 1, . . . ,1 such thatLk is an ideal inLk+1, k = 1, . . . , r −1.

Theorem 4.2. A Lie algebraLr is solvable if and only if its derived algebra of a

finite ordersvanishes:L(s)r = 0, 0 < s< ∞.

Proof. Let Lr be a Lie algebra such thatL(s)r = 0. Consider the sequence of succes-

sive derived algebras:

Lr ⊃ L(1)r ⊃ L(2)

r · · · ⊃ L(s)r = 0. (4.2.10)

By the definition of derived algebras,L(k)r and subspaces ofL(k−1)

r containingL(k)r

are ideals ofL(k−1)r . Hence, one can construct a link in the chain (4.2.9) between

L(k−1)r andL(k)

r merely by adding toL(k)r one operator fromL(k−1)

r , then one more,

etc. if dimL(k)r < dimL(k−1)

r −1. Proceeding in this manner for all parts of (4.2.10),one accomplishes the construction of (4.2.9).

Conversely, letLr be a solvable Lie algebra. It follows from the chain (4.2.9) thatthe derived algebra of orderr is zero. Indeed,

L(r)r =

(L(1)

r

)(r−1)⊂ L(r−1)

r−1 ⊂ ·· · ⊂ L(1)1 = 0.

Example 4.8. Let us examine the Lie algebraL6 spanned by symmetries (3.5.26).One finds the derived algebras by inspecting Table (4.1.6) and arrives at the se-quence (4.2.10):

L6 ⊃ L(1)6 ⊃ L(2)

6 ⊃ L(3)6 = 0,

where


L6 = 〈X1,X2,X3,X4,X5,X6〉,

L(1)6 = 〈X1,X2,X3,X4〉,

L(2)6 = 〈X3〉.

Hence, the Lie algebraL6 is solvable. The chain (4.2.9),

L6 ⊃ L5 ⊃ ·· · ⊃ L1,

given by the proof of Theorem 4.2, comprises the subalgebras

L5 = 〈X1,X2,X3,X4,X5〉,

L4 = 〈X1,X2,X3,X4〉,L3 = 〈X1,X2,X3〉,

L2 = 〈X2,X3〉,L1 = 〈X3〉.

4.3 Isomorphism and similarity

Lie’s group classification of ordinary differential equations is based on the enumer-ation of all possible Lie algebras (infinitesimal groups in Lie’s terminology) in the(x,y) plane. In this enumeration, the algebras are maximally simplified by a properchoice of bases and by means of a suitable change of variables. Associated withthese two types of simplifying transformations are two distinctly different notions:isomorphicandsimilar (or equivalent) Lie algebras.

4.3.1 Isomorphic Lie algebras

An isomorphism between Lie algebras is commonly defined as a linear mappingpreserving the commutators in the Lie algebras in question.We will use the follow-ing equivalent definition.

Definition 4.7. Let L and K be two finite-dimensional Lie algebras such thatdimL = dimK. The algebrasL and K are said to beisomorphicif one can findbases (real or complex) inL andK such that the algebras have, in these bases, equalstructure constants, i.e. the same table of commutators.

Example 4.9. (See [19], Chapter 21,§3, Example 1). LetL andK be the three-dimensional Lie algebras spanned by the operators

4.3 Isomorphism and similarity 85

X1 =∂∂x

+∂∂y

, X2 = x∂∂x

+y∂∂y

, X3 = x2 ∂∂x

+y2 ∂∂y

(4.3.1)

and

Y1 =∂∂x

, Y2 = sinx∂∂x

+cosx∂∂y

, Y3 = cosx∂∂x

−sinx∂∂y

, (4.3.2)

respectively. The operators (4.3.1) and (4.3.2) have the following commutators:

[X1,X2] = X1, [X1,X3] = 2X2, [X2,X3] = X3, (4.3.3)

[Y1,Y2] = Y3, [Y1,Y3] = −Y2, [Y2,Y3] = −Y1. (4.3.4)

If we take inK the new basis

Y′1 = Y1 +Y3, Y′

2 = Y2, Y′3 = Y1−Y3,

then the commutator table (4.3.4) takes the form (4.3.3). Hence, the Lie algebrasLandK are isomorphic.

Example 4.10. Let L andK be the three-dimensional Lie algebras spanned by theoperators

X1 =∂∂x

, X2 = y∂∂x

, X3 = xy∂∂x

+(1+y2)∂∂y

(4.3.5)

and

Y1 =∂∂x

, Y2 =∂∂y

, Y3 = x∂∂x

−y∂∂y

, (4.3.6)

respectively. The operators (4.3.5) and (4.3.6) have the following commutators:

[X1,X2] = 0, [X1,X3] = X2, [X2,X3] = −X1, (4.3.7)

[Y1,Y2] = 0, [Y1,Y3] = Y1, [Y2,Y3] = −Y2. (4.3.8)

Now the algebrasL andK are not isomorphic over the reals, but they are isomorphicin the complex domain. Namely, if we make inL the complex change of the basis(see [19], Chapter 21,§3, Example 2)

X′1 = X1− iX2, X′

2 = iX1−X2, X′3 = −iX3,

i.e. take the following new basis:

X′1 = (1− iy)

∂∂x

, X′2 = (i −y)

∂∂x

, X′3 = −ixy

∂∂x

− i(1+y2)∂∂y

,

then the commutator table (4.3.7) takes the form (4.3.8). Hence, the Lie algebrasLandK are complex isomorphic.


4.3.2 Similar Lie algebras

Definition 4.8. The Lie algebrasL andL are said to besimilar (or equivalent) ifone is obtained from the other by a change of variables (real or complex). It meansthat the operators

X = ξ i(x)∂

∂xi

and

X = ξ ′i(x′)∂

∂x′i

of L andL, respectively, are related by Eqs. (2.2.6) and (2.2.7), i.e.by

x′i = x′i(x), ξ ′i = ξ k ∂x′i

∂xk , i = 1, . . . ,n. (4.3.9)

Lemma 4.7. If two r-dimensional Lie algebrasLr andLr with the same number ofvariables are similar, they are isomorphic.

Proof. Let the structure constantscγαβ of the algebraLr be defined by relations

(4.1.18),[Xα ,Xβ ] = cγαβ Xγ . Let bases ofLr andLr be given by the operators

Xα = ξ iα(x)

∂∂xi and Xα = ξ ′i

α(x′)∂

∂x′i, α = 1, . . . , r, (4.3.10)

respectively, whereXα are obtained fromXα by transformation (4.3.9). Then Eq.(4.1.10) from Lemma 4.2 yields:

[Xα ,Xβ

]= [Xα ,Xβ ] = cγ

αβ Xγ = cγαβ Xγ .

Thus,Lr andLr have the same structure constants in the bases (4.3.10), andhencethey are isomorphic.

Remark 4.3. The converse is not true: two Lie algebras may be isomorphic but notsimilar. Therefore, there exist, e.g.four non-similar two-dimensional Lie algebrasL2 in the plane, but onlytwo non-isomorphicL2.

It is precisely similarity (and not simply isomorphism) that is of use in groupanalysis as a criterion of reducibility of one differentialequation to another by asuitable change of variables. Nonetheless, establishing isomorphism is important asa first (and necessary due to Lemma 4.7) step for the determination of similarity.

Let Lr and Lr be two isomorphicr-dimensional Lie algebras with the bases(4.3.10) in whichLr andLr have equal structure constants. We shall consider herethe case whenr > n and assume that

rank(ξ iα(x)) = rank(ξ ′i

α (x′)) = n.

4.3 Isomorphism and similarity 87

Renumbering the indices, if necessary, one can assume that this is the value of therank of the square matrices(ξ i

h), (ξ ′ih ) (i,h = 1, . . . ,n). Then the following condi-

tions hold:

ξ ip = ϕh

pξ ih, ξ ′i

p = ϕ ′hp ξ ′i

h , i = 1, . . . ,n; p = n+1, . . . , r, (4.3.11)

whereϕhp andϕ ′h

p are functions ofx andx′, respectively, and where we sum overhfrom 1 ton. Under the assumptions made, we have the following theorem.

Theorem 4.3. A necessary and sufficient condition that the Lie algebrasLr andLr

with the same structure constants and the same number of variablesxi andx′i besimilar is that the functionsϕh

p, ϕ ′hp satisfy the equations

ϕ ′hp (x′) = ϕh

p(x), h = 1, . . . ,n; p = n+1, . . . , r, (4.3.12)

and that these equations do not lead to relations between thex’s alone or thex′’salone.

Sketch of the proof.(for a detailed proof, see [9],§22) Recall that the similarity ofthe algebrasLr andLr means that the coordinates of the bases (4.3.10) are relatedby (4.3.9), i.e. there is a change of variablesx′i = ϕ i(x), such that

ξ ′iα = ξ k

α∂x′i

∂xk, i = 1, . . . ,n; α = 1, . . . , r. (4.3.13)

Necessity. Let the algebrasLr andLr be similar. Then (4.3.13) and (4.3.11) yieldn(r − n) equations (4.3.12). By construction, equations (4.3.12) are compatible.Therefore it is impossible to eliminate all variablesx′i from and arrive at relationsbetween thex’s alone, and vice versa.

Sufficiency. Let Eqs. (4.3.12) be consistent and not lead to relations between thex’s alone or thex′’s alone. Let us consider Eqs. (4.3.13) forα = 1, . . . ,n. Under theassumptions on the ranks, these equations can be solved for derivatives and written:

∂x′i

∂xk= ξ ′i

h (x′)ξ hk (x), i,k = 1, . . . ,n, (4.3.14)

where(ξ hk ) is the square matrix inverse to(ξ i

h), i.e.ξ hk ξ i

h = δ ik, ξ h

i ξ il = δ h

l with i,k,h, l =1, . . . ,n. It follows from the conditions (4.3.12) that the system of Eqs. (4.3.14) (andhence the system of Eqs. (4.3.13)) is integrable. This meansthat there is a changeof variables that establishes the similarity of the algebrasLr andLr .

Determination of the change of variables. The similarity transformationx′i =ϕ i(x) is determined by solving, with respect to thex′’s, the mixed system of equa-tions comprising the functional equations (4.3.12) and thedifferential equations(4.3.14), the latter being integrable in view of Eqs. (4.3.12).

Example 4.11. Let us check if the Lie algebraL3 spanned by (4.3.1),


X1 =∂∂x

+∂∂y

, X2 = x∂∂x

+y∂∂y

, X3 = x2 ∂∂x

+y2 ∂∂y

,

is similar to the Lie algebraL3 spanned by

X1 =∂

∂x′, X2 = x′

∂∂x′

+y′

2∂

∂y′, X3 = x′2

∂∂x′

+x′y′∂

∂y′· (4.3.15)

The commutator tables of the operators (4.3.1) and (4.3.15)are identical. They havethe form (4.3.3):

[X1,X2] = X1, [X1,X3] = 2X2, [X2,X3] = X3,

[X1,X2] = X1, [X1,X3] = 2X2, [X2,X3] = X3.

Hence,L3 andL3 are isomorphic and have the same structure constants. Equations(4.3.11) written in the form

ξ k3 =

2

∑h=1

ϕhξ kh , ξ ′k

3 =2

∑h=1

ϕ ′hξ ′kh

yield:x2 = ϕ1 +xϕ2, x′2 = ϕ ′1 +x′ϕ ′2,

y2 = ϕ1 +yϕ2, 2x′y′ = y′ϕ ′2.

Hence,ϕ1 = −xy, ϕ2 = x+y, ϕ ′1 = −x′2, ϕ ′2 = 2x′.

With these functionsϕh andϕ ′h, equations (4.3.12) assume the form

x′2 = xy, 2x′ = x+y,

whence, after eliminatingx′, one arrives at the relationx− y = 0. According toTheorem 4.3,L3 andL3 are not similar (neither over the reals, nor over the complexnumbers), though they are isomorphic.

4.4 Low-dimensional Lie algebras

4.4.1 One-dimensional algebras

Any one-dimensional Lie algebraL1 in IRn is spanned by one operatorX1, and henceit is Abelian. Thus, according to Theorem 2.7 from Section 2.2.2, the structure andrealization ofL1 can be taken in canonical variables to be

[X1,X1] = 0, X1 =∂

∂x1 · (4.4.1)

4.4 Low-dimensional Lie algebras 89

4.4.2 Two-dimensional algebras in the plane

Let L2 be a two-dimensional Lie algebra with a basis

X1 = ξ1(x,y)∂∂x

+ η1(x,y)∂∂y

,

X2 = ξ2(x,y)∂∂x

+ η2(x,y)∂∂y

·(4.4.2)

The structure ofL2 is defined by the commutator (4.1.3) which is written:

[X1,X2] =(

X1(ξ2)−X2(ξ1)) ∂

∂x+

(X1(η2)−X2(η1)

) ∂∂y

· (4.4.3)

The operators (4.4.2) may be linearly connected (see [17], Section 4.5),

λ1(x,y)X1 + λ2(x,y)X2 = 0,

or not. Namely, the operatorsX1 andX2 (4.4.2) are linearly connected if and only ifthe determinant

det

(ξ1 η1

ξ2 η2

)= ξ1η2−η1ξ2 (4.4.4)

vanishes. We will call the determinant (4.4.4) thepseudoscalar productof the oper-ators (4.4.2) and denote it byX1∨X2 :

X1∨X2 = ξ1η2− ξ2η1. (4.4.5)

Lemma 4.8. Let us introduce inL2 a new basis

X′1 = α1X1 + α2X2, X′

2 = β1X1 + β2X2 (4.4.6)

with constant coefficientsα1,α2 andβ1,β2 satisfying the condition

α1β2−α2β1 6= 0 (4.4.7)

for the linear independence ofX′1 andX′

2. Then the commutator (4.4.3) of the basicoperators (4.4.2) undergoes the following transformation:

[X′1,X

′2] = (α1β2−α2β1) [X1,X2]. (4.4.8)

Proof. Since the commutator is bilinear and anticommutative, we have:

[X′1,X

′2] = α1β2[X1,X2]+ α2β1[X2,X1] = (α1β2−α2β1) [X1,X2].

Lemma 4.9. In any two-dimensional Lie algebra one can chose a basisX1, X2 suchthat

[X1,X2] = 0 or [X1,X2] = X1. (4.4.9)


Proof. According to Lemma 4.8, the equation[X1,X2] = 0 does not depend on achoice of a basis inL2 : if this equation is satisfied in one basis, it holds in any otherbasis. Therefore, let us consider the case[X1,X2] 6= 0. Let

[X1,X2] = α1X1 + α2X2 6= 0 (4.4.10)

with certain constant coefficientsα1,α2. In order to obtain a new basisX′1,X

′2 in

which the second equation of (4.4.9) holds,[X′1,X

′2] = X′

1, we set

X′1 = α1X1 + α2X2 (4.4.11)

and chose a new basis (4.4.6) with any constantsβ1,β2 satisfying the condition

α1β2−α2β1 = 1. (4.4.12)

Then Eqs. (4.4.8), (4.4.12), (4.4.10), and (4.4.11) yield:

[X′1,X

′2] = [X1,X2] = X′

1. (4.4.13)

This is precisely the second equation of (4.4.9) written in the basisX′1,X

′2.

Remark 4.4. If in Eq. (4.4.10)α2 = 0, i.e. if Eq. (4.4.10) has the form

[X1,X2] = α1X1,

one can satisfy Eq. (4.4.12) by takingβ2 = 1/α1 andβ1 = 0. Then a new basis(4.4.6) satisfying Eq. (4.4.13) is given by

X′1 = α1X1, X′

2 =1

α1X2.

If α2 6= 0, we takeβ1 = −1/α2, β2 = 0 and, invoking Eq. (4.4.11), obtain thefollowing basis satisfying Eq. (4.4.13):

X′1 = α1X1 + α2X2, X′

2 = − 1α2

X1.

Lemma 4.10. (See Lemma 4.2 in Section 4.1.2). The commutator (4.4.3) is invari-ant under any change of variables

x = x(x,y), y = y(x,y). (4.4.14)

Namely, ifX1, X2, [X1,X2]

denote the operatorsX1, X2, [X1,X2] written in the new variables ¯x, y, then the fol-lowing equation holds:

[X1,X2] = [X1,X2]. (4.4.15)

Proof. We know (see Eq. (2.1.20)) that an operator


X = ξ (x,y)∂∂x

+ η(x,y)∂∂y

(4.4.16)

is written in the new variables (4.4.14) as follows:

X = X(x(x,y)

) ∂∂ x

+X(y(x,y)

) ∂∂ y

· (4.4.17)

We will write Eq. (2.1.19) in the form

X(F(x, y)

)= X

(F(x(x,y), y(x,y))

), (4.4.18)

where the operatorX in the left-hand side acts on a function of the variables ¯x, y,whereas the operatorX in the right-hand side acts on a composite function of thevariablesx, y. Equation (4.4.18) means thatthe result of the action of X is indepen-dent on a choice of coordinates.

After the transformation (4.4.17) the operators (4.4.2) become

X1 = ξ 1∂∂ x

+ η1∂∂ y

, X2 = ξ 2∂∂ x

+ η2∂∂ y

,

where

ξ 1 = X1(x(x,y)

), η1 = X1

(y(x,y)

),

ξ 2 = X2(x(x,y)

), η2 = X2

(y(x,y)

).

(4.4.19)

The commutator (4.4.3) ofX1 andX2 is written

[X1,X2] =(

X1(ξ 2)−X2(ξ 1)) ∂

∂ x+

(X1(η2)−X2(η1)

) ∂∂ y

·

Invoking Eqs. (4.4.19) we have:

X1(ξ 2) = X1X2(x(x,y)

).

Applying Eq. (4.4.18) toF = X2(x(x,y)

)we obtain:

X1(ξ 2) = X1X2(x(x,y)

)= X1X2

(x(x,y)

).

Acting likewise we get:

X2(ξ 1) = X2X1(x(x,y)

)= X2X1

(x(x,y)

).

HenceX1(ξ 2)−X2(ξ 1) = [X1,X2]

(x(x,y)

),

and likewiseX1(η2)−X2(η1) = [X1,X2]

(y(x,y)

).


Finally:

[X1,X2] = [X1,X2](x(x,y)

) ∂∂ x

+[X1,X2](y(x,y)

) ∂∂ y

· (4.4.20)

Equation (4.4.17) written forX = [X1,X2] demonstrates that the right-hand side ofEq. (4.4.17) is identical with[X1,X2] thus proving Eq. (4.4.15).

Corollary 4.1. The equations (4.4.9) are invariant under the change of variables(4.4.14). Indeed, application of Eq. (4.4.15) to Eqs. (4.4.9) yields

[X1,X2] = 0, [X1,X2] = X1.

Lemma 4.11. The pseudoscalar product of (4.4.2) transforms under the change(4.4.6) of a basis inL2 according to

X′1∨X′

2 = (α1β2−α2β1)(X1∨X2

), (4.4.21)

and under a change of variables (4.4.14) it undergoes the transformation

X1∨X2 =∂ (x, y)∂ (x,y)

(X1∨X2

), (4.4.22)

where∂ (x, y)∂ (x,y)

=∂ x(x,y)

∂x∂ y(x,y)

∂y− ∂ x(x,y)

∂y∂ y(x,y)

∂x

is the Jacobian of the change of variables (4.4.14).

Proof. It is manifest that the pseudoscalar product (4.4.5) isbilinear:

(αX + βY)∨Z = α(X∨Z)+ β (Y∨Z),

X∨ (αY + βZ) = α(X∨Y)+ β (X∨Z),

and anticommutative:

X2∨X1 = −(X1∨X2), X1∨X1 = 0, X2∨X2 = 0.

Therefore the change (4.4.6) of the basis leads to Eq. (4.4.21):

X′1∨X′

2 = α1β2(X1∨X2

)+ α2β1

(X2∨X1

)

= (α1β2−α2β1)(X1∨X2

).

To prove Eq. (4.4.22), we rewrite Eqs. (4.4.19) in the form


ξ 1 = ξ1∂ x∂x

+ η1∂ x∂y

, η1 = ξ1∂ y∂x

+ η1∂ y∂y

,

ξ 2 = ξ2∂ x∂x

+ η2∂ x∂y

, η2 = ξ2∂ y∂x

+ η2∂ y∂y

,

substitute these expression in

X1∨X2 = ξ 1η2− ξ2η1

and obtain:

X1∨X2 = ξ1η2

(∂ x∂x

∂ y∂y

− ∂ x∂y

∂ y∂x

)− ξ2η1

(∂ x∂x

∂ y∂y

− ∂ x∂y

∂ y∂x

),

and hence Eq. (4.4.22).Since the Jacobian of the change of variables (4.4.14) does not vanish, equation

(4.4.22) implies the following statement.

Corollary 4.2. The equation

X1∨X2 ≡ ξ1η2− ξ2η1 = 0 (4.4.23)

is invariant under the change of variables (4.4.14).

Summarizing the results of Lemma 4.9 and Corollaries 4.1 and4.2, we arrive atthe following statement.

Theorem 4.4. All two-dimensional Lie algebrasL2 in the plane fall intofour typesdetermined by the followingcanonical structures:

I. [X1,X2] = 0, ξ1η2−η1ξ2 6= 0,

II . [X1,X2] = 0, ξ1η2−η1ξ2 = 0, (4.4.24)

III . [X1,X2] = X1, ξ1η2−η1ξ2 6= 0,

IV . [X1,X2] = X1, ξ1η2−η1ξ2 = 0.

Theorem 4.4 provides a basis for enumeration of all non-similar realizations oftwo-dimensional Lie algebras in the plane ([19], Chap. 18,§1; see also [14], Section12.2.2). The result is as follows.

Theorem 4.5. A basis of any two-dimensional Lie algebras in the plane can betransformed by a change of variables

t = t(x,y), u = u(x,y) (4.4.25)

into the followingcanonical formsarranged according to the types (4.4.24):


I. X1 =∂∂ t

, X2 =∂

∂u;

II . X1 =∂

∂u, X2 = t

∂∂u

;

III . X1 =∂

∂u, X2 = t

∂∂ t

+u∂∂u

;

IV . X1 =∂

∂u, X2 = u

∂∂u

·

(4.4.26)

The corresponding variablest,u are calledcanonical variablesfor L2.

Proof. We first transform one of basis operators to a translation generator by usingEqs. (4.4.14) and (4.4.17). Then we simplify the second operator of the basis. Wewill consider separately each type of Eqs. (4.4.24).

Type I.Let us choose new variables (4.4.14) such that one of basis operatorsbecomesX1 = ∂/∂ x. Let

X2 = ξ (x, y)∂∂ x

+ η(x, y)∂∂ y

be the second basic operator. Denoting ¯x, y again byx,y, we can assume that weconsider a Lie algebraL2 of type I having a basis of the form

X1 =∂∂x

, X2 = ξ (x,y)∂∂x

+ η(x,y)∂∂y

·

Then[X1,X2] = 0 yieldsξx = ηx = 0, while the conditionξ1η2−η1ξ2 6= 0 impliesthatη(y) 6= 0. Thus, in any canonical variables forX1, a basis ofL2 of type I has theform

X1 =∂∂x

, X2 = ξ (y)∂∂x

+ η(y)∂∂y

, η(y) 6= 0. (4.4.27)

Canonical variables forX1 can be determined in many different ways, which factcan be used for transformingX2 into the required form. Let us find the general formof the transformations (4.4.25) leaving unaltered the formof the first operator inEqs. (4.4.27), i.e.

X1 =∂ t∂x

∂∂ t

+∂u∂x

∂∂u

=∂∂ t

·

It follows that∂ t∂x

= 1,∂u∂x

= 0,

whence

t = x+ f (y), u = g(y). (4.4.28)

After this transformation, the second operator (4.4.27) becomes


X2 =(

ξ (y)+ η(y) f ′(y)) ∂

∂ t+ η(y)g′(y)

∂∂u

·

It takes the form

X2 =∂

∂u

if the functionsf andg solve the equations

ξ (y)+ η(y) f ′(y) = 0, η(y)g′(y) = 1.

We have arrived at the change of variables (4.4.25) given by

t = x−∫ ξ (y)

η(y)dy, u =

∫dy

η(y)(4.4.29)

transforming the operators (4.4.27) to the first canonical form of (4.4.26):

X1 =∂∂ t

, X2 =∂∂u

·

Type II.Here we start by choosing the canonical variablesx,y such that the firstoperator has the formX1 = ∂/∂y. Then we adopt the reasoning used in the previouscase. The conditions

[X1,X2] = 0, ξ1η2−η1ξ2 = 0

yield (cf. Eqs. (4.4.27)):

X1 =∂∂y

, X2 = η(x)∂∂y

· (4.4.30)

The most general transformation preserving theX1 in Eq. (4.4.30) is obtained fromEqs. (4.4.28) merely by interchangingx andy as well ast andu:

t = f (x), u = y+g(x). (4.4.31)

We set heref (x) = η(x), g(x) = 0 to obtain the change of variables

t = η(x), u = y (4.4.32)

transforming the operators (4.4.30) to the canonical form II:

X1 =∂

∂u, X2 = t

∂∂u

·

Type III. We take againX1 = ∂/∂y and employ transformations (4.4.31). Theconditions[X1,X2] = X1, ξ1η2−η1ξ2 6= 0 yield:


X1 =∂∂y

, X2 = ξ (x)∂∂x

+(

y+ η(x)) ∂

∂y, ξ (x) 6= 0. (4.4.33)

After transformation (4.4.31) the second operator of Eqs. (4.4.33) becomes

X2 = ξ (x) f ′(x)∂∂ t

+(

y+ η(x)+ ξ (x)g′(x)) ∂

∂u·

It becomes

X2 = t∂∂ t

+u∂∂u

≡ f (x)∂∂ t

+(y+g(x)

) ∂∂u

if

ξ (x) f ′(x) = f (x), ξ (x)g′(x)+ η(x) = g(x). (4.4.34)

Integrating the linear first-order ordinary differential equations (4.4.34) for unknownfunctions f andg and substituting into Eqs. (4.4.31), we transform the operators(4.4.33) to the canonical form III:

X1 =∂

∂u, X2 = t

∂∂ t

+u∂∂u

·

Type IV.As in the previous case we obtain using the conditions

[X1,X2] = X1, ξ1η2−η1ξ2 = 0

that

X1 =∂∂y

, X2 =(

y+ η(x)) ∂

∂y· (4.4.35)

It is transparent that the change of variables

t = x, u = y+ η(x) (4.4.36)

transforms (4.4.35) to the canonical form IV:

X1 =∂

∂u, X2 = u

∂∂u

·

Remark 4.5. For the practical construction ofcanonical variablesthere is no needto follow the procedure used in proving Theorem 4.5. One has merely to determinethe type ofL2 by calculating the commutator[X1,X2] and the pseudoscalar productX1∨X2 of the basisX1,X2. If [X1,X2] does not vanish, but[X1,X2] 6= X1, it is nec-essary to find by Lemma 4.9 and Remark 4.4 a new basis satisfying the equation[X1,X2] = X1. Denoting the new basis again byX1,X2 and determining the type ac-cording to the table (4.4.24), one findscanonical variables t,u by solving (for eachtype) the followingcompatible overdeterminedsystems offour first-order partialdifferential equations (Eqs. (6.5.15) in [17]):


I. X1(t) = 1, X2(t) = 0; X1(u) = 0, X2(u) = 1.

II . X1(t) = 0, X2(t) = 0; X1(u) = 1, X2(u) = t. (4.4.37)

III . X1(t) = 0, X2(t) = t; X1(u) = 1, X2(u) = u.

IV . X1(t) = 0, X2(t) = 0; X1(u) = 1, X2(u) = u.

Thus, we have arrive at the following table of Lie’s enumeration of the non-similar realizations of two-dimensional Lie algebras in the plane.

Table 4.4.1. Structure and non-similar realizations ofL2

Type Structure ofL2 Canonical forms ofL2

I [X1, X2] = 0, ξ1η2−η1ξ2 6= 0 X1 =∂∂x

, X2 =∂∂y

II [X1, X2] = 0, ξ1η2−η1ξ2 = 0 X1 =∂∂y

, X2 = x∂∂y

III [X1, X2] = X1, ξ1η2−η1ξ2 6= 0 X1 =∂∂y

, X2 = x∂∂x

+y∂∂y

IV [X1, X2] = X1, ξ1η2−η1ξ2 = 0 X1 =∂∂y

, X2 = y∂∂y

4.4.3 Three-dimensional algebras in the plane

Two non-isomorphic possibilities, [X1,X2] 6= 0 or [X1,X2] = 0, for two-dimensional

Lie algebrasL2 mean that the first derivativeL(1)2 has the dimension 1 or 0. In the

case of three-dimensional Lie algebrasL3 the derived algebraL(1)3 may have the

dimension 3, 2, 1, or 0. Lie (see [19], Chapter 21,§2 and§3) classified the non-

isomorphic structures ofL3 according to the dimension ofL(1)3 and obtained the

following result.

Theorem 4.6. The structure of any three-dimensional algebraL3 can be trans-formed, by acomplexchange of its basisX1,X2,X3, to one and only one of thefollowing seven forms, where A, B, C, and D indicate that the first derived algebra

L(1)3 has the dimension 3, 2, 1, and 0, respectively:

A. (1) [X1,X2] = X1, [X1,X3] = 2X2, [X2,X3] = X3,

B. (2) [X1,X2] = 0, [X1,X3] = X1, [X2,X3] = cX2 (c 6= 0, 6= 1),

(2′) [X1,X2] = 0, [X1,X3] = X1, [X2,X3] = X2,

(3) [X1,X2] = 0, [X1,X3] = X1, [X2,X3] = X1 +X2,

C. (4) [X1,X2] = 0, [X1,X3] = X1, [X2,X3] = 0,

(5) [X1,X2] = 0, [X1,X3] = 0, [X2,X3] = X1,

D. (6) [X1,X2] = 0, [X1,X3] = 0, [X2,X3] = 0.


Type (2′) is distinguished from type (2) due to its autonomous significance.

For two-dimensional Lie algebras, as well as for one-dimensional ones, the clas-sification over the reals is the same as that over the complex numbers. In the case ofthe three-dimensional algebras the situation is different. L. Bianchi [5] investigatedthe non-isomorphic structures ofL3 in the real domain. He showed that, in addi-tion to Lie’s seven types given in Theorem 4.6, there exist inthe real domain twomore types that can be reduced to Lie’s table by a complex change of their bases.Bianchi’s classification is well known in the literature, see, e.g. [14], Section 37.3.8.

Lie’s complex classification ([19], Chapter 22,§3) of all non-similar three-dimensional algebras in the plain is given by the following theorem.

Theorem 4.7. Any three-dimensional algebra of operators in two variables can betransformed, by a complex change of variables, to one and only one of the following13 distinctly differentstandard formsof L3.

A. The first derived algebra has dimension three :

(1) X1 =∂∂x

+∂∂y

, X2 = x∂∂x

+y∂∂y

,

X3 = x2 ∂∂x

+y2 ∂∂y

,

(2) X1 =∂∂x

, X2 = 2x∂∂x

+y∂∂y

, X3 = x2 ∂∂x

+xy∂∂y

,

(3) X1 =∂∂y

, X2 = y∂∂y

, X3 = y2 ∂∂y

·

B. The first derived algebra has dimension two :

(4) X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

+cy∂∂y

(c 6= 0, 6= 1),

(5) X1 =∂∂y

, X2 = x∂∂y

,

X3 = (1−c)x∂∂x

+y∂∂y

(c 6= 0, 6= 1),

(6) X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

+y∂∂y

,

(7) X1 =∂∂y

, X2 = x∂∂y

, X3 = y∂∂y

,

(8) X1 =∂∂x

, X2 =∂∂y

, X3 = (x+y)∂∂x

+y∂∂y

,

(9) X1 =∂∂y

, X2 = x∂∂y

, X3 =∂∂x

+y∂∂y

·


C. The first derived algebra has dimension one :

(10) X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂x

,

(11) X1 =∂∂y

, X2 = x∂∂y

, X3 = x∂∂x

+y∂∂y

,

(12) X1 =∂∂x

, X2 =∂∂y

, X3 = x∂∂y

·

D. The first derived algebra has dimension zero :

(13) X1 =∂∂y

, X2 = x∂∂y

, X3 = p(x)∂∂y

·

Remark 4.6. Classification over the reals provides ([21], see also [12],Chapter 8written by N. H. Ibragimov and F. M. Mahomed) the additionalL3 :

(14) X1 =∂∂y

, X2 = x∂∂x

+y∂∂y

,

X3 = 2xy∂∂x

+(y2−x2)∂∂y

,

(15) X1 = (1+x2)∂∂x

+xy∂∂y

, X2 = y∂∂x

−x∂∂y

,

X3 = xy∂∂x

+(1+y2)∂∂y

,

(16) X1 =∂∂x

, X2 =∂∂y

, X3 = (bx+y)∂∂x

+(by−x)∂∂y

,

(17) X1 =∂∂y

, X2 = x∂∂y

, X3 = (1+x2)∂∂x

+(x+b)y∂∂y

·

4.4.4 Three-dimensional algebras inIR3

We will consider now 1, 2, and 3-dimensional Lie algebras of operators (2.1.17) inthe three-dimensional space IR3 of variablesx,y,z. For the sake of brevity, we willuse the notation

∂x =∂∂x

, ∂y =∂∂y

, ∂z =∂∂z

and write the operator (2.1.17) in the form

X = ξ (x,y,z)∂x + η(x,y,z)∂y + ζ (x,y,z)∂z . (4.4.38)


First we summarize, in the notation (4.4.38), the results onLie algebras in the(x,y) plane discussed in Sections 4.4.1, 4.4.2 and 4.4.3. According to Eq. (4.4.1),all one-dimensional Lie algebras are similar toL1 spanned by

X1 = ∂x . (4.4.39)

Furthermore, all two-dimensional Lie algebras in the planeare similar to one of fournon-similar algebrasL2 with the following bases (see Table 4.4.1):

I. X1 = ∂x , X2 = ∂y ;

II . X1 = ∂y , X2 = x∂y ;

III . X1 = ∂y , X2 = x∂x +y∂y ;

IV . X1 = ∂y , X2 = y∂y ·

(4.4.40)

Finally, the non-similarL3 in the real domain are given by Eqs. (1)–(17) from The-orem 4.7 and Remark 4.6.

Now we return to the three dimensional space IR3 and provide the result of classi-fication ofL3 in the real domaingiven in [18]. The Lie algebras in IR3 of dimensions1 and 2 are again given by (4.4.39) and (4.4.40). The algebrasof dimension 3 areenumerated in Table 4.4.2.

Table 4.4.2. Non-similar three-dimensional real Lie algebras in IR3

1 X1 = ∂z, X2 = ∂x, X3 = ∂y

2 X1 = ∂z, X2 = ∂x, X3 = y∂x +a(y)∂z

3 X1 = ∂z, X2 = x∂z, X3 = y∂z

4 X1 = ∂z, X2 = x∂z, X3 = a(x)∂z, a′′(x) 6= 0

5 X1 = ∂z, X2 = ∂x, X3 = ∂y +x∂z

6 X1 = ∂z, X2 = ∂x, X3 = y∂x +x∂z

7 X1 = ∂z, X2 = ∂x, X3 = x∂z

8 X1 = ∂z, X2 = ∂x, X3 = qx∂x +∂y +z∂z

9 X1 = ∂z, X2 = ∂x, X3 = qx∂x +z∂z

10 X1 = ∂z, X2 = ∂x, X3 = y∂x +z∂z,

11 X1 = ∂z, X2 = x∂z, X3 = (1−q)x∂x +z∂z

12 X1 = ∂z, X2 = x∂z, X3 = ∂y +z∂z,

13 X1 = ∂z, X2 = ∂x, X3 = x∂x +∂y +(z+x)∂z

14 X1 = ∂z, X2 = ∂x, X3 = x∂x +(z+x)∂z

15 X1 = ∂z, X2 = x∂z, X3 = ∂x−z∂z− ε∂y, ε = 0 or 1

16 X1 = ∂z, X2 = ∂x, X3 = (qx+z)∂x +∂y−x∂z

17 X1 = ∂z, X2 = ∂x, X3 = (qx+z)∂x −x∂z

18 X1 = ∂z, X2 = x∂z, X3 = (x2−qx+1)∂x +xz∂z

4.5 Lie algebras and multi-parameter groups 101

Continued

19 X1 = ∂z, X2 = ∂x +z∂z, X3 = 2z∂x +ex∂y +z2∂z

20 X1 = ∂z, X2 = ∂x +z∂z, X3 = 2z∂x +(z2±e2x)∂z

21 X1 = ∂z, X2 = z∂z, X3 = z2∂z

22 X1 = ∂z, X2 = sinz∂x +coszcosx

∂y− tanxcosz∂z,

X3 = cosz∂x−sinzcosx

∂y + tanxsinz∂z

23 X1 = ∂z, X2 = sinz∂x− tanxcosz∂z, X3 = cosz∂x + tanxsinz∂z

4.5 Lie algebras and multi-parameter groups

4.5.1 Definition of multi-parameter groups

We will use the notation and assumptions of Section 2.1.1, but the parametera willbe replaced now by the vector-parametera = (a1, . . . ,ar). We consider invertibletransformationsTa : IRn → IRn,

x = f (x,a), (4.5.1)

defined in a neighborhood of

a = 0≡ (0, . . . ,0).

Heref = ( f 1, . . . , f n)

is a vector valued function whose componentsf i = f i(x,a) are at least three timescontinuously differentiable with respect to the variables

x1, . . . ,xn,a1, . . . ,ar .

We impose the initial condition:

f∣∣a=0 = x. (4.5.2)

Definition 4.9. A set of transformations (4.5.1) is called anr-parameter local groupGr if

f ( f (x,a),b) = f (x,c) (4.5.3)

for all valuesa andb of the parameter sufficiently close toa = 0. Herec = c(a,b)is a vector-function with components

cα = φα(a,b), α = 1, . . . , r, (4.5.4)


defined and thrice continuously differentiable for sufficiently smalla andb (in thesame sense as in Definition 2.1). For brevity,Gr is termed amulti-parameter group.

The functions (4.5.4) define a composition law in the groupGr . It is assumed thatthe system of equations

φα (a,b) = 0, α = 1, . . . , r (4.5.5)

has a unique solutionb = (b1, . . . ,br) for any smalla. Givena, the solutionb of thesystem (4.5.5) is denoted bya−1. Hence, the inverse transformationT−1

a is

x = f (x,a−1).

4.5.2 Construction of multi-parameter groups

The following theorem is sufficient to meet the needs of applied group analysiswhen the practical construction of multi-parameter groupsis desired.

Theorem 4.8. Let Lr be anr-dimensional vector space spanned by the operators

Xα = ξ iα(x)

∂∂xi , α = 1, . . . , r. (4.5.6)

The compositionTa = Tar · · ·Ta1 of r one-parameter groups of transformationsTaα

generated individually by each of the base operatorsXα via the Lie equations

dx i

daα = ξ iα(x), x i

∣∣aα=0 = xi , i = 1, . . . ,n, (4.5.7)

is anr-parameter (local) groupGr if and only if Lr is a Lie algebra. By applyingthe same construction to anys-dimensional subalgebra ofLr , one generates ans-parameter subgroup of the groupGr .

Remark 4.7. The above construction ofGr depends upon the choice of a basis inLr . Therefore, the theory of Lie groups considers more generalconstructions anddifferent representations of Lie groups with a given Lie algebra. However, all theserepresentations are equivalent (similar).

Example 4.12. Consider a three-dimensional Lie algebra spanned by

X1 =∂∂x

, X2 =∂∂y

, X3 = y∂∂x

·

Solution of the Lie equations (4.5.7) for these operators provides the following threeone-parameter groups with the respective parametersa1,a2,a3 :

4.5 Lie algebras and multi-parameter groups 103

Ta1 : x = x+a1, y = y;

Ta2 : x = x, y = y+a2;

Ta3 : x = x+ya3, y = y.

Their compositionTa = Ta3Ta2Ta1,

wherea = (a1,a2,a3), has the form:

x = x+ya3+a1+a2a3,

y = y+a2.(4.5.8)

The consecutive application ofTa andTb, whereb= (b1,b2,b3), yields the transfor-mationTbTa :

x = x+y(a3+b3)+a2(a3 +b3)+b2b3 +a1+b1,

y = y+a2+b2.

The equationTbTa = Tc leads to the equations

x = x+yc3+c1+c2c3,

y = y+c2,

whence

c1 = a1 +b1−b2a3,

c2 = a2 +b2,

c3 = a3 +b3.

Thus, the transformation (4.5.8) provides, in accordance with Theorem 4.8, a three-parameter group with the composition law (4.5.4) given by the following equations:

φ1(a,b) = a1 +b1−b2a3,

φ2(a,b) = a2 +b2,

φ3(a,b) = a3 +b3.

(4.5.9)

Example 4.13. Let L2 be the two-dimensional vector space spanned by

X1 =∂∂x

, X2 = x∂∂y

· (4.5.10)

The operatorsX1 andX2 generate the following two one-parameter groups:


Ta1 : x = x+a1, y = y;

Ta2 : x = x, y = y+a2x.

Their compositionTa = Ta2Ta1,

wherea = (a1,a2), has the form:

x = x+a1,

y = y+xa2+a1a2.(4.5.11)

The consecutive application ofTa andTb, whereb = (b1,b2), yields the transforma-tionsTbTa :

x = x+a1+b1,

y = y+(x+a1)(a2 +b2)+b1b2.(4.5.12)

The equationTbTa = Tc implies that

x = x+c1,

y = y+xc2+c1c2.(4.5.13)

Equations (4.5.12) and (4.5.13) yield

c1 = a1 +b1,

c2 = a2 +b2,

c1c2 = a1(a2 +b2)+b1b2.

(4.5.14)

Substitution ofc1 andc2 given by the first two equations from (4.5.14) into the thirdequation yieldsb1a2 = 0. Since the latter equation cannot be satisfied for arbitrarya andb, the two-parameter family of transformations (4.5.11) doesnot satisfy thegroup property (4.5.3). This conclusion agrees with Theorem 4.8 because the vectorspaceL2 spanned by the operators (4.5.10) is not a Lie algebra.


Exercise 4.1.Make the commutator table for the operators (3.5.11).

Exercise 4.2.Check Eq. (4.1.11) for the operators from Table (4.1.5).

Exercise 4.3.Make the commutator table for the operators (3.5.11) and verify thatthese operators span an 8-dimensional Lie algebra.


Exercise 4.4.Consider again the operators (3.5.11). Verify the Jacobi identity(4.1.9) by takingX = X2, Y = X4, Z = X7.

Exercise 4.5.Find the structure constants of the three-dimensional Lie algebraL3

spanned by the operators (3.5.13).

Exercise 4.6.Prove Lemma 4.6.

Exercise 4.7.Show that the derived algebraL′ is an ideal ofL.

Exercise 4.8.Make the commutator table for the Lie algebraL5 spanned by theoperators

X1 =∂∂ t

, X2 =∂∂x

, X3 = t∂∂ t

+x∂∂x

,

X4 = t2 ∂∂ t

+ tu∂∂u

, X5 = 2x∂∂x

−u∂∂u

and find the structure constants. InvestigateL5 for solvability.

Exercise 4.9. Investigate for solvability the Lie algebraL3 spanned by

X1 =(1+x2) ∂

∂x+xy

∂∂y

,

X2 = xy∂∂x

+(1+y2) ∂

∂y,

X3 = y∂∂x

−x∂∂y

·

Exercise 4.10.Using the commutator table (4.1.5) and Theorem 4.2 show thattheLie algebraL3 = 〈X1,X2,X3〉 spanned by the operators (3.5.13) is solvable and con-struct the chain (4.2.9),L3 ⊃ L2 ⊃ L1.

Exercise 4.11.Find the derived algebraL′3 and determine dimL′

3 for the Lie algebraL3 spanned by the operators (15) from Remark 4.6:

X1 = (1+x2)∂∂x

+xy∂∂y

, X2 = y∂∂x

−x∂∂y

,

X3 = xy∂∂x

+(1+y2)∂∂y

·

Exercise 4.12.Find the derived algebraL′3 and determine dimL′

3 for the Lie algebraL3 spanned by the operators (16) from Remark 4.6:

X1 =∂∂x

, X2 =∂∂y

, X3 = (bx+y)∂∂x

+(by−x)∂∂y

·


Exercise 4.13.Construct the commutator table and find the derived algebra for theLie algebraL3 spanned by the operators (23) from Table 4.4.2:

X1 =∂∂z

, X2 = sinz∂∂x

− tanxcosz∂∂z

,

X3 = cosz∂∂x

+ tanxsinz∂∂z

·

Exercise 4.14.Determine the number of linearly unconnected operators among theoperatorsX1, X2, X3 from Exercise 4.13. (See [17], Section 4.5, Definition 4.5.1and Lemma 4.5.1.)

Exercise 4.15.Construct the commutator table and find the derived algebra for theLie algebraL3 spanned by the operators (22) from Table 4.4.2:

X1 =∂∂z

, X2 = sinz∂∂x

+coszcosx

∂∂y

− tanxcosz∂∂z

,

X3 = cosz∂∂x

− sinzcosx

∂∂y

+ tanxsinz∂∂z

·

Exercise 4.16.Determine the number of linearly unconnected operators among theoperatorsX1, X2, X3 from Exercise 4.15.

Chapter 5Galois groups via symmetries

This chapter provides a bridge between Lie symmetry groups for differential equa-tions and Galois groups for algebraic equations. Namely, the Galois groups are con-structed as the restriction of Lie symmetries of algebraic equations on the roots ofthe equations under consideration. The approach is illustrated by several examples.

5.1 Preliminaries

We will construct the Galois groups for several simple algebraic equations by firstcalculating their Lie symmetries and then restricting the symmetry group to the rootsof the equation in question. This procedure can be applied todifferential equationsas well. As a result we obtain another representation of Lie symmetries called theGalois representation of symmetries of differential equations (see [14], pp. 251–254,[16] and the references therein).

The approach will be illustrated by applying it to the following equations:

x2 +1 = 0, (5.1.1)

x4−x2 +1 = 0, (5.1.2)

x4 +x3 +x2+x+1= 0. (5.1.3)

It is known from the literature on Galois groups that the Galois groupG of Eq.(5.1.1) is

G = {1,(x1,x2)}, (5.1.4)

wherex1,x2 are the roots of Eq. (5.1.1),(x1,x2) is the permutation of the roots, and1 is the identical permutation (unit of the group). It is alsoknown that the Galoisgroups of Eqs. (5.1.2) and (5.1.3) are

107

108 5 Galois groups via symmetries

G = {1,(x1,x2)(x3,x4),(x1,x3)(x2,x4),(x1,x4)(x2,x3)} (5.1.5)

andG = {1,(x1,x2,x4,x3),(x1,x3,x4,x2),(x1,x4)(x2,x3)}, (5.1.6)

respectively. In Eq. (5.1.5),x1,x2,x3,x4 denote the roots of Eq. (5.1.2), and in Eq.(5.1.6) the roots of Eq. (5.1.3).

5.2 Symmetries of algebraic equations

5.2.1 Determining equation

Let us consider algebraic equations of thenth degree:

Pn(x) ≡C0xn +C1xn−1 +C2xn−2 + · · ·+Cn−1x+Cn = 0. (5.2.1)

The symmetries of algebraic equations are defined as in the case of differential equa-tions. Namely, let a transformation

x = f (x) (5.2.2)

convert an equation (5.2.1) of thenth degree into an algebraic equation

Pn(x) ≡ C0xn +C1xn−1 +C2xn−2 + · · ·+Cn−1x+Cn = 0. (5.2.3)

In general, the coefficientsCi in Eq. (5.2.3) will not coincide with the coefficientsCi in the original equation (5.2.1). If they coincide, we will say that the transfor-mation (5.2.2) is a symmetry (a Lie symmetry) of Eq. (5.2.1).We can define thesymmetry of algebraic equations as follows.

Definition 5.1. The transformation (5.2.2) is called a symmetry of Eq. (5.2.1) if Eq.(5.2.3) coincides with the equationPn(x) = 0 wheneverx solves Eq. (5.2.1).

Thedetermining equationfor symmetries of algebraic equations can also be writ-ten, as for differential equations, either in the form (2.3.3):

Pn(x)∣∣Pn(x)=0 = 0, (5.2.4)

or in the form (2.3.8):Pn(x) = µ(x)Pn(x), (5.2.5)

wherePn(x) = C0xn +C1xn−1 +C2xn−2 + · · ·+Cn−1x+Cn.

The requirement that the transformation (5.2.2) maps any algebraic equationinto an algebraic equation is rather restrictive. In particular, if one considers only

5.2 Symmetries of algebraic equations 109

uniquely invertible transformations, then one can show that the general form of in-vertible transformations (5.2.2) convertingeveryequation (5.2.1) into an algebraicequation (5.2.3) is provided by the linear fractional transformation

x =ax+ εb+ δx

(5.2.6)

with complex coefficients satisfying the invertibility condition

ab− εδ 6= 0. (5.2.7)

If we do not require existence of the uniquely determined inverse transformation,we can use transformations (5.2.2) given by the rational fractions

x =A0xr +A1xr−1 + · · ·+Ar

B0xs+B1xs−1 + · · ·+Bs· (5.2.8)

The transformation (5.2.8) maps algebraic equations into algebraic equations (ingeneral, not of the same degree). It was considered by E. W. Tschirnhausen in 1683and is known asTschirnhausen’s transformation.

In what follows, we will use the following simple result concerning Eqs. (5.2.1)that are symmetric in their coefficients.

Lemma 5.1. Equations (5.2.1) whose coefficients satisfy the conditions

Cn = C0, Cn−1 = C1, . . . ,Ci = Cn−i (5.2.9)

have the symmetry

x =1x· (5.2.10)

Proof. Indeed,

Pn(x) = C01xn +C1

1xn−1 + · · ·+Cn−1

1x

+Cn

= x−n(C0 +C1x+ · · ·+Cn−1xn−1 +Cnxn

).

Therefore the conditions (5.2.9) yield

Pn(x) = x−nPn(x).

Hence, the determining equation (5.2.5) is satisfied withµ(x) = x−n.

5.2.2 First example

Let us find the symmetries of the linear fractional form (5.2.6) for the quadraticequation (5.1.1),


x2 +1 = 0.

The determining equation (5.2.4) is written

(x2 +1)∣∣x2=−1 = 0. (5.2.11)

Substituting (5.2.6) in ¯x2 +1, we have:

x2 +1 = (δx+b)−2[(a2 + δ 2)x2 +2(aε +bδ )x+b2+ ε2].

Therefore the determining equation (5.2.11) becomes

2(aε +bδ )x+b2+ ε2−a2− δ 2 = 0

and yields the following system of two equations:

aε +bδ = 0, b2 + ε2−a2− δ 2 = 0. (5.2.12)

In the caseδ = 0 we obtain, using the first equation of (5.2.12) and the condition(5.2.7), thatε = 0. Then the second equation of (5.2.12) becomesb = ±a. Hence,we have obtained the following two symmetries of the form (5.2.6) withδ = 0 :

x = x and x = −x. (5.2.13)

If δ 6= 0, equations (5.2.12) are written:

b = −aεδ

, (a2 + δ 2)(ε2− δ 2) = 0.

These equations together with the condition (5.2.7) yieldε2− δ 2 = 0. Hence,

δ = ε, b = −a, or δ = −ε, b = a,

whereε 6= 0. Thus, we have two types of transformations:

x =ax+ εa− εx

and x =ax+ εεx−a

· (5.2.14)

If a = 0, these transformations reduce to (see also Lemma 5.1)

x =1x

and x = −1x· (5.2.15)

If a 6= 0, the first transformation of (5.2.14) provides a one-parameter local group:

Tα : x =x+ α1−αx

, (5.2.16)

while the second transformation of (5.2.14) can be written in the form

5.2 Symmetries of algebraic equations 111

Sβ : x =x+ β

βx−1. (5.2.17)

Note that (5.2.13) are obtained from (5.2.16)–(5.2.17) by letting α = β = 0,whereas (5.2.15) can be obtained from (5.2.16)–(5.2.17) bylettingα = β = ∞. Thetransformations (5.2.16)–(5.2.17) form a group. Indeed, the compositionSβ ◦ Tαacts as follows:

Sβ(Tα(x)

)=

x+α1−αx + β

β x+α1−αx −1

=(1−αβ )x+ α + β(α + β )x+ αβ −1

=x+ α+β

1−αβα+β1−αβ x−1

·

Hence,

Sβ ◦Tα = Sγ , γ =α + β1−αβ

·

The similar calculations show that

Tα ◦Sβ = Sδ , δ =β −α1+ αβ

·

Other group properties are obviously satisfied. Thus, we have obtained the followingresult.

Theorem 5.1. The group of symmetries of the linear fractional form (5.2.6) for Eq.(5.1.1) is generated by the transformations (5.2.16)–(5.2.17), where the parametersα andβ range over the extended complex plane.

5.2.3 Second example

Let us find the symmetries of the form (5.2.6) for Eq. (5.1.2),

x4−x2 +1 = 0.

The determining equation (5.2.4) is written

(x4− x2+1)∣∣x4=x2−1 = 0.

Inspecting this equation as in the first example, we obtain the following four sym-metries of the form (5.2.6):

I : x = x; S: x = −x; R : x =1x

; T : x = −1x· (5.2.18)

Let us verify that, e.g. the transformationR is a symmetry. We have:

x4− x2 +1 =1x4 −

1x2 +1 =

1x4

(x4−x2+1

).


It follows that x4− x2 +1 = 0 wheneverx4−x2 +1 = 0.The transformations (5.2.18) form a group. Indeed:

S◦R= R◦S= T, S◦T = T ◦S= R, T ◦R= R◦T = S

andS−1 = S, R−1 = R, T−1 = T.

5.2.4 Third example

Consider now Eq. (5.1.3),

P4(x) ≡ x4 +x3 +x2+x+1= 0. (5.2.19)

This equation satisfies the conditions (5.2.9) and therefore has the symmetry (5.2.7),x= x−1. The reckoning shows that the transformation (5.2.7) together with the iden-tical transformation ¯x = x are the only symmetries of the form (5.2.6). In order tofind more symmetries, we note that the transformation

x = xn (5.2.20)

with any integern indivisible by 5 is a symmetry for Eq. (5.2.19).Indeed, we have to substitute ¯x = xn in the determining equation (5.2.4):

P4(x)∣∣P4(x)=0 ≡ (x4 + x3+ x2 + x+1)

∣∣x4=−(x3+x2+x+1)

= 0. (5.2.21)

Here we should take into account not only the equation

x4 = −(x3 +x2+x+1) (5.2.22)

but also the equations obtained from (5.2.22) by repeated multiplication byx :

x5 = x ·x4 = −(x4 +x3 +x2+x) = 1, x6 = x ·x5 = x, . . . .

Thus, we extend (5.2.22) as follows:

x5 = 1, x6 = x, x7 = x2, x8 = x3, x9 = x4,

x10 = 1, x11 = x, x12 = x2, x13 = x3, x14 = x4,x15 = 1, · · · . · · · . · · · . · · · .

(5.2.23)

Let us verify that the transformation

x = x2

satisfies the determining equation (5.2.4). Using Eqs. (5.2.22)–(5.2.23), we have:

5.3 Construction of Galois groups 113

P4(x)∣∣P4(x)=0 = (x8 +x6 +x4+x2 +1)

∣∣P4(x)=0

= x3 +x+x4+x2 +1 = 0.

Likewise, we verify that the transformation

x = x3

also satisfies the determining equation (5.2.4). In this case we have:

P4(x)∣∣P4(x)=0 = (x12+x9+x6 +x3+1)

∣∣P4(x)=0

= x2 +x4+x+x3+1 = 0.

The transformationx = x4

is a symmetry because it is the repeated action of the symmetry transformationx = x2.

The transformationx = x5

is not a symmetry because for this transformation Eqs. (5.2.23) yield

P4(x)∣∣P4(x)=0 = x20+x15+x10+x5 +1 = 5.

The composition of the above symmetries shows that the transformations (5.2.20)with all positiven 6= 5m (m = 0,1,2, . . .) are symmetries for Eq. (5.2.19). Finally,the proof for the transformation (5.2.20) with the negativevalues ofn is obtained bytaking the composition of the transformations (5.2.20) with the positiven and thesymmetry (5.2.10).

Thus, we have demonstrated that the transformation (5.2.20) is a symmetry forEq. (5.2.19), provided thatn 6= 5m (m= 0,±1,±2, . . .).

5.3 Construction of Galois groups

5.3.1 First example

Let us construct the Galois group for Eq. (5.1.1),

x2 +1 = 0,

using its symmetry groupG consisting of the transformations (5.2.16) and (5.2.17).Since Eq. (5.1.1) is invariant under the groupG, the roots

x1 = i, x2 = −i


of Eq. (5.1.1) are merely permuted among themselves (or individually unaltered) bythe transformations (5.2.16)–(5.2.17). Consequently, the restriction of the groupGon the set{x1,x2} is well defined. It is manifest that this restriction is a group. It willbe called theinduced symmetry group, or briefly, theinduced groupand denoted byG . The induced group comprises permutations of the rootsx1,x2. Let us find thesepermutations.

The action of the transformations (5.2.16) and (5.2.17) on the roots are as fol-lows:

Tα(x1) =i + α1−α i

= i = x1, Tα(x2) =−i + α1+ α i

= −i = x2,

Sβ (x1) =i + β

β i −1= −i = x2, Sβ (x2) =

−i + β−β i −1

= i = x1.

We see that the restriction ofTα on the roots is the identical transformation (theunit of the groupG ) which is denoted by 1. We also see thatSβ permutes the roots.This permutation is denoted by(x1,x2). Thus, the induced groupG comprises twoelements:

G = {1, (x1,x2)}. (5.3.1)

Comparison with (5.1.4) shows that theGalois groupof Eq. (5.1.1) coincides withthe induced group (5.3.1).


Let us construct the Galois group for Eq. (5.1.2),

x4−x2 +1 = 0,

using its symmetry groupG consisting of the transformationsI ,S,R,T given in(5.2.18). The roots of Eq. (5.1.2) are

x1 =

√12(1+ i

√3), x2 = −x1, x3 =

√12(1− i

√3), x4 = −x3.

Denoting byI , S,R, T the restriction ofI ,S,R,T on the roots, we obtainI = 1 (theunit) and the following permutations:

S=

(x1 x2 x3 x4

x2 x1 x4 x3

), R=

(x1 x2 x3 x4

x3 x4 x1 x2

),

T =

(x1 x2 x3 x4

x4 x3 x2 x1

).

They are also denoted by

5.3 Construction of Galois groups 115

(x1,x2)(x3,x4), (x1,x3)(x2,x4), (x1,x4)(x2,x3).

Thus the induced groupG in this case comprises four elements:

G = {1, (x1,x2)(x3,x4), (x1,x3)(x2,x4), (x1,x4)(x2,x3)}. (5.3.2)


5.3.3 Third example

Consider Eq. (5.1.3),x4 +x3 +x2+x+1= 0.

We know that Eq. (5.1.3) has the infinite groupG of symmetries given by the trans-formation (5.2.20). Let us take a finite subgroup of this infinite group, namely thesubgroup comprising the transformations

x = x, x = x2, x = x3, x =1x

, (5.3.3)

i.e. the transformations (5.2.20) withn = 1,2,3,−1. The roots of Eq. (5.1.3) are

x1 = ε, x2 = ε2, x3 = ε3, x4 = ε4, (5.3.4)

whereε = e2π i/5.

The reckoning shows that the restriction of the transformations (5.3.3) to the roots(5.3.4) yields four permutations, namely the unit 1 and the permutations

(x1 x2 x3 x4

x2 x4 x1 x3

),

(x1 x2 x3 x4

x3 x1 x4 x2

),

(x1 x2 x3 x4

x4 x3 x2 x1

)

denoted by(x1,x2,x4,x3), (x1,x3,x4,x2), (x1,x4)(x2,x3).

One can verify that the restriction of the arbitrary transformations (5.2.20) to theroots (5.3.4) does not give new permutations. Hence, in thiscase also the inducedsymmetry group comprises four elements:

G = {1, (x1,x2,x4,x3), (x1,x3,x4,x2), (x1,x4)(x2,x3)}. (5.3.5)



5.3.4 Concluding remarks

I summarize the above observations in the following definition of Galois groups foralgebraic equations.

Definition 5.2. Let G be the group of symmetries (5.2.2) of an algebraic equation(5.2.1). The restriction of the groupG to the roots of Eq. (5.2.1) is called the Galoisgroup of Eq. (5.2.1).

This definition provides a parallel between the role of symmetries in solvabilityof differential equations by quadrature and solvability ofalgebraic equation in radi-cals. Namely, we know due to S. Lie that if annth-order ordinary differential equa-tions admits a solvablen-parameter symmetry group, then the equation in questioncan be integrated by quadratures. On the other hand, it is known from the theoryof Galois groups that an algebraic equation of degreen is solvable by radicals if itsGalois group is annth-order solvable group. Now we can formulate this statementin terms of symmetry groups:

An algebraic equation of degree n is solvable by radicals if it has a symmetrygroup G such that the induced groupG is an nth-order solvable group.

Assignment to Part I

Problems

1. Find the one-parameter group with the generator

X = y∂∂x

−x∂∂y

+(v+x

) ∂∂u

−(u+y

) ∂∂v

·

2. Let x,y be two independent variables andv,u be two dependent variables. Con-sider the operatorX from Problem 1. Find its first prolongation

X(1) = y∂∂x

−x∂∂y

+(v+x

) ∂∂u

−(u+y

) ∂∂v

+ ζ 11

∂∂ux

+ ζ 12

∂∂uy

+ ζ 21

∂∂vx

+ ζ 22

∂∂vy

·

3. Make the commutator table for the five-dimensional Lie algebra L5 spanned bythe following operators in the three-dimensional space(t,x,y) :

X1 =∂∂ t

, X2 =∂∂x

, X3 = t∂∂ t

+x∂∂x

,

X4 = t2 ∂∂ t

+ ty∂∂y

, X5 = 2x∂∂x

−y∂∂y

·

Find the first derived algebraL′5 and determine its dimension.

4. Investigate the Lie algebraL5 from Problem 3 for solvability.

5. Make the commutator table for the three-dimensional Lie algebraL3 spanned bythe following operators in the three-dimensional space(x,y,z) :

117

118 Assignment to Part I

X1 =∂∂z

,

X2 = sinz∂∂x

+coszcosx

∂∂y

− tanxcosz∂∂z

,

X3 = cosz∂∂x

− sinzcosx

∂∂y

+ tanxsinz∂∂z

·

6. Investigate the Lie algebraL3 from Problem 5 for solvability.

7. Find the one-parameter group with the generator

X = e−z(

sinx∂∂x

−cosx∂∂z

)·

8. Let L3 be the three-dimensional Lie algebra spanned by the following operatorsin the three-dimensional space(x,y,z) :

X1 =∂∂x

, X2 =∂∂z

, X3 =∂∂y

+x∂∂z

·

Find the transformationsTa1,Ta2,Ta3 generated byX1,X2,X3, respectively, and con-struct the three-parameter groupG3 by taking the composition

Ta = Ta3Ta2Ta1,

wherea = (a1,a2,a3). Find the composition law inG3.

9. Find a basis of invariants for the one-parameter group with the generator

X = (1+ t2)∂∂ t

+ tr∂∂ r

+(r − tvr)∂

∂vr− tvθ

∂∂vθ

−2tρ∂

∂ρ−4t p

∂∂ p

,

wherevr ,vθ are the components of the velocity in the polar coordinatesr,θ (seeSection 2.2.3, Example 2.9).

Solutions

1. Solving the Lie equations (2.1.24) for ¯x, y we obtain the rotation

x = xcosa+ysina, y = ycosa−xsina.

Therefore the Lie equations for ¯u, v are written

Assignment to Part I 119

duda

= v+xcosa+ysina, u|a=0 = u,

dvda

= −u+xsina−ycosa, v|a=0 = v.

(1)

Differentiating the first equation of (1) and using the second one we get

d2uda2 = −u,

whenceu = C1 cosa+C2sina. (2)

Substituting Eq. (2) in the first equation of (1) we find

v = C2cosa−C1sina−xcosa−ysina. (3)

Letting in Eqs. (2) and (3)a = 0 and using the initial conditions from Eqs. (1) wegetC1 = u, C2 = v+x and obtain

u = ucosa+vsina+xsina, v = vcosa−usina−ysina.

Answer. The group is given by the transformations

x = xcosa+ysina, y = ycosa−xsina,

u = ucosa+vsina+xsina, v = vcosa−usina−ysina.

2. The prolongation formula (3.3.21) yields:

X(1) = y∂∂x

−x∂∂y

+(v+x

) ∂∂u

−(u+y

) ∂∂v

+(1+vx+uy

) ∂∂ux

+(vy−ux

) ∂∂uy

+(vy−ux

) ∂∂vx

−(1+vx+uy

) ∂∂vy

·

3. The commutator table has the form

X1 X2 X3 X4 X5

X1 0 0 X1 2X3−X5 0

X2 0 0 X2 0 2X2

X3 −X1 −X2 0 X4 0

X4 −2X3 +X5 0 −X4 0 0

X5 0 −2X2 0 0 0

This table shows that the derived algebra isL′5 =< X1,X2,Y3, X4 >, i.e. it is a four-

dimensional Lie algebra spanned by the operatorsX1,X2,X4 and


Y3 = 2X3−X5.

4. The commutator table for the first derived algebraL′5 from Problem 3 is

X1 X2 Y3 X4

X1 0 0 2X1 Y3

X2 0 0 0 0

Y3 −2X1 0 0 X4

X4 −Y3 0 −X4 0

This table shows that the second derived algebra isL′′5 =< X1,Y3,X4 > . It follows

further thatL′′′5 =< X1,Y3,X4 >, i.e.L′′′

5 = L′′5, and henceL(n)

5 = L′′5 for anyn≥ 2. It

means thatL5 is not solvable.

5. The reckoning shows that the commutator table has the form

X1 X2 X3

X1 0 X3 −X2

X2 −X3 0 X1

X3 X2 −X1 0

6. The commutator table from Problem 5 yieldsL′3 = L3. Hence,L3 is not solvable.

7. The Lie equations (2.1.23) are written as follows:

dxda

= e−zsinx,dzda

= −e−zcosx.

Dividing the second equation by the first one, we obtain

dzdx

= −cosxsinx

,

or

dz= −cosxsinx

dx = −dsinxsinx

·

Thus, we have the first integral

ezsinx = C. (4)

Now we write the first Lie equation in the form

da =ez

sinxdx =

C

sin2 xdx,


integrate it and obtain:

−Ccosxsinx

= a+K,

or, substituting the expression (4) forC :

−ezcosx = a+K.

Using the initial conditions, ¯x = x, z= z whena = 0, we find the constant of inte-grationK and obtain:

ezcosx = ezcosx−a. (5)

DeterminingC in Eq. (4) by using the initial conditions we conclude that the grouptransformation is defined implicitly by the equations

ez sinx = ez sinx, ez cosx = ez cosx−a. (6)

Checking.The group composition law is verified as follows:

e¯z sin ¯x = ez sinx = ez sinx,

e¯z cos¯x−b = ez cosx−b = ez cosx− (a+b).

In addition, we can obtain from Eqs. (6) the coordinates of the generatorX. To thisend, we write the first equation of Eqs. (6) for the infinitesimal transformation

x = x+aξ , z= z+aη

and obtain:

ez sinx = ez+aη sin(x+aξ ) = ezeaη(sinxcosaξ +cosxsinaξ )

≈ ez(1+aη)(sinx+aξ cosx)

≈ ez sinx+aez(ξ cosx+ η sinx),

whenceξ cosx+ η sinx = 0. (7)

Dealing likewise with the second equation of Eqs. (6) we obtain

ez cosx−a= ez+aη cos(x+aξ )

≈ ez cosx+aez(η cosx− ξ sinx),

whenceez(η cosx− ξ sinx) = −1. (8)

Solving Eqs. (7)–(8) forξ ,η we get the coordinates of our operatorX.

Answer. The group transformation is defined implicitly by Eqs. (6).

8. Following the construction described in Section 4.5.2 we obtain:


Ta1 : x = x+a1, y = y, z= z;

Ta2 : x = x, y = y, z= z+a2;

Ta3 : x = x, y = y+a3, z= z+xa3.

Their compositionTa = Ta3Ta2Ta1 has the form:

x = x+a1, y = y+a3,

z= z+a2+xa3+a1a3.

The consecutive application ofTa andTb, whereb= (b1,b2,b3), yields the transfor-mationTbTa :

x = x+a1+b1, y = y+a3+b3,

z= z+a2+b2+x(a3+b3)+a1(a3 +b3)+b1b3.

The equationTbTa = Tc leads to the equations

x = x+c1, y = y+c3,

z= z+c2+xc3+c1c3,

whence

c1 = a1 +b1,

c3 = a3 +b3,

c2 = a2 +b2−a3b1.

Answer. The transformationTa has the form

x = x+a1, y = y+a3, z= z+a2+xa3+a1a3.

The composition law is

φ1(a,b) = a1 +b1,

φ2(a,b) = a2 +b2−a3b1,

φ3(a,b) = a3 +b3.

9. The equation (2.2.1) for the invariantJ(t, r,vr ,vθ ,ρ , p) is written

X(J) ≡2t(1+ t2)∂J∂ t

+ tr∂J∂ r

+(r − tvr)∂J∂vr

− tvθ∂J∂vθ

−2tρ∂J∂ρ

−4t p∂J∂ p

= 0.


We will use its characteristic system:

dt1+ t2 =

drtr

=dvr

r − tvr= −dvθ

tvθ= − dρ

2tρ= − dp

4t p·

Integration of the first equation of this system,

dt1+ t2 =

drtr

,

gives ln(1+ t2)+ lnC1 = ln r. It yields r = C1√

1+ t2, whence solving with respectto the constant of integrationC1 we obtain the first integral, i.e. the invariant

λ =r√

1+ t2·

Let us integrate now the second equation, namely the equation

dt1+ t2 =

dvr

r − tvr·

We write it in the formdvr

dt= − tvr

1+ t2 +r

1+ t2 ,

substitute herer = λ√

1+ t2 and obtain the following non-homogeneous linear first-order equation:

dvr

dt= − t

1+ t2 vr +λ√

1+ t2, λ = const.

It can be readily solved, e.g. by the method of variation of parameter (see, e.g. [17],Section 3.2.7). The homogeneous equation

dvr

dt= − t

1+ t2 vr

yields ln|vr | = − 12 ln(1+ t2)+ lnC, C =const., whence

vr =C√

1+ t2·

Thus, we set

vr =C(t)√1+ t2

substitute this expression and its derivative

dvr

dt= − tC(t)

(1+ t2)√

1+ t2+

C′(t)√1+ t2


in the non-homogeneous equation and obtain:

− tC(t)

(1+ t2)√

1+ t2+

C′(t)√1+ t2

= − t1+ t2

C(t)√1+ t2

+λ√

1+ t2,

whenceC′(t) = λ , and henceC(t) = λ t +K, K = const. Thus,

vr =λ t +K√

1+ t2·

Let us solve this equation forK. Using the equationr = λ√

1+ t2, we get

K = vr

√1+ t2−λ t =

1λ

(rvr −λ 2t

)=

1λ

(rvr −

tr2

1+ t2

).

Sinceλ is a constant on the solutions of the characteristic system,it follows thatKλ = const. Hence, settingU = Kλ we arrive at the invariant

U = rvr −tr2

1+ t2 ·

The equationdrtr

= −dvθtvθ

yields ln|t| = lnC2− ln |vθ |, C2 = const., whence the invariant isV = rvθ .Other invariants are obtained likewise. The equation

dt1+ t2 = − dρ

2tρ

yields the invariantR= (1+ t2)ρ and the last equation

dt1+ t2 −

dp4t p

,

the invariantP = (1+ t2)2p.

Answer. The following five invariants provide a basis:

λ =r√

1+ t2, U = rvr −

tr2

1+ t2 ,

V = rvθ , R= (1+ t2)ρ , P = (1+ t2)2p.

Part IIApproximate Transformation Groups

Theory of approximate transformation groups is a new direction in Lie group analy-sis of differential equations. It has been developed in [2],[3] for tackling differentialequations with a small parameter.

The challenge has been as follows. The classical Lie group analysis allows oneto single out equations with remarkable symmetry properties among differentialequations used in mathematical modelling. However, small perturbations of modelequations may break the admissible group thus reducing the value of group methodsin general applied sciences. For example, it is well known that the Burgers equation

ut = uux +uxx

and the Korteweg-de Vries equation

ut = uux +uxxx,

considered separately, have infinite sets of Lie-Backlundsymmetries. On the otherhand, the combined Burgers–Korteweg-de Vries equation

ut = uux + ε(auxxx+buxx)

with non-vanishing parametersa,b,ε does not have this remarkable property. There-fore,development of methods of group analysis that are stable with respect to smallperturbations of equations has become vital.An attempt to develop such methodshas been undertaken in [2].

Our approach is based on the concept of approximate transformations and a rep-resentation of local Lie groups by approximate transformations. The extension ofLie’s infinitesimal characterization of local groups to approximate transformationgroups leads to approximate Lie algebras and approximate symmetries of differen-tial equations with small parameters. This extension augments possibilities of theclassical Lie group analysis by adding stability with respect to small perturbationsof mathematical models. For instance, the infinite series ofsymmetries of the Burg-ers equation and the Korteweg-de Vries equation do not disappear when we com-bine them together as above, but pass from the category of “exact” symmetries tothe new category of “approximate” symmetries [3]. Numerousexamples show thatsymmetries of many other physically interesting differential equations are stablewith respect to small perturbations (see [12], Chapter 9).

The new theory maintains the basic attributes of Lie group analysis. In particular,in the classical Lie group theory, one-parameter transformation groups with giveninfinitesimal generators are obtained either by solving theLie equations or applyingthe exponential map. The same constructions apply to approximate transformationgroups by adding the adjectiveapproximateto thegenerators, Lie equationsandexponential map.

126

Chapter 6Preliminaries

6.1 Motivation

The initiation and subsequent development of the theory of approximate transfor-mation groups were inspired by the following circumstances.

A variety of differential equations recognized as mathematical models in engi-neering and physical sciences, involve empirical parameters or constitutive laws.Therefore coefficients of model equations are defined approximately with an in-evitable error. Consequently, differential equations depending on a small parameteroccur frequently in applications. Unfortunately, any small perturbation of coeffi-cients of a differential equation disturbs its symmetry properties. Thisinstability ofLie symmetrieswas the first circumstance that led us [2] to the idea of modifyingLie’s theory by introducing the concept of approximate transformation groups.

The second circumstance is that utilization ofLie group techniques may lead tounjustified complexities.This happens, e.g. in developing the theory of relativity inthe de Sitter space-time (see [15]) with the metric

ds2 = −(1+ εσ2)−2

4

∑µ=1

(dxµ)2, σ2 =4

∑µ=1

(xµ)2.

Here the following notation is used:

(x1,x2,x3,x4) = (x,y,z, ict), i =√−1,

wherec= 2.99793×1010cm/s is the velocity of light in empty space,ε = K/4 withK denoting the curvature of the de Sitter universe. Accordingto cosmological data,the curvatureK is a small constant (∼ 10−54 cm−2), and henceε can be treated as asmall parameter. We assume here thatε ≥ 0. Thede Sitter group(i.e., the group ofisometric motions in the de Sitter space-time) differs fromthePoincare group(i.e.,the group of isometries in the Minkowski space-time) in thatthe usual translationsof space-time coordinatesxµ are replaced by more complicated transformations,

127

128 6 Preliminaries

the so-called “generalized translations” in the de Sitter space-time. The generalizedtranslation, e.g. along thex1 axis has the generator

X =(

1+ ε[(x1)2− (x2)2− (x3)2− (x4)2]) ∂

∂x1

+2εx1(

x2 ∂∂x2 +x3 ∂

∂x3 +x4 ∂∂x4

).

The corresponding group transformations (with the group parametera) have theform

x1 = 2x1 cos(2a

√ε)+

1− εσ2

2√

εsin

(2a

√ε)

1+ εσ2+(1− εσ2)cos(2a√

ε)−2x1√

ε sin(2a√

ε), (6.1.1)

x j = 2x j

1+ εσ2+(1− εσ2)cos(2a√

ε)−2x1√

ε sin(2a√

ε),

j = 2,3,4.

However, sinceε is small, it is sufficient to use an approximate expression ofthese transformations, e.g. by expanding them in powers ofε and considering onlythe leading terms of the first order. That is, to consider the de Sitter group as aperturbation of the Poincare group by the curvatureK. Then the result is given bythe following simple approximate transformation:

x1 ≈ x1 +a+ ε([(x1)2− (x2)2− (x3)2− (x4)2]a+x1a2 +

13

a3),

x j ≈ x j + ε(2ax1+a2)x j , j = 2,3,4.

(6.1.2)

The question naturally arises of how to calculate this perturbation directly, with-out using the complicated group transformation (6.1.1). The theory of approximategroups gives a method of calculation. Namely, one can use either theapproximateLie equationsor theapproximate exponential map.We calculate the one-parameterapproximate transformation group (6.1.2) in Example 7.6 byapplying Theorem 7.2on the approximate exponential map to the above generatorX.

Thus, the approximate transformation group techniques often simplify calcula-tions significantly. Moreover, approximate symmetries canhelp to solve with thedesired accuracy differential equations that have no Lie point symmetries and hencecannot be solved by Lie group methods. An example of this kind, namely the equa-tion

y′′−x− εy2 = 0

will be discussed in Section 9.1.1.

6.2 A sketch on Lie transformation groups 129

6.2 A sketch on Lie transformation groups

A brief sketch of the Lie equations and the exponential map isgiven here to draw aparallel between the classical and approximate group theories.

6.2.1 One-parameter transformation groups

Let x = (x1, . . . ,xn) ∈ IRn. Consider a one-parameter family of invertible transfor-mationTa:

x = f (x,a), (6.2.1)

or in coordinates,xi = f i(x,a), i = 1, . . . ,n.

The transformationTa carries the pointx∈ IRn to the point ¯x∈ IRn.Here the parametera ranges over all real numbers from a neighborhoodU ⊂ IR

of a = 0, and we impose the condition that Eq. (6.2.1) is the identity transformationif and only if a = 0, i.e.

f (x,0) = x, (6.2.2)

and, conversely, the equationf (x,a) = x with a∈U impliesa = 0.

Definition 6.1. A set G of transformations (6.2.1) is called a continuous one-parameter transformation group in IRn if the functionsf i(x,a) satisfy the condition(6.2.2) and thegroup property

f i( f (x,a),b) = f i(x,c), i = 1, . . . ,n, (6.2.3)

for all a,b∈U , wherec∈U is a certain (smooth) function ofa andb:

c = φ(a,b), (6.2.4)

such that the equationφ(a,b) = 0 (6.2.5)

has a unique solutionb∈U for anya∈U . Givena, the solutionb of Eq. (6.2.5) isdenoted bya−1. The functionφ(a,b) is termed a group composition law.

According to Definition 6.1, a continuous groupG contains the (unique) identitytransformationI = T0. Further, the group property (6.2.3) means that any two trans-formationsTa,Tb ∈G carried out one after the other result in a transformation whichalso belongs toG:

TbTa = Tc, c = φ(a,b) ∈U,

for any a,b ∈ U . The solvability of Eq. (6.2.5), together with the group property(6.2.3), provides the inverse transformationT−1

a = Ta−1 ∈ G to Ta ∈ G:

130 6 Preliminaries

Ta−1Ta = TaTa−1 = I

for anya∈U .

6.2.2 Canonical parameter

Definition 6.2. The group parametera is said to becanonicalif the compositionlaw (6.2.4) has the form

φ(a,b) = a+b, (6.2.6)

i.e., if the group property is written

f i( f (x,a),b) = f i(x,a+b), i = 1, . . . ,n. (6.2.7)

Theorem 6.1. Given an arbitrary group composition law (6.2.4), there exists thecanonical parameter ˜a. Namely, it is defined by the formula

a =

∫ a

0

daA(a)

, (6.2.8)

where

A(a) =∂φ(a,b)

∂b

∣∣∣∣b=0

. (6.2.9)

Example 6.1. We letn = 1 and consider the transformation

x = x+ax.

Using Eqs. (6.2.3) one can verify that this transformation provides a one-parametergroup where the composition law (6.2.4) has the form

φ(a,b) = a+b+ab.

The parametera is not canonical. Therefore we apply Theorem 6.1, Equation (6.2.9)yields:

A(a) = 1+a,

and hence the canonical parameter (6.2.8) is written

a =

∫ a

0

da1+a

= ln(1+a).

In what follows, we will adopt the canonical parameter when referring to one-parameter Lie groups as well as approximate transformationgroups.


6.2.3 Group generator and Lie equations

Let G be a group of transformations (6.2.1) withf (x,a) satisfying the condition(6.2.2) and the group property (6.2.7). Its infinitesimal transformation

xi ≈ xi +aξ i(x), (6.2.10)

where

ξ i(x) =∂ f i(x,a)

∂a

∣∣∣∣a=0

, (6.2.11)

is associated with the followinggeneratorof the groupG :

X = ξ i(x)∂

∂xi · (6.2.12)

One-parameter groups are determined by their generators according to the followingtheorem due to Lie.

Theorem 6.2. The transformation (6.2.1), ¯x = f (x,a), satisfies the group property(6.2.7) and has the operator (6.2.12) as its generator if andonly if it solves theordinary differential equations (called theLie equations)

dxi

da= ξ i(x), i = 1, . . . ,n, (6.2.13)

with the initial conditions

xi∣∣a=0 = xi , i = 1, . . . ,n. (6.2.14)

Proof. Let us assume thatf (x,a) satisfies the group property (6.2.7):

f i(x,a+b) = f i( f (x,a),b).

Let us considerb as an increment ofa, i.e. setb = ∆a, and expand both sides of Eq.(6.2.7) in powers of∆a keeping only the principal parts. We have:

f i(x,a+b)≈ f i(x,a)+∂ f i(x,a)

∂a∆a

and

f i( f (x,a),b) ≈ f i(x,a)+∂ f i

(f (x,a),b

)

∂b

∣∣∣∣b=0

∆a.

The substitution in Eq. (6.2.7) yields:

∂ f i(x,a)

∂a=

∂ f i(

f (x,a),b)

∂b

∣∣∣∣b=0

. (6.2.15)

132 6 Preliminaries

According to Eq. (6.2.11), the right-hand side of Eq. (6.2.15) is equal toξ i( f (x,a)).Invoking that f (x,a) = x and considering Eq. (6.2.15) for any fixedx, we arrive atEqs. (6.2.13).

Let us prove now that the solution ¯x = f (x,a) of Eqs. (6.2.13)–(6.2.14) sat-isfies the group property (6.2.7). Consider, for any fixedx and a, the functionsu = (u1, . . . ,un) andv = (v1, . . . ,vn) of the variableb defined as follows:

u(b) = f (x,b) ≡ f ( f (x,a),b), v(b) = f (x,a+b).

To prove the group property (6.2.7), it suffices to show thatu(b) = v(b) in a neigh-borhood ofb = 0. Since f (x,a) solves Eqs. (6.2.13)–(6.2.14), one has:

dudb

≡ d f (x,b)

db= ξ (u), u

∣∣b=0 = f (x,a);

anddvdb

≡ d f (x,a+b)

db= ξ (v), v

∣∣b=0 = f (x,a).

Hence, bothz= u(b) andz= v(b) solve one and the same Cauchy problem:

dzdb

= ξ (z), z∣∣b=0 = f (x,a).

The uniqueness of the solution of the Cauchy problem guarantees that, in a neigh-borhood ofb = 0, we haveu(b) = v(b), i.e.

f ( f (x,a),b) = f (x,a+b).

This completes the proof.

Example 6.2. Theorem 6.2 allows one to obtain a one-parameter group of trans-formations (6.2.1) by solving the Lie equations (6.2.13) with the initial conditions(6.2.14) for any given operator (6.2.12) or the corresponding infinitesimal transfor-mation (6.2.10).

Consider, e.g. the infinitesimal transformation

x≈ x+ax2, y≈ y+axy

in the(x,y) plane. The corresponding differential operator (6.2.12) is

X = x2 ∂∂x

+xy∂∂y

and the Lie equations (6.2.13) have the form

dxda

= x2,dyda

= xy.

Integrating these equations, one obtains


x =1

C1−a, y =

C2

C1−a,

whereC1 andC2 are arbitrary constants. The initial conditions (6.2.14) provide

x =1

C1, y =

C2

C1,

whence

C1 =1x

, C2 =yx·

Thus the one-parameter group of transformations has the form

x =x

1−ax, y =

y1−ax

· (6.2.16)

6.2.4 Exponential map

The solution of the Lie equations (6.2.13)–(6.2.14) can be represented explicitly bythe exponential map,

xi = eaX(xi), i = 1, . . . ,n, (6.2.17)

where the exponent is given by the infinite sum:

eaX = 1+aX+a2

2!X2 +

a3

3!X3 + · · · . (6.2.18)

Example 6.3. The exponential map (6.2.17) is written in the(x,y) plane in theform:

x = eaX(x), y = eaX(y), (6.2.19)

where eaX is given by the series (6.2.18). Let us apply the exponentialmap to thegenerator

X = x2 ∂∂x

+xy∂∂y

considered in Example 6.2. We have:

X(x) = x2, X2(x) = X(X(x)) = X(x2) = 2!x3,

X3(x) = 3!x4, . . . .

These equations hint the general formula

Xn(x) = n!xn+1, n = 1,2, . . . .

The proof is given by induction:

134 6 Preliminaries

Xn+1(x) = X(n!xn+1) = (n+1)!x2xn = (n+1)!xn+2.

It follows:eaX(x) = x+ax2+ · · ·+anxn+1 + · · · .

One can rewrite the right-hand side of this equation in the form

x(1+ax+ · · ·+anxn + · · ·) =x

1−ax,

where the well known Taylor expansion of the function(1−ax)−1 is used providedthat|ax| < 1. Hence,

eaX(x) =x

1−ax·

Similarly,X(y) = xy,

X2(y) = X(xy) = yX(x)+xX(y) = y(x2)+y(xy) = 2!yx2,

X3(y) = 2![yX(x2)+x2X(y)] = 2![y(2x3)+x2(xy)] = 3!yx3.

This hints the general formula

Xn(y) = n!yxn, n = 1,2, . . .

that can be readily verified by induction:

Xn+1(y) = n!X(yxn) = n![nyxn+1+xn(xy)] = (n+1)!yxn+1.

It follows:

eaX(y) = y+ayx+a2yx2 + · · ·+anyxn + · · ·

= y(1+ax+ · · ·+anxn + · · ·) =y

1−ax·

Thus, we arrive at the transformation (6.2.16):

x =x

1−ax, y =

y1−ax

·

6.3 Approximate Cauchy problem

6.3.1 Notation

Consider functionsf (x,ε) depending on variablesx = (x1, . . . ,xn) and a parameterε. These functions are defined in a neighborhood ofε = 0 and are assumed to be

6.3 Approximate Cauchy problem 135

continuous inx,ε together with derivatives of an order required in the subsequentdiscussion.

Definition 6.3. Let p≥ 1 be an integer. A functionf (x,ε) is said to beof order lessthanε p and is written

f (x,ε) = o(ε p) (6.3.1)

if it satisfies the condition

limε→0

f (x,ε)

ε p = 0. (6.3.2)

Remark 6.1. Equation (6.3.2) is satisfied if there exists a constantC > 0 such that

| f (x,ε)| ≤C|ε|p+1. (6.3.3)

It is also satisfied if there exists a functionχ(x,ε) that is analytic nearε = 0 andsuch that

f (x,ε) = ε p+1χ(x,ε). (6.3.4)

Definition 6.4. Iff (x,ε)−g(x,ε) = o(ε p), (6.3.5)

the functionsf andg are said to beapproximately equal(with an erroro(ε p)) andwritten

f (x,ε) = g(x,ε)+o(ε p), (6.3.6)

or brieflyf ≈ g (6.3.7)

when there is no ambiguity.

The approximate equality defines an equivalence relation, and we join functionsinto equivalence classes by lettingf (x,ε) andg(x,ε) be members of the same classif and only if f ≈ g.

Given a functionf (x,ε), let

f0(x)+ ε f1(x)+ · · ·+ ε p fp(x) (6.3.8)

be the approximating polynomial of degreep in ε obtained via the Taylor series ex-pansion off (x,ε) in powers ofε aboutε = 0. Then any functiong≈ f (in particular,the functionf itself) has the form

g(x,ε) = f0(x)+ ε f1(x)+ · · ·+ ε p fp(x)+o(ε p). (6.3.9)

Consequently the function (6.3.8) is called acanonical representativeof the equiv-alence class of functions containingf .

Thus, the equivalence class of functionsg(x,ε) ≈ f (x,ε) is determined by theordered set ofp+1 functions

f0(x), f1(x), . . . , fp(x).

136 6 Preliminaries

6.3.2 Definition of the approximate Cauchy problem

Theorem 6.3. Let functions f (x,ε), g(x,ε) be analytic near the point(x0,ε) andsatisfy the condition

g(x,ε) = f (x,ε)+o(ε p). (6.3.10)

Let x = x(t,ε) andx = x(t,ε) solve the problems

dxdt

= f (x,ε), x|t=0 = α(ε)

anddxdt

= g(x,ε), x|t=0 = β (ε),

respectively, where

α(0) = β (0) = x0, β (ε) = α(ε)+o(ε p).

Then,x(t,ε) = x(t,ε)+o(ε p). (6.3.11)

In other words, the solutions of two Cauchy problems with approximately equalright-hand sides and initial conditions, are equal in the same order of approximation.

Proof. The functionu(t,ε) = x(t,ε)− x(t,ε) satisfies the conditions

u(0,ε) = o(ε p), (6.3.12)∣∣∣∣dudt

∣∣∣∣ ≤ | f (x,ε)−g(x,ε)| ≤ | f (x,ε)−g(x,ε)|+ |g(x,ε)−g(x,ε)|. (6.3.13)

Using Eq. (6.3.10) written in the form of the inequality (6.3.3)

|g(x,ε)− f (x,ε)| ≤Cε p+1, C = const.,

and the Lipschitz condition

|g(x,ε)−g(x,ε)| ≤ K|x− x|, K = const.,

one obtains from Eq. (6.3.13) the following inequality:∣∣∣∣dudt

∣∣∣∣ ≤ K|u|+Cε p+1. (6.3.14)

For every fixedε there exists suchtε , that the functionu(t,ε) has a constant sign onthe interval form 0 totε . Then inequality (6.3.14) yields that on this interval one has

ddt|u| ≤ K|u|+Cε p+1.

6.3 Approximate Cauchy problem 137

Dividing this inequality by

|u|+ CK

ε p+1

and integrating from 0 tot with |t| ≤ |tε |, one obtains

|u(t,ε)| ≤ CK

(eKtε −1

)ε p+1 +u(0,ε)eKtε .

Whence, using Eq. (6.3.12), one obtains the required Eq. (6.3.11).

Definition 6.5. An approximate Cauchy problem

dzdt

≈ f (z,ε), (6.3.15)

z|t=0 ≈ α(ε), (6.3.16)

is determined as follows. The approximate differential equation (6.3.15) is inter-preted as a family of differential equations

dzdt

= g(z,ε) with g(z,ε) ≈ f (z,ε). (6.3.17)

The approximate initial condition (6.3.16) is interpretedlikewise:

z|t=0 = β (ε) with β (ε) ≈ α(ε). (6.3.18)

The approximate equations in (6.3.17) and (6.3.18) have theaccuracyp as in Eqs.(6.3.15) and (6.3.16).

Definition 6.6. The solution of the approximate initial value problem (6.3.15) and(6.3.16) is the equivalence class (see Section 6.3.1) of thesolutions of the exactCauchy problems (6.3.17) and (6.3.18). In other words, the solution of the approxi-mate Cauchy problem (6.3.15) and (6.3.16) is the solution ofany problem (6.3.17)and (6.3.18) considered with the accuracy up too(ε p).

Theorem 6.4. The solution to the approximate Cauchy problem (6.3.15) and(6.3.16)is unique.

Proof. The uniqueness of the solution follows from Theorem 6.3. Indeed, accord-ing to this theorem, the solutions of all problems of the form(6.3.17) and (6.3.18)coincide with the mentioned accuracy.

Chapter 7Approximate transformations

In the rest of the book we will consider the approximation in the first order of preci-sion inε. In other words, equation (6.3.5) for approximate equality will be consid-ered forp = 1.

7.1 Approximate transformations defined

Definition 7.1. An approximate transformation

x≈ f (x,a,ε) (7.1.1)

in IRn is written in the first-order of precision in the form

xi ≈ f i0(x,a)+ ε f i

1(x,a), i = 1, . . . ,n, (7.1.2)

and is defined as the set of all invertible transformations

x i = gi(x,a,ε) (7.1.3)

such thatgi(x,a,ε) ≈ f i

0(x,a)+ ε f i1(x,a).

We will deal in what follows with approximate transformations (7.1.1) obeyingthe initial condition ¯x|a=0 ≈ x, or in the coordinate form

xi∣∣a=0 ≈ xi , i = 1, . . . ,n. (7.1.4)

Accordingly, we will assume that the functionsgi(x,a,ε) in the representations(7.1.3) of the approximate transformation (7.1.2) are defined in a neighborhood ofa = 0 and satisfy the conditions

gi(x,a,ε) ≈ xi , i = 1, . . . ,n,

139

140 7 Approximate transformations

if and only if a = 0.

7.2 Approximate one-parameter groups

7.2.1 Introductory remark

Consider transformationsx = f (x,a,ε)

depending on a small parameterε and defining a one-parameter group with respectto a, so that the vector-functionf = ( f 1, . . . , f n) satisfies the following equations:

f ( f (x,a,ε),b,ε) = f (x,a+b,ε),

f (x,0,ε) = x.(7.2.1)

Let a vector-functiong = (g1, . . . ,gn) be approximately equal tof , i.e.

g(x,a,ε) = f (x,a,ε)+o(ε). (7.2.2)

Consider now, together with ¯x, the “neighboring points” ˜x determined by

x = g(x,a,ε). (7.2.3)

Substituting Eq. (7.2.2) into Eq. (7.2.1), we obtain the following approximate equa-tions in the first-order of precision inε :

g(g(x,a,ε),b,ε) ≈ g(x,a+b,ε),

g(x,0,ε) ≈ x.(7.2.4)

7.2.2 Definition of one-parameter approximate transformationgroups

Definition 7.2. An approximate transformation(7.1.1),

x≈ f (x,a,ε)

satisfying the initial condition (7.1.4),

x∣∣a=0 ≈ x, (7.2.5)

is called a one-parameter approximate transformation group in IRn if the group prop-erty is satisfied with the accuracyo(ε) :

7.2 Approximate one-parameter groups 141

f ( f (x,a,ε),b,ε) ≈ f (x,a+b,ε). (7.2.6)

Remark 7.1. For approximate transformations written in the form (7.1.2),


1(x,a), i = 1, . . . ,n,

the definition means that the vector-functionsg = (g1, . . . ,gn) in all exact represen-tations(7.1.3) of form (7.1.2):

xi = gi(x,a,ε), i = 1, . . . ,n, (7.2.7)

satisfy the approximate equation (7.2.6):

g(g(x,a,ε),b,ε) ≈ g(x,a+b,ε). (7.2.8)

Remark 7.2. In Eqs. (7.2.6) and (7.2.8), unlike the exact group property(6.2.7) inthe classical Lie group theory,f andg, respectively, do not necessarily denote thesame functions at each occurrence. They can be replaced by any equivalent functions(see the next example).

Example 7.1. Let us taken = 1 and consider the functions

f (x,a,ε) = x+a(

1+ εx+12

εa)

and

g(x,a,ε) = x+a(1+ εx)(

1+12

εa).

They are equal in the first order of precision, namely Eq. (6.3.4) is satisfied:

g(x,a,ε) = f (x,a,ε)+ ε2χ(x,a), χ(x,a) =12

a2x.

They also obey the approximate group property (7.2.6). Indeed,

f (g(x,a,ε),b,ε) = f (x,a+b,ε)+ ε2 χ(x,a,b,ε), (7.2.9)

where

χ(x,a,b,ε) =12

a(ax+ab+2bx+ εabx).

7.2.3 Generator of approximate transformation group

Definition 7.3. The generator of an approximate transformation group(7.1.2) isthe set of all first-order linear differential operators

X = ξ i(x,ε)∂

∂xi (7.2.10)


with the coefficients

ξ i(x,ε) ≈ ξ i0(x)+ εξ i

1(x), i = 1, . . . ,n,

where

ξ i0(x) =

∂ f i0(x,a)

∂a

∣∣∣∣a=0

, ξ i1(x) =

∂ f i1(x,a)

∂a

∣∣∣∣a=0

. (7.2.11)

According to this definition, the generator of an approximate transformationgroup (7.1.1) is anapproximate differential operator

X ≈ ξ i(x,ε)∂

∂xi , (7.2.12)

whereξ i(x,ε) are the components of the vector

ξ (x,ε) ≈ ∂ f (x,a,ε)

∂a

∣∣∣∣a=0

. (7.2.13)

In practice, it is convenient to identifyX with its canonical representative:

X =(ξ i

0(x)+ εξ i1(x)

) ∂∂xi · (7.2.14)

7.3 Infinitesimal description

7.3.1 Approximate Lie equations

Lemma 7.1. Let an approximate transformation (7.1.1) in IRn,

x≈ f (x,a,ε),

satisfy the approximate group property (7.2.6) as well as the initial condition

x|a=0 ≈ x

and have the approximate generator (7.2.12) withξ defined by Eq. (7.2.13):

ξ (x,ε) ≈ ∂ f (x,a,ε)

∂a

∣∣∣∣a=0

.

Then the vector-functionf (x,a,ε) solves the approximate equation

∂ f (x,a,ε)

∂a≈ ξ

(f (x,a,ε),ε

). (7.3.1)

7.3 Infinitesimal description 143

Conversely, given any smooth vector-functionξ (x,ε), an approximate transfor-mation (7.1.1) obtained by solving the approximate Cauchy problem

dxda

≈ ξ (x,ε), x∣∣a=0 ≈ x (7.3.2)

satisfies the approximate group property (7.2.6).

Proof.Let us assume thatf (x,a,ε) satisfies the approximate equation (7.2.6) writtenin the form (6.3.4):

f (x,a+b,ε) = f ( f (x,a,ε),b,ε)+ ε2 χ(x,a,b,ε). (7.3.3)

Let us setb = ∆a and expand both sides of Eq. (7.3.3) in powers of∆a keeping onlythe principal parts. We have:

f (x,a+b,ε) ≈ f (x,a,ε)+∂ f (x,a,ε)

∂a∆a

and

f ( f (x,a,ε),b,ε) ≈ f (x,a,ε)+∂ f i

(f (x,a,ε),b,ε

)

∂b

∣∣∣∣b=0

∆a.

The substitution in Eq. (7.3.3) yields:

∂ f (x,a,ε)

∂a=

∂ f(

f (x,a,ε),b,ε)

∂b

∣∣∣∣b=0

+ ε2 χ(x,a,b,ε). (7.3.4)

Invoking Eq. (7.2.13), we replace the first term in the right-hand side of Eq. (7.3.4)by ξ ( f (x,a,ε),ε) and arrive at Eq. (7.3.1).

Let us prove now that the solution ¯x≈ f (x,a,ε) of the Cauchy problem (7.3.2)satisfies the group property (7.2.6). Consider, for any fixedx anda, the functionsu(b,ε) andv(b,ε) defined as follows:

u(b,ε) = f (x,b,ε) ≡ f ( f (x,a,ε),b,ε), v(b,ε) = f (x,a+b,ε).

To prove the group property (7.2.6), it suffices to show thatu(b,ε) ≈ v(b,ε) in aneighborhood ofb = 0. Since f (x,a,ε) solves Eqs. (7.3.2), one has:

dudb

≡ d f (x,b,ε)

db≈ ξ (u,ε), u

∣∣b=0 ≈ f (x,a,ε);

anddvdb

≡ d f (x,a+b,ε)

db≈ ξ (v,ε), v

∣∣b=0 ≈ f (x,a,ε).

Hence, bothz= u(b,ε) andz= v(b,ε) solve one and the same Cauchy problem:

dzdb

≈ ξ (z,ε), z∣∣b=0 ≈ f (x,a,ε).


Theorem 6.4 on uniqueness of the solution of the approximateCauchy problemguarantees that in a neighborhood ofb = 0 we haveu(b,ε) ≈ v(b,ε), i.e. equation(7.2.6):

f ( f (x,a,ε),b,ε) ≈ f (x,a+b,ε).

This completes the proof.In computations, it is convenient to use Eqs. (7.3.2) in a modified form given

below. Let us write the approximate transformation (7.1.2),


1(x,a), i = 1, . . . ,n,

in the following form:x≈ x0 + ε x1, (7.3.5)

wherex0 = f0(x,a), x1 = f1(x,a).

Accordingly, the vector (7.2.13) is written

ξ (x,ε) ≈ ξ0(x)+ εξ1(x), (7.3.6)

whereξ0(x) andξ1(x) are defined by Eqs. (7.2.11).We will also write the canonical representation (7.2.14) ofthe approximate gen-

erator in the formX = X0 + εX1 (7.3.7)

with

X0 = ξ i0(x)

∂∂xi , X1 = ξ i

1(x)∂

∂xi .

Theorem 7.1. The approximate transformation (7.3.5), ¯x ≈ x0 + ε x1, satisfies theapproximate group property (7.2.6) and has the operator (7.3.7) as its approximategenerator if and only if the following ordinary differential equations with initialconditions are satisfied:

dxi0

da= ξ i

0(x0), xi0

∣∣a=0 = xi , i = 1, . . . ,n, (7.3.8)

dxi1

da=

n

∑k=1

∂ξ i0(x)

∂xk

∣∣∣∣∣x=x0

xk1

+ ξ i1(x0), xi

1

∣∣a=0 = 0. (7.3.9)

Equations (7.3.8)–(7.3.9) are called theapproximate Lie equations.

Proof. Substituting the representations (7.3.5) and (7.3.6) for ¯x andξ (x,ε), respec-tively, in Eqs. (7.3.2), we have:

dx0

da+ ε

dx1

da≈ ξ0(x0 + ε x1)+ εξ0(x0 + ε x1),

(x0 + ε x1)∣∣a=0 ≈ x.

(7.3.10)


Equations (7.3.8) and (7.3.9) are obtained from Eqs. (7.3.10) by singling out theterms not containingε and the terms that are linear inε.

Example 7.2. Let n = 1 and let

X = (1+ εx)∂∂x

·

Here ξ0(x) = 1, ξ1(x) = x, and Eqs. (7.3.8)–(7.3.9) are written:

dx0

da= 1, x0

∣∣a=0 = x, (7.3.11)

dx1

da= x0, x1

∣∣a=0 = 0. (7.3.12)

The solution of Eq. (7.3.11) is ¯x0 = x+a. Substituting it in the differential equationin (7.3.12) we have:

dx1

da= x+a.

Integrating and using the initial condition we obtain the following solution of thesystem (7.3.11)–(7.3.12).

x0 = x+a, x1 = ax+a2

2·

Hence, the approximate transformation group is given by

x≈ x+a+ ε(

ax+a2

2

).

Example 7.3. Let n = 2 and let

X = (1+ εx2)∂∂x

+ εxy∂∂y

· (7.3.13)

Here ξ0(x,y) = (1,0), ξ1(x,y) = (x2,xy), and Eqs. (7.3.8)–(7.3.9) are written:

dx0

da= 1,

dy0

da= 0, x0

∣∣a=0 = x, y0

∣∣a=0 = y,

dx1

da= (x0)

2,dy1

da= x0y0, x1

∣∣a=0 = 0, y1

∣∣a=0 = 0.

The integration gives the following approximate transformation group:

x≈ x+a+ ε(

ax2 +a2x+a3

3

), y≈ y+ ε

(axy+

a2

2y).


7.3.2 Approximate exponential map

Theorem 7.2. Given an operator

X = X0 + εX1 (7.3.14)

with a small parameterε, where

X0 = ξ i0(x)

∂∂xi , X1 = ξ i

1(x)∂

∂xi , (7.3.15)

the corresponding approximate group transformation

xi = xi0 + ε xi

1, i = 1, . . . ,n, (7.3.16)

is determined by the following equations:

xi0 = eaX0(xi), xi

1 = 〈〈aX0,aX1〉〉(xi0), i = 1, . . . ,n, (7.3.17)

where

eaX0 = 1+aX0+a2

2!X2

0 +a3

3!X3

0 + · · · (7.3.18)

and

〈〈aX0,aX1〉〉 = aX1+a2

2![X0,X1]+

a3

3![X0, [X0,X1]]+ · · · . (7.3.19)

In other words, the approximate operatorX = X0+εX1 generates the one-parameterapproximate transformation group given by the followingapproximate exponentialmap:

xi =(1+ ε〈〈aX0,aX1〉〉

)eaX0(xi), i = 1, . . . ,n. (7.3.20)

Proof. (see [13]) The substitution of the operator (7.3.14),

X0 + εX1

in the definition (6.2.18) of the exponent yields:

ea(X0+εX1) = 1+a(X0 + εX1

)+

a2

2!

(X0 + εX1

)2+

a3

3!

(X0 + εX1

)3+ · · · .

Now we single out the sum of terms up to the first degree inε and obtain:

ea(X0+εX1)

≈1+aX0+a2

2!X2

0 +a3

3!X3

0 + · · ·

+ ε[aX1+

a2

2!

(X0X1 +X1X0

)+

a3

3!

(X2

0 X1 +X0X1X0 +X1X20

)

+a4

4!

(X3

0 X1 +X20X1X0 +X0X1X2

0 +X1X30

)+ · · ·

]. (7.3.21)


By using the identities

X0X1 = X1X0 +[X0,X1],

X20 X1 +X0X1X0 = 2X1X2

0 +3[X0,X1]X0 +[X0, [X0,X1]], . . .

we rewrite Eq. (7.3.21) in the form:

ea(X0+εX1)

≈1+aX0+a2

2!X2

0 +a3

3!X3

0 + · · ·

+ ε[aX1

(1+aX0+

a2

2!X2

0 +a3

3!X3

0 + · · ·)

+a2

2![X0,X1]

(1+aX0+

a2

2!X2

0 +a3

3!X3

0 + · · ·)

+a3

3![X0, [X0,X1]]

(1+aX0+

a2

2!X2

0 +a3

3!X3

0 + · · ·)

+ · · ·].

Whence, using the exponent (6.2.18) we have:

ea(X0+εX1) ≈(1+ ε〈〈aX0,aX1〉〉

)eaX0. (7.3.22)

In other words, the exponential map

xi = eaX(xi)

written for the operator (7.3.14) and evaluated in the first order of precision withrespect toε has the form (7.3.20). Taking into account Eq. (7.3.16), onearrives atEqs. (7.3.17). This completes the proof of the theorem.

Example 7.4. Let us apply Theorem 7.2 to the operator

X = (1+ εx)∂∂x

considered in Example 7.2. Here

X0 =∂∂x

, X1 = x∂∂x

.

ThereforeX0(x) = 1, X2

0 (x) = X30 (x) = · · · = 0,

and


[X0,X1] =∂∂x

= X0,

[X0, [X0,X1]] = [X0,X0] = 0, . . . .

Consequently,x0 = eaX0(x) = x+a,

and

〈〈aX0,aX1〉〉 =(

ax+a2

2!

) ∂∂x

,

whence

x1 = 〈〈aX0,aX1〉〉(x0) =(

ax+a2

2!

) ∂∂x

(x+a) = ax+a2

2!.

Hence,

x≈ x+a+ ε(

ax+a2

2

).

Example 7.5. Let us apply Theorem 7.2 to the operator (7.3.13) from Example 7.3.In this case we have

X0 =∂∂x

, X1 = x2 ∂∂x

+xy∂∂y

.

Therefore,x0 = eaX0(x) = x+a,

and

[X0,X1] = 2x∂∂x

+y∂∂y

,

[X0, [X0,X1]] = 2∂∂x

,

[X0, [X0, [X0,X1]]] = 0, . . . .

Consequently,

〈〈aX0,aX1〉〉 = aX1+a2

2!

(2x

∂∂x

+y∂∂y

)+2

a3

3!∂∂x

=(

ax2 +a2x+a3

3

) ∂∂x

+(

axy+a2

2y) ∂

∂y.

Whence

x1 = 〈〈aX0,aX1〉〉(x0) =(

ax2 +a2x+a3

3

) ∂∂x

(x+a),

y1 = 〈〈aX0,aX1〉〉(y0) =(

axy+a2

2y) ∂

∂y(y).


Hence,

x1 = ax2 +a2x+a3

3, y1 = axy+

a2

2y.

We thus arrive at the result of Example 7.3:

x≈ x+a+ ε(

ax2 +a2x+a3

3

),

y≈ y+ ε(

axy+a2

2y).

Example 7.6. Let us apply the approximate exponential map to the following op-erator:

X =(

1+ ε[(x1)2− (x2)2− (x3)2− (x4)2]) ∂

∂x1

+2εx1(

x2 ∂∂x2 +x3 ∂

∂x3 +x4 ∂∂x4

).

HereX = X0 + εX1 with

X0 =∂

∂x1 ,

X1 =((x1)2− (x2)2− (x3)2− (x4)2

) ∂∂x1

+ 2x1(

x2 ∂∂x2 +x3 ∂

∂x3 +x4 ∂∂x4

).

The operatorX0 generates the translation group:

x10 = x1 +a, x j

0 = x j , j = 2,3,4.

We have:

[X0,X1] = 2(

x1 ∂∂x1 +x2 ∂

∂x2 +x3 ∂∂x3 +x4 ∂

∂x4

),

[X0, [X0,X1]] = 2∂

∂x1 , [X0, [X0, [X0,X1]]] = 0, . . . .

Consequently, equation (7.3.19) takes the form:

〈〈aX0,aX1〉〉 =([(x1)2− (x2)2− (x3)2− (x4)2]a+x1a2 +

13

a3) ∂

∂x1

+(2ax1+a2)(

x2 ∂∂x2 +x3 ∂

∂x3 +x4 ∂∂x4

).

Therefore Eq. (7.3.17) yields


x11 = 〈〈aX0,aX1〉〉(x1

0) = [(x1)2− (x2)2− (x3)2− (x4)2]a+x1a2 +13

a3,

x j1 = (2ax1 +a2)x j , j = 2,3,4.

We have arrived at the following approximate transformation group:

x1 ≈ x10 + ε x1

1 = x1 +a

+ ε([(x1)2− (x2)2− (x3)2− (x4)2]a+x1a2 +

13

a3),

x j ≈ x j0 + ε x j

1 = x j + ε(2ax1+a2)x j , j = 2,3,4.


Exercise 7.1.Derive Eq. (7.2.9) in Example 7.1.

Exercise 7.2.Provide details of the proof of Theorem 7.1.

Exercise 7.3.Rewrite the approximate Lie equations (7.3.8)–(7.3.9) in the case ofone variablex.

Exercise 7.4.Rewrite the approximate Lie equations (7.3.8)–(7.3.9) forapproxi-mate transformation groups on the(x,y) plane.

Exercise 7.5.Solve the approximate Lie equations and find the approximatetrans-formation group for the generator

X =(

t +ε6

t2) ∂

∂ t−

(u+

ε3

tu) ∂

∂u· (1)

Exercise 7.6.Solve the approximate Lie equations (7.3.8)–(7.3.9) for the operator

X =(

1+ ε[(x1)2− (x2)2− (x3)2− (x4)2]) ∂

∂x1

+2εx1(

x2 ∂∂x2 +x3 ∂

∂x3 +x4 ∂∂x4

)

from Example 7.6.

Chapter 8Approximate symmetries

8.1 Definition of approximate symmetries

Definition 8.1. Let G be a one-parameter approximate transformation group con-sidered in the first order of precision:

zi ≈ f (z,a,ε) ≡ f i0(z,a)+ ε f i

1(z,a), i = 1, . . . ,N. (8.1.1)

Let

F(z,ε) ≈ F0(z)+ εF1(z)

and

F(z,ε) ≈ F( f (z,a,ε),ε).

An approximate equation

F(z,ε) ≈ 0 (8.1.2)

is said to beapproximately invariantwith respect toG (or admits G) if

F(z,ε) ≈ 0

wheneverz= (z1, . . . ,zN) satisfies Eq. (8.1.2).

If z= (x,u,u(1), . . . ,u(k)), then equation (8.1.2) becomes an approximate differ-ential equation of orderk, andG is anapproximate symmetry groupof the differen-tial equation. Recall that in our approximationF(z,ε) ≈ 0 meansF(z,ε) = o(ε).

A different approach to approximate symmetries, based on decomposing differ-ential equations with a small parameter, has been proposed in [10]. For a comparisonof two approaches I refer the reader to [11].

151

152 8 Approximate symmetries

8.2 Calculation of approximate symmetries

8.2.1 Determining equations

Theorem 8.1. Equation (8.1.2) is approximately invariant under the approximatetransformation group (8.1.1) with the generator

X = X0+ εX1 ≡ ξ i0(z)

∂∂zi + εξ i

1(z)∂

∂zi , (8.2.1)

if and only if [XF(z,ε)

]F≈0 = o(ε),

or [X0F0(z)+ ε

(X1F0(z)+X0F1(z)

)](8.1.2)

= o(ε). (8.2.2)

Proof. Equation (8.2.2) is obtained by substituting Eqs. (8.1.2) and (8.2.1) in thedetermining equation [

XF(z,ε)]

F=0 = 0

and considering the result in the first-order of precision with respect toε.The operator (8.2.1) satisfying Eq. (8.2.2) is called an infinitesimalapproximate

symmetryof, or an approximate operator admittedby Eq. (8.1.2). Accordingly,equation (8.2.2) is termed thedetermining equationfor approximate symmetries.

Remark 8.1. The determining equation (8.2.2) can be written as follows:

X0F0(z) = λ (z)F0(z), (8.2.3)

X1F0(z)+X0F1(z) = λ (z)F1(z). (8.2.4)

The factorλ (z) is determined by Eq. (8.2.3) and then substituted in Eq. (8.2.4). Thelatter equation must hold for all solutions ofF0(z) = 0.

8.2.2 Stable symmetries

Comparing Eq. (8.2.3) with the determining equation of exact symmetries, we ob-tain the following statement.

Theorem 8.2. If equation (8.1.2) admits an approximate transformation group withthe generatorX = X0 + εX1, whereX0 6= 0, then the operator

X0 = ξ i0(z)

∂∂zi (8.2.5)

is an exact symmetry of the equation

8.2 Calculation of approximate symmetries 153

F0(z) = 0. (8.2.6)

Remark 8.2. It is manifest from Eqs. (8.2.3), (8.2.4) that ifX0 is anexactsymmetryof Eq. (8.2.6) thenX = εX0 is anapproximatesymmetry of Eq. (8.1.2). Even thoughsuch approximate are “trivial”, they are significant for theapproximate Lie algebrastructure(see Section 8.3.2).

Definition 8.2. Equations (8.2.6) and (8.1.2) are termed anunperturbed equationand aperturbed equation,respectively. Under the conditions of Theorem 8.2, theoperatorX0 is called astable symmetryof the unperturbed equation (8.2.6). Thecorresponding approximate symmetry generatorX = X0 + εX1 for the perturbedequation (8.2.5) is called adeformation of the infinitesimal symmetry X0 of Eq.(8.2.6) caused by the perturbationεF1(z). In particular, if the most general symmetryLie algebra of Eq. (8.2.6) is stable, we say that the perturbed equation (8.1.2)inheritsthe symmetries of the unperturbed equation.

8.2.3 Algorithm for calculation

Remark 8.1 and Theorem 8.2 provide an infinitesimal method for calculating ap-proximate symmetries (8.2.1) for differential equations with a small parameter. Im-plementation of the method requires the following three steps.

1st step.Calculation of the exact symmetriesX0 of the unperturbed equation(8.2.6), e.g. by solving the determining equation

X0F0(z)∣∣∣F0(z)=0

= 0. (8.2.7)

2nd step.Determination of theauxiliary function Hby virtue of Eqs. (8.2.3),(8.2.4) and (8.1.2), i.e. by the equation

H =1ε

[X0

(F0(z)+ εF1(z)

)∣∣∣F0(z)+εF1(z)=0

](8.2.8)

with knownX0 andF1(z).3rd step. Calculation of the operatorsX1 by solving thedetermining equation

for deformations:

X1F0(z)∣∣∣F0(z)=0

+H = 0. (8.2.9)

Note that Eq. (8.2.9), unlike the determining equation (8.2.7) for exact symme-tries, isinhomogeneous.


8.3 Examples

8.3.1 First example

Example 8.1. Let us find the approximate symmetries of the followingperturbednonlinear wave equation:

utt −(u2ux

)x + εut = 0. (8.3.1)

We write the approximate operators admitted by Eq. (8.3.1) in the form

X = X0 + εX1 ≡ (τ0 + ετ1)∂∂ t

+(ξ0 + εξ1)∂∂x

+(η0 + εη1)∂

∂u, (8.3.2)

whereτν , ξν , andην (ν = 0,1) are unknown functions oft,x andu.1st step.Solving the determining equation (8.2.7) for the exact symmetriesX0

of the unperturbed equationutt −

(u2ux

)x = 0 (8.3.3)

one obtains (it follows from the group classification given in [1])

τ0 = C1 +C3t, ξ0 = C2 +(C3 +C4)x, η0 = C4u, (8.3.4)

whereC1, . . . ,C4 are arbitrary constants. Hence,

X0 = (C1 +C3t)∂∂ t

+(C2+C3x+C4x)∂∂x

+C4u∂

∂u· (8.3.5)

In other words, equation (8.3.3) admits the four-dimensional Lie algebraL4 with thebasis

X01 =

∂∂ t

, X02 =

∂∂x

, X03 = t

∂∂ t

+x∂∂x

, X04 = x

∂∂x

+u∂∂u

· (8.3.6)

2nd step.Substituting the expression (8.3.5) of the generatorX0 in Eq. (8.2.8)we obtain the auxiliary function

H = C3ut .

3rd step.Now the determining equation (8.2.9) for deformations is written

X1(utt −u2uxx−2uu2

x

)∣∣∣(8.3.3)

+C3ut = 0, (8.3.7)

whereX1 denotes the prolongation of the operator

X1 = τ1∂∂ t

+ ξ1∂∂x

+ η1∂∂u

8.3 Examples 155

to the derivatives ofu involved in Eq. (8.3.1). Upon settingutt =(u2ux

)x the left-

hand side of Eq. (8.3.7) becomes a polynomial in the variables utx, uxx, ut , ux.Equating to zero its coefficients we obtain:

τ1 = τ1(t), ξ1 = ξ1(x), 3τ ′′1 = C3,

ξ ′′1 = 0, η1 = (ξ ′

1(x)− τ ′1(t))u.

Hence,

τ1 = A1 +A3t +16

C3 t2,

ξ1 = A2 +(A3 +A4)x, (8.3.8)

η1 =(

A4−13

C3t)

u.

Substituting Eqs. (8.3.4) and (8.3.8) in Eq. (8.3.2), we obtain the following approx-imate symmetries for Eq. (8.3.1):

X1 =∂∂ t

, X2 =∂∂x

,

X3 = t∂∂ t

+x∂∂x

+ε6

(t2 ∂

∂ t−2tu

∂∂u

), (8.3.9)

X4 = x∂∂x

+u∂∂u

, X5 = εX1, X6 = εX2,

X7 = εX4, X8 = εX3.

Remark 8.3. Equations (8.3.8) upon lettingA1 = C3 = A2 = A4 = 0 yield

X8 = ε(

t∂∂ t

+x∂∂x

).

However, in the first-order of precision, the operatorX8 can be written in the formgiven in Eqs. (8.3.9).

Remark 8.4. Equations (8.3.9) show that all symmetries (8.3.6) of Eq. (8.3.3) arestable. It means that the Lie algebraL4 spanned by the operators (8.3.6) is stable.Hence, theperturbed equation(8.3.1) inherits the symmetries of the unperturbedequation(8.3.3).

8.3.2 Approximate commutator and Lie algebras

The following table of commutators, evaluated in the first-order of precision, showsthat the operators (8.3.9) span an eight-dimensionalapproximate Lie algebra L8, and


hence generate an eight-parameter approximate transformations group. For the sakeof brevity, we will write only the off-diagonal elements of the table of commutators.

Table 8.3.1 shows that although the approximate symmetriesX5,X6,X7 andX8 are“trivial” according to Remark 8.2, they are necessary for the Lie algebra structure,and hence for constructing multi-parameter approximate transformation groups.

Table 8.3.1. Approximate commutators

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 0 X1 + 13(X8−X7) 0 0 0 0 X5

X2 0 X2 X2 0 0 X6 X6

X3 0 0 −X5 −X6 0 0

X4 0 0 −X6 0 0

X5 0 0 0 0

X6 0 0 0

X7 0 0

X8 0


Example 8.2. Consider the equation

utt −(u−4ux

)x + εut = 0. (8.3.10)

1st step.The unperturbed equation

utt −(u−4ux

)x = 0 (8.3.11)

admits the five-dimensional Lie algebraL5 consisting of the operators

X0 =(C1 +C3t +C5t

2) ∂

∂ t+

(C2 +C3x+C4x

) ∂∂x

+(− 1

2C4 +C5t

)u

∂∂u

·

A basis of this algebra is

X01 =

∂∂ t

, X02 =

∂∂x

, X03 = t

∂∂ t

+x∂∂x

,

X04 = x

∂∂x

− 12

u∂

∂u, X0

5 = t2 ∂∂ t

+ tu∂∂u

·(8.3.12)

8.3 Examples 157

2nd step.Equation (8.2.8) provides the following auxiliary function:

H = C3ut +2C5tut +C5u.

3rd step. The determining equation (8.2.9) for deformations yields that C3 =C5 = 0. Proceeding further as in the previous example, we obtain thefollowingapproximate symmetries of the perturbed equation (8.3.10):

X1 =∂∂ t

, X2 =∂∂x

,

X3 = ε(

t∂∂ t

+x∂∂x

), X4 = x

∂∂x

− 12

u∂

∂u,

X5 = εX1, X6 = εX2,

X7 = εX4, X8 = ε(

t2 ∂∂ t

+ tu∂∂u

). (8.3.13)

Equations (8.3.13) show that not all symmetries (8.3.12) ofEq. (8.3.11) are stable.Namely, the operators

X03 = t

∂∂ t

+x∂∂x

and X05 = t2 ∂

∂ t+ tu

∂∂u

from Eqs. (8.3.12) are unstable. Hence, the perturbed equation (8.3.10) does notinherit the symmetries of the unperturbed equation (8.3.11).

8.3.4 Third example

Example 8.3. Consider the equation

utt −(u−4/3ux

)x + εut = 0. (8.3.14)

The unperturbed equation

utt −(

u−4/3ux

)x= 0 (8.3.15)

admits the five-dimensional Lie algebraL5 spanned by the operators

X01 =

∂∂ t

, X02 =

∂∂x

, X03 = t

∂∂ t

+x∂∂x

,

X04 = x

∂∂x

− 32

u∂∂u

, X05 = x2 ∂

∂x−3xu

∂∂u

·(8.3.16)


Proceeding as before, one arrives at the following approximate symmetries of theperturbed equation (8.3.14):

X1 =∂∂ t

, X2 =∂∂x

, X3 =(

t − ε4

t2) ∂

∂ t+x

∂∂x

− 34

εtu∂∂u

,

X4 = x∂∂x

− 32

u∂∂u

, X5 = x2 ∂∂x

−3xu∂∂u

, X6 = εX1, (8.3.17)

X7 = εX2, X8 = εX03 , X9 = εX4, X10 = εX5.

They span a ten-dimensional approximate Lie algebraL10. We see that in this ex-ample the algebraL5 is stable, i.e. the perturbed equation (8.3.14) inherits the sym-metries of the unperturbed equation.

Remark 8.5. Equations (8.3.1) and (8.3.10) with an arbitrary parameterε 6= 0 haveonly three exact symmetries(see [12], page 227, Section 9.2.1.1.2). Namely, theexact symmetries of Eq. (8.3.1) are the operators

X1 =∂∂ t

, X2 =∂∂x

, X4 = x∂∂x

+u∂∂u

from (8.3.9), and the exact symmetries of Eq. (8.3.10) are the operators

X1 =∂∂ t

, X2 =∂∂x

, X4 = x∂∂x

− 12

u∂∂u

from (8.3.13).

More examples can be found in [12], Chapters 2 and 9.


Exercise 8.1.Verify that the operatorX3 from (8.3.9),

X3 = t∂∂ t

+x∂∂x

+ε6

(t2 ∂

∂ t−2tu

∂∂u

),

is an approximate symmetry for the perturbed equation (8.3.1).

Exercise 8.2.Make the table of approximate commutators for the operators(8.3.13).

Exercise 8.3.Verify that the operatorX3 from (8.3.17),

X3 = t∂∂ t

+x∂∂x

− ε4

(t2 ∂

∂ t+3tu

∂∂u

),

is an approximate symmetry for the perturbed equation (8.3.14).


Exercise 8.4.Make the table of approximate commutators for the operators(8.3.17).

Exercise 8.5.Construct the one-parameter approximate transformation group withthe generatorX3 from Exercise 8.3.

Chapter 9Applications

9.1 Integration of equations with a small parameter usingapproximate symmetries

Let us discuss, by way of examples, group methods of integration of differentialequations with a small parameter with known approximate symmetries.

9.1.1 Equation having no exact point symmetries

Example 9.1. The second-order equation

y′′−x− εy2 = 0 (9.1.1)

has no exact point symmetries ifε 6= 0 is regarded as a constant coefficient, andhence cannot be integrated by the Lie method. Moreover, thisequation cannot be in-tegrated by quadrature. However, it possesses approximatesymmetries ifε is treatedas a small parameter, e.g.

X1 =∂∂y

+ε3

[2x3 ∂

∂x+

(3yx2 +

1120

x5) ∂

∂y

],

X2 = x∂∂y

+ε6

[x4 ∂

∂x+

(2yx3 +

730

x6) ∂

∂y

].

(9.1.2)

The commutator of the operators (9.1.2) is[X1,X2] ≈ 0. Hence they span a two-dimensional approximate Abelian Lie algebra and can be usedfor consecutive inte-gration of Eq. (9.1.1) as follows ([14], Section 12.4).

EquationsX1(t) ≈ 1, X1(u) ≈ 0 yield the canonical variables

t = y− ε(1

2x2y2 +

1160

yx5), u = x− 2

3εyx3 (9.1.3)

161

162 9 Applications

for X1 from (9.1.2). Thus we haveX1 ≈ ∂/∂ t, while Eq. (9.1.1) reads

u′′ +uu′3+ ε[3u2u′ +

16(u2u′)2− 11

60(u2u′)3

]= 0.

Integration by the standard substitutionu′ = p(u) yields:

p′ +up2+ ε(

3u2 +16

u4p− 1160

u6p2)

= 0. (9.1.4)

The second operator of (9.1.2) is written

X2 = p2 ∂∂ p

+ ε[

12

u4 ∂∂u

+(

2u3p− 1315

u5p2)]

∂∂ p

and becomes the infinitesimal translation

X2 ≈∂∂v

upon introducing the new independent variablez and dependent variablev definedby the equations

z= u+ εu4

2p, v = −1

p+ ε

(u3

p2 −13u5

15p

). (9.1.5)

In the variables of (9.1.5), equation (9.1.4) is written

v′ +z+1160

εz6 = 0

and yields

v = −1160

z2− 11420

εz7 +C.

Substituting this expression forv in Eqs. (9.1.5) and eliminatingz, one obtains thefunctionp(u). The quadrature

∫du

p(u)= t +C

followed by the substitution in Eqs. (9.1.3), completes theapproximate integrationof Eq. (9.1.1).

9.1.2 Utilization of stable symmetries

Let us consider the van der Pol equation

9.1 Integration of equations with a small parameter using approximate symmetries 163

y′′ +y= ε(y′−y′3), ε = const. (9.1.6)

If ε is treated as an arbitrary constant, equation (9.1.6) has only one exactpointsymmetry, namely the translation of the independent variable x with the generator

X =∂∂x

·

On the other hand, it can be shown that Eq. (9.1.6) with a smallparameterε inheritsall 8 point symmetries of the unperturbed equation

y′′ +y= 0. (9.1.7)

The stability (see Definition 8.2.1) of all symmetries is a necessary condition forexistence of an approximate transformation connecting perturbed and unperturbedequations. Let us use this circumstance for the integrationof the van der Pol equationin the first order of precision.

The unperturbed equation (9.1.7) is homogeneous, and henceadmits the dilationgenerator

X0 = y∂∂y

· (9.1.8)

The reckoning shows that although the perturbed equation (9.1.6) is not homoge-neous, it admits, as an approximate symmetry, the followingdeformation of theoperator (9.1.8):

X =(

y− ε4

[y2y′ +3xy(y2+y′2)

]) ∂∂y

· (9.1.9)

We will write Eq. (9.1.7) and its symmetry (9.1.8) in the form

z′′ +z= 0 (9.1.10)

and

X0 = z∂∂z

, (9.1.11)

respectively, and look for an approximate transformation

y = z+ ε f

connecting the van der Pol equation (9.1.6) with Eq. (9.1.10).We can simplify the calculations by using the stability of the dilation generator

(9.1.8). Namely, we will begin with constructing the approximate transformationsmapping the operator (9.1.11) into the operator (9.1.9). Then we will subject theresulting transformations to the condition that they connect Eqs. (9.1.10) and (9.1.6).

Noting that the approximate symmetry (9.1.9) depends on thefirst-order deriva-tive y′, we will search the approximate transformation of (9.1.11) into (9.1.9) in theform

y = z+ ε f (x,z,z′). (9.1.12)

164 9 Applications

The operator (9.1.11) is written in the variabley given by (9.1.12) as follows:

X 0 = X0(y)∂∂y

,

where the prolongation ofX0 to the derivativey′ is understood. So we have:

X0(y) =

(z

∂∂z

+z′∂

∂z′

)(z+ ε f (x,z,z′)

)= z+ ε

(z

∂ f∂z

+z′∂ f∂z′

).

SinceX 0 should be identical with the operator (9.1.9), we have:

z+ ε(

z∂ f∂z

+z′∂ f∂z′

)= y− ε

4

[y2y′ +3xy(y2+y′2)

].

Substituting in the right-hand side of this equation the expression (9.1.12) fory andnoting thaty = zup to terms of orderε according to Eq. (9.1.12), we get

z+ ε(

z∂ f∂z

+z′∂ f∂z′

)= z+ ε

(f − 1

4

[z2z′ +3xz(z2 +z′2)

]),

whence

z∂ f∂z

+z′∂ f∂z′

= f − 14

[z2z′ +3xz(z2 +z′2)

]. (9.1.13)

Let us solve the non-homogeneous linear first-order partialdifferential equation(9.1.13). The characteristic system of Eq. (9.1.13) is

dzz

=dz′

z′=

d f

f − 14

[z2z′ +3xz(z2 +z′2)

] ·

The first equation of this system yields the first integralz′/z= λ = const. Writingthe second equation of the characteristic system in the form

d fdz

=1z

(f − 1

4

[z2z′ +3xz(z2 +z′2)

]),

and substitutingz′ = λz, we obtain the following linear first-order ordinary differ-ential equation forf :

d fdz

=1z

f − 14

[λz2 +3x(1+ λ 2)z2].

The integration yields

f = −18

[K(x)z+ λz3 +3xz(1+ λ 2)z2],

9.1 Integration of equations with a small parameter using approximate symmetries 165

where the “constant of integration”K(x) is an arbitrary function ofx. Substitutingλz= z′, we obtain

f = −18

[K(x)z+z2z′ +3xz(z2 +z′2)

]

and finally arrive at the following transformation (9.1.12)mapping the operator(9.1.11) into (9.1.9):

y = z− ε8

[K(x)z+z2z′ +3xz(z2 +z′2)

]. (9.1.14)

Sincey = z up to terms of orderε, the inverse to (9.1.11) in the first order of ap-proximation is given by

z= y+ε8

[K(x)y+y2y′ +3xy(y2+y′2)

]. (9.1.15)

Now we will determineK(x) using the requirement that (9.1.14) maps Eq.(9.1.10) into Eq. (9.1.6). Let us substitute (9.1.15) in Eq.(9.1.10). Sincey = zup toterms of orderε, the equationz′′ = −zyields that

y′′ = −y

up to terms of orderε., Therefore, differentiating (9.1.15), we have:

z′ = y′ +ε8

[K(x)y′ +K′(x)y+2y3+5yy′2+3x(y2y′ +y′3)

]

and

z′′ = y′′ +ε8

[2K′(x)y′ +K′′(x)y−K(x)y−y2y′ +8y′3−3xy(y2+y′2)

].

Consequently,

z′′ +z= y′′ +y+ε8

[8y′3 +2K′(x)y′ +K′′(x)y

].

Hence, we obtain Eq. (9.1.6) if we letK(x) = −4x. Thus, we have arrived at thefollowing transformation mapping Eq. (9.1.10) into Eq. (9.1.6):

y = z+ε8

[4xz−z2z′−3xz

(z2 +z′2

)]. (9.1.16)

Substituting in (9.1.16) the general solutionz= Acost + Bsint of Eq. (9.1.10)one obtains the followinggeneral approximate solutionof the van der Pol equation(9.1.6):

166 9 Applications

y =Acosx+Bsinx+ε8

[(4−3(A2+B2)

)x(Acosx+Bsinx)

+ (Asinx−Bcosx)(Acosx+Bsinx)2]. (9.1.17)

Letting hereA = 1,B = 0 and thenA = 0,B = 1, i.e. taking a fundamental set ofsolutionsz1 = cosx andz2 = sinx of the linear equationz′′ +z= 0, one obtains thefollowing particular approximate solutionsof Eq. (9.1.6):

y1 = cosx+ε8

[xcosx+sinx cos2x

],

y2 = sinx+ε8

[xsinx−cosx sin2x

].

9.2 Approximately invariant solutions

9.2.1 Nonlinear wave equation

Example 9.2. Consider Eq. (8.3.1),

utt −(u2ux

)x + εut = 0, (9.2.1)

with known approximate symmetries (8.3.9) and find an approximately invariantsolution based on the approximate symmetryX = X3−X4 with X3, X4 from (8.3.9).We have:

X = t∂∂ t

−u∂∂u

+ε6

(t2 ∂

∂ t−2tu

∂∂u

). (9.2.2)

The approximate invariants for the operator (9.2.2) are written in the form

J(t,x,u,ε) = J0(t,x,u)+ εJ1(t,x,u)+o(ε).

They are determined by the equation

X(J) = o(ε).

Using for the operator (9.2.2) the notation

X = X0+ εX1,

where

X0 = t∂∂ t

−u∂∂u

, X1 =16

(t2 ∂

∂ t−2tu

∂∂u

),

we write the determining equationX(J) = o(ε) for the approximate invariants in theform

9.2 Approximately invariant solutions 167

X0(J0)+ ε[X0(J1)+X1(J0)

]= 0,

whenceX0(J0) = 0, X0(J1)+X1(J0) = 0,

or

t∂J0

∂ t−u

∂J0

∂u= 0,

t∂J1

∂ t−u

∂J1

∂u= −1

6

(t2 ∂J0

∂ t−2tu

∂J0

∂u

).

(9.2.3)

Solving Eqs. (9.2.3), we will find two functionally independent invariants

J1 = J01(t,x,u)+ εJ1

1(t,x,u),

J2 = J02(t,x,u)+ εJ1

2(t,x,u)(9.2.4)

for the operator (9.2.2).

Remark 9.1. Recall that functions (9.2.4) are said to be functionally dependent ifJ2 = Ψ(J1), in other words if the equation

J02(t,x,u)+ εJ1

2(t,x,u) = Ψ(J0

1(t,x,u)+ εJ11(t,x,u)

)+o(ε)

with a certain functionΨ holds identically int,x,u. If such a functionΨ does notexist,the functions (9.2.4) are said to befunctionally independent.It is manifest thatif J0

1(t,x,u) andJ02(t,x,u) are functionally independent, then so are the functions

(9.2.4) as well.

Let us proceed to the integration of the system (9.2.3). The first equation from(9.2.3) has precisely two functionally independent solutions, namely

J01 = x, J0

2 = tu.

SubstitutingJ01 = x in the second equation of (9.2.3) and taking its simplest solution

J11 = 0, we obtain one invariant of (9.2.4),

J1 = x. (9.2.5)

Note that it does not involve the dependent variableu. Now we substitute the solu-tion J0

2 = tu of the first equation of (9.2.3) into the second equation of (9.2.3) andobtain the following non-homogeneous linear equation:

t∂J1

2

∂ t−u

∂J12

∂u=

16

t2u.

The first equation of the characteristic system

168 9 Applications

dtt

= −duu

= 6dJ1

2

t2u

yields the first integraltu = λ = const. Therefore the second equation

dtt

= 6dJ1

2

t2u

of the characteristic system, upon replacingtu by λ , is written λ dt = 6dJ12 and

yields

J12 =

16

tλ +C =16

t2u+C.

Letting hereC = 0, we obtain the second invariant (9.2.4),

J2 = tu+ε6

t2u. (9.2.6)

By Remark 9.1 the invariants (9.2.5), (9.2.6) are functionally independent.LettingJ2 = ϕ(J1), i.e. (

1+ε6

t)

tu = ϕ(x)

and solving fortu in the first order of precision,

tu =(

1+ε t6

)−1ϕ(x) =

(1− ε t

6

)ϕ(x)+o(ε),

we obtain the following form for the approximately invariant solutions:

u =

(1t− ε

6

)ϕ(x). (9.2.7)

Now we substitute (9.2.7) in Eq. (9.2.1). Differentiation of (9.2.7) yields:

ut = − 1t2 ϕ , utt =

2t3 ϕ , ux =

(1t− ε

6

)ϕ ′.

Furthermore,

u2ux =

(1t− ε

6

)3

ϕ2ϕ ′ =

(1t3 −

ε2t2

)ϕ2ϕ ′ +o(ε),

and hence we have in our approximation:

(u2ux)x =

(1t3 −

ε2t2

)(ϕ2ϕ ′)′.

Therefore


utt −(u2ux

)x + εut =

1t3

(1− ε t

2

)[2ϕ − (ϕ2ϕ ′)′

].

Thus, equation (9.2.1) yields

(ϕ2ϕ ′)′ = 2ϕ . (9.2.8)

Let us integrate Eq. (9.2.8). We have

ϕ2ϕ ′′ +2ϕϕ ′2 = 2ϕ ,

or

ϕϕ ′′ +2ϕ ′2 = 2.

The latter equation is reduced to a first-order equation by the standard substitutionϕ ′ = p(ϕ). Sinceϕ ′′ = pp′, we have

ϕ pp′ +2p2 = 2,

or

ϕdp2

dϕ+4p2 = 4.

Now we denotep2 = v, integrate the resulting linear equation

ϕdvdϕ

+4v= 4

and obtainv = 1+Cϕ−4. The equationp±√v yields

p = ±√

1+Cϕ−4 .

Thus,dϕdx

= ±√

1+Cϕ−4

and the integration provides the general solution to Eq. (9.2.8) in the following im-plicit form involving one quadrature:

∫ ϕ2dϕ√ϕ4 +C

= C±x. (9.2.9)

Substitutingϕ(x) given by (9.2.9) in (9.2.7) we obtain the invariant solutionto Eq.(9.2.1). For example, settingC = 0 we have from (9.2.9)ϕ(x) = ±x, and the invari-ant solution (9.2.7) is

u = ±(x

t− ε

x6

)·

170 9 Applications

9.2.2 Approximate travelling waves of KdV equation

Example 9.3. Let us find theapproximate travelling wavesfor the following per-turbed Korteweg-de Vries equation:

ut = uux +uxxx+ εu. (9.2.10)

We will begin with the well-known solitary wave solutionu = U0,

U0 = 3κ sech2z, (9.2.11)

of the Korteweg-de Vries equation

ut = uux +uxxx. (9.2.12)

Here

sechz=1

coshz=

2ez+e−z ,

and

z=

√κ

2

(x+ κt

)

is an invariant of the group with the generator

X0 =∂∂ t

−κ∂∂x

, κ = const. (9.2.13)

The functionU0 given by Eq. (9.2.11) is invariant under the operator (9.2.13). Letus check thatu = U0 is an invariant solution, i.e. that it solves Eq. (9.2.12). Usingthe equations

(sechz)′ = −sech2z ·sinhz, (sech2z)′ = −2sech3z ·sinhz,

where

sinhz=ez−e−z

2,

we have:(U0)t = −3κ5/2sech3z ·sinhz,

(U0)x = −3κ3/2sech3z ·sinhz,

(U0)xx =92

κ2sech4z ·sinh2 z− 32

κ2sech2 z,

(U0)xxx = 3κ5/2(3sech2z−1

)sech3z·sinhz.

(9.2.14)

In the last equation of (9.2.14) the following identity has been used:

sinh3 z=(

cosh2 z−1)sinhz.


It is manifest from Eqs. (9.2.14) that (9.2.11) solves Eq. (9.2.12):

(U0)t = U0(U0)x +(U0)xxx.

Using the generators of the Galilean and scaling transformations,

X1 = t∂∂x

− ∂∂u

, X2 = 3t∂∂ t

+x∂∂x

−2u∂∂u

admitted by the Korteweg-de Vries equation (9.2.12), and invoking Remark 8.2,we consider the approximate symmetryXε = X0 + ε

(κX1 −X2

)of the perturbed

equation (9.2.10). Thus we will take the approximate symmetry

Xε =∂∂ t

−κ∂∂x

− ε[3t

∂∂ t

+(x−κ t

) ∂∂x

+(κ −2u

) ∂∂u

](9.2.15)

and use it for finding an approximately invariant solution bylooking for the invariantperturbation of the solitary wave (9.2.11):

u = U0 + εv(t,x). (9.2.16)

Let us write the operator (9.2.15) in the form

Xε = X0 + εX1,

whereX0 is the operator (9.2.13) andX1 is given by

X1 = −[3t

∂∂ t

+(x−κ t

) ∂∂x

+(κ −2u

) ∂∂u

]. (9.2.17)

Then the invariant equation test for Eq. (9.2.16),

Xε(u−U0− εv)∣∣∣(9.2.16)

= o(ε),

is written (X0(u−U0)+ ε

[X1(u−U0)−X0(v)

])

u=U0= 0. (9.2.18)

Note thatX0(u−U0) vanishes identically since the operatorX0 does not contain thedifferentiation inu and since the functionU0 is an invariant because it depends onlyon the invariantz. Therefore Eq. (9.2.18) becomes

[X1(u−U0)−X0(v)

]u=U0

= 0,

whence, invoking definitions (9.2.13) and (9.2.17) ofX0 andX1, respectively, weobtain the following differential equation forv(t,x) :

vt −κvx = 3t(U0)t +(x−κt)(U0)x +2U0−κ . (9.2.19)

172 9 Applications

SinceU0 = U0(z), it is easy to integrate Eq. (9.2.19) in the “natural” variables

z=

√κ

2

(x+ κt

), y = t.

Then, denoting byU ′0 the derivative ofU0 with respect toz, we have:

(U0)t =12

κ3/2U ′0, (U0)x =

12

κ1/2U ′0,

and Eq. (9.2.19) becomes

vy =(

z+12

κ3/2y)U ′

0 +2U0−κ .

The integration yields:

v =(

zy+14

κ3/2y2)U ′

0 +(2U0−κ)y+g(z).

Returning to the variablest,x, we have:

v =(

xt +32

t2)(U0)x +(2U0−κ)t +h(x+ κt).

Inserting thisv in (9.2.16) and substituting in the perturbed KdV equation (9.2.10)we obtainh = x+ κt.

Thus, the approximate symmetry (9.2.15) provides the following approximatelyinvariant solution (approximate travelling wave):

u = U0 + ε[x+2tU0+

(xt+

32

t2)(U0)x

], (9.2.20)

or invoking Eqs. (9.2.11) and (9.2.14):

u = 3κ sech2z+ ε[x+6κt sech2z−3κ

32

(tx+

32

κ t2)

sech3z ·sinhz].

9.3 Approximate conservation laws

Noether’s theorem can be generalized to equations with a small parameter admittingapproximate transformation groups. For the sake of brevity, we restrict the discus-sion to first-order LagrangiansL(x,u,u(1),ε) and to approximate point transforma-tion groups. Then one can prove the following statement on approximate conserva-tion laws [4] (see also [12], Section 2.6).

Theorem 9.1. Let approximate Euler-Lagrange equations

9.3 Approximate conservation laws 173

δLδuα ≡ ∂L

∂uα −Di

(∂L

∂uαi

)= o(ε) (9.3.1)

possess an approximate symmetry

X =[ξ i

0(x,u)+ εξ i1(x,u)

] ∂∂xi +

[ηα

0 (x,u)+ εηα1 (x,u)

] ∂∂uα

such that the following holds:

XL+LDi(ξ i) = Di(Bi)+o(ε). (9.3.2)

Then the functionsCi(x,u,u(1),ε) defined by

Ci = Lξ i +(ηα − ξ juαj )

∂L∂uα

i−Bi +o(ε) (9.3.3)

satisfy theapproximate conservation lawfor Eq. (9.3.1), i.e.

Di(Ci)∣∣∣(9.3.1)

= o(ε). (9.3.4)

The following notation is used in Eqs. (9.3.2)–(9.3.3):

ξ i = ξ i0(x,u)+ εξ i

1(x,u),

ηα = ηα0 (x,u)+ εηα

1 (x,u).

Theorem 9.1 on approximate conservation laws can be extended to Euler-Lagrangeequations with higher-order Lagrangians.

Example 9.4. Equation (9.1.1),

y′′−x− εy2 = 0,

has the Lagrangian

L =12

(y′2−x2y′

)+

ε3

y3.

Consider an approximate symmetry of Eq. (9.1.1), e.g. the operatorX1 from (9.1.2).Let us verify that the condition (9.3.2) is satisfied. Takingthe first prolongation ofthe operatorX1 :

X1 =23

εx3 ∂∂x

+[1+ ε

(yx2 +

1160

x5)] ∂

∂y+ ε

[2xy−x2y′ +

1112

x4] ∂

∂y′

we have, ignoring the higher-order terms inε :

174 9 Applications

X1L+LDx(ξ ) = ε(

2xyy′ +y2−x3y− 14

x4y′− 1124

x6)

= εDx

(xy2− 1

4x4y− 11

24·7x7).

Hence, the condition (9.3.2) holds with

B = ε(

xy2− 14

x4y− 1124·7x7

).

The reckoning shows that

Lξ +(η − ξ y′)∂L∂y′

= y′− x2

2+ ε

(x2yy′− 1

3x3y′2 +

1160

x5y′− 12

x4y− 1160·2x7

).

Therefore, the formula (9.3.3) and the conservation equation (9.3.4) provide thefollowing approximate first integralfor Eq. (9.1.1):

y′− x2

2+ ε

(x2yy′− 1

3x3y′2−xy2− 1

4x4y+

1160

x5y′− 1135·12

x7)

= C. (9.3.5)

Thus, we have reduced the second-order equationy′′−x− εy2 = 0 to the first-orderequation (9.3.5) with an arbitrary constantC.


Exercise 9.1.The unperturbed equation (9.1.1),

y′′−x= 0,

has, as any linear second-order ordinary differential equation, eight symmetries.Find them.

Exercise 9.2.Verify that the operators (9.1.2) are approximate symmetries for Eq.(9.1.1).

Exercise 9.3.Find the one-parameter approximate transformation groupsfor eachoperator of (9.1.2):

X1 =∂∂y

+ε3

[2x3 ∂

∂x+

(3yx2 +

1120

x5) ∂

∂y

],

X2 = x∂∂y

+ε6

[x4 ∂

∂x+

(2yx3 +

730

x6) ∂

∂y

].

Assignment to Part II

Problems

1. Find the approximate transformation group for the generator

X = t∂∂ t

+x∂∂x

+ε6

(t2 ∂

∂ t−2tu

∂∂u

)· (1)

2.Verify that the operator (1) is an approximate symmetry for the following equationwith the small parameterε :

utt −(u2ux

)x + εut = 0. (2)

3. Make the table of approximate commutators for the operators

X1 =∂∂ t

, X2 =∂∂x

, X3 = ε(

t∂∂ t

+x∂∂x

), X4 = x

∂∂x

− 12

u∂∂u

,

X5 = εX1, X6 = εX2, X7 = εX4, X8 = ε(

t2 ∂∂ t

+ tu∂∂u

). (3)

4. Show that the approximate transformation

y = z+ε8

[4xz−z2z′−3xz

(z2 +z′2

)]+o(ε) (4)

with a small parameterε maps the equation

z′′ +z= 0 (5)

into the van der Pol equation

y′′ +y= ε(y′−y′3). (6)

5. The approximate transformation (4) maps the symmetries

175

176 Assignment to Part II

X01 =

∂∂x

, X02 = z

∂∂z

(7)

of Eq. (5) into approximate symmetries

X1 = X01 + εX1

1 , X2 = X02 + εX1

2

of the perturbed equation (6). Find these approximate symmetries.

Solutions

1. We have to solve the approximate Lie equations (7.3.8)–(7.3.9) for the operator(1). Heren = 3 and

ξ0(t,x,u) = (t,x,0), ξ1(t,x,u) =

(16

t2, 0, −13

tu

).

Accordingly, equations (7.3.8) have the form

dt0da

= t0,dx0

da= x0,

du0

da= 0;

a = 0 : t0 = t, x0 = x, u0 = u

and yieldt0 = t ea, x0 = xea, u0 = u. (8)

Therefore Eqs. (7.3.9) are written:

dt1da

= t1 +16

t2e2a,dx1

da= x1,

du1

da= −1

3t uea;

a = 0 : t1 = x1 = u1 = 0.

(9)

The first differential equation in Eqs. (9) is a non-homogeneous linear first-orderequation fort1. Its integration yields

t1 =16

t2e2a +Kea.

Using the initial conditiont1 = 0 ata = 0, we find the constantK :

K = −16

t2.

Hence,

t1 =16

t2(ea−1)ea. (10)

Assignment to Part II 177

The solution to the second differential equation in Eqs. (9)is given byx1 = Cea.The initial condition, ¯x1 = 0 ata = 0, yieldsC = 0. Hence,

x1 = 0. (11)

The solution to the third differential equation in Eqs. (9) is given by

u1 = −13

tuea+C.

The initial condition, ¯u1 = 0 ata = 0, yieldsC = tu/3. Hence,

u1 =13

tu(1−ea). (12)

Collecting Eqs. (8), (10), (11) and (12), we conclude that the approximate trans-formation group with the generator (1) has the form

t =[t +

ε6

t2(ea−1)]

ea, x = xea, u = u+ε3

tu(1−ea).

2. First we prolong the operator (1) to the derivatives involved in Eq. (2), i.e. wewrite it in the form

X =(

t +ε6

t2) ∂

∂ t+x

∂∂x

− ε3

tu∂∂u

+ ζ1∂

∂ut

+ζ2∂

∂ux+ ζ11

∂∂utt

+ ζ22∂

∂uxx,

where

ζ1 = Dt

(−ε

3tu

)−utDt

(t +

ε6

t2)

= −ut −ε3

(u+2tut),

ζ2 = Dx

(−ε

3tu

)−uxDx(x) = −

(1+

ε3

t)

ux,

ζ11 = Dt(ζ1

)−uttDt

(t +

ε6

t2)

= −2utt − ε (ut + tutt),

ζ22 = Dx(ζ2

)−uxxDx(x) = −

(2+

ε3

t)

uxx.

Thus, the prolonged operator (1) has the form

X =(

t +ε6

t2) ∂

∂ t+x

∂∂x

− ε3

tu∂∂u

−[ut +

ε3

(u+2tut)] ∂

∂ut

−(

1+ε3

t)

ux∂

∂ux− [2utt + ε (ut + tutt)]

∂∂utt

−(

2+ε3

t)

uxx∂

∂uxx·

178 Assignment to Part II

The calculation shows that the approximate invariance condition (8.2.2) for Eq. (2)is satisfied:

X(utt −u2uxx−2uu2

x + εut)

= −2(utt −u2uxx−2uu2

x + εut)

−εt(utt −u2uxx−2uu2

x

)= ε2tut = o(ε).

Hence the operator (1) is an approximate symmetry for Eq. (2).

3. The calculation leads to the following table of approximatecommutators for theoperators (3):

X1 X2 X3 X4 X5 X6 X7 X8

X1 0 0 X5 0 0 0 0 2(X3−X7)

X2 0 X6 X2 0 0 X6 0

X3 0 0 0 0 0 0

X4 0 0 −X6 0 0

X5 0 0 0 0

X6 0 0 0

X7 0 0

X8 0

4. The inverse to the transformation (4) is given, in the first order of precision, by

z= y− ε8

[4xy−y2y′−3xy

(y2 +y′2

)]. (13)

Equation (6) yields:y′′ = −y+O(ε).

Therefore, we have:

z′ = y′− ε8

[4y−2y3−5yy′ 2 +4xy′−3xy2y′−3xy′ 3

]+o(ε)

andz′′ = y′′− ε

8

[8y′ +y2y′−8y′ 3−4xy+3xy3+3xyy′ 2

]+o(ε). (14)

Substituting Eqs. (13) and (14) in the left-hand side of Eq. (5) we obtain

z′′ +z= y′′ +y− ε(y′−y′3)+o(ε).

Hence Eq. (5) leads to Eq. (6).

Assignment to Part II 179

5. After the approximate transformation (4) the operatorX01 from Eqs. (7) becomes

X1 =∂∂x

+ ε[

12

y− 38

y(y2 +y′2

)] ∂∂y

·

Turning to the operatorX02 from Eqs. (7), we write it in the prolonged form

X02 = z

∂∂z

+z′∂

∂z′

and act on Eq. (4) to obtain

X02 (y) = z+

ε8

[4xz−z2z′−3xz

(z2 +z′2

)]

−ε8

[2z2z′ +6xz

(z2 +z′2

)]+o(ε)

= y− ε4

[y2y′ +3xy

(y2 +y′2

)]+o(ε).

Hence, the operatorX02 becomes

X2 =(

y− ε4

[y2y′ +3xy(y2+y′2)

]) ∂∂y

·

Bibliography

1. AMES, W. F., LOHNER, R. J.,AND ADAMS, E. Group properties ofutt = [ f (u)ux]x. Internat.J. Nonlinear Mech. 16(1981), 439–447.

2. BAIKOV , V. A., GAZIZOV , R. K., AND IBRAGIMOV , N. H. Approximate symmetries ofequations with a small parameter.Preprint No. 150, Institute of Applied Mathematics, Acad.Sci. USSR,Moscow (1987), 1–28. English transl. published in: N. H. Ibragimov,SelectedWorks,Vol. III, ALGA Publications, Karlskrona. 2008, Paper 2.

3. BAIKOV , V. A., GAZIZOV, R. K., AND IBRAGIMOV, N. H. Approximate symmetries.Math.Sbornik 136 (178), No.3(1988), 435–450. English transl.:Math. USSR Sb. 64, No.2(1989),427–441. Reprinted in: N.H. Ibragimov,Selected Works,Vol. II, ALGA Publications, Karl-skrona. 2006, Paper 6.

4. BAIKOV , V. A., GAZIZOV , R. K., AND IBRAGIMOV , N. H. Approximate symmetries andconservation laws.Trudy Matem. Inst. Steklova 200(1991), 35–45. English transl. by AMS,Proceedings of the Steklov Institute of Mathematics, No.2(1993), 35–47.

5. BIANCHI , L. Lezioni sulla teoria dei gruppi continui finiti di transformazioni. Enrico Spoerri,Pisa, 1918.

6. CAMPBELL , J. E. Introductory treaties on Lie’s theory of finite continuous transformationgroups. Clarendon Press, Oxford, 1903.

7. COHEN, A. An introduction to the Lie theory of one-parameter groups with applications tothe solution of differential equations. D.C. Heath, New York, 1911.

8. DICKSON, L. E. Differential equations from the group standpoint.Ann. of Math. 25(1924),287 –378.

9. EISENHART, L. P. Continuous groups of transformations. Princeton Univ. Press, Princeton,N.J., 1933.

10. FUSHCHICH, V. I., AND SHTELEN, W. M. On approximate symmetry and approximatesolutions of the nonlinear wave equation with a small parameter. J. Phys. A: Math. Gen. 22(1989), L887–L890.

11. GAZIZOV, R. K. Symmetries of differential equations with a small parameter: A comparisonof two aproaches. InModern Group Analysis VII,Eds. N. H. Ibragimov, K. Razi Naqvi, E.Straume, Proc. Int. Conf., Sophus Lie center, Nordfjordeid, Norway, 30 June – 5 July, 1997,MARS Publishers, Trondhem, 1999, 107–114.

12. IBRAGIMOV, N. H., Ed. CRC Handbook of Lie group analysis of differential equations.Vol.3: New trends in theoretical developments and computational methods. CRC Press Inc., BocaRaton, 1996.

13. IBRAGIMOV, N. H. Perturbation methods in group analysis. InDifferential equations andchaos(New Delhi, 1996), New Age International Publishers, 41–60. Reprinted in: N. H.Ibragimov,Selected Works,Vol. II, ALGA Publications, Karlskrona. 2006, Paper 12.

14. IBRAGIMOV, N. H. Elementary Lie group analysis and ordinary differential equations. JohnWiley & Sons, Chichester, 1999.

15. IBRAGIMOV, N. H. Tensors and Riemannian geometry. ALGA Publications, Karlskrona,2008.

16. IBRAGIMOV , N. H. A bridge between Lie symmetries and Galois groups. In“DifferentialEquations: Geometry, Symmetries and Integrability.” The Abel Symposium 2008. Proc. of FifthAbel Symposium, Tromso, Norway, June 17-22, 2008.Eds. B. Kruglikov, V. Lychagin, E.Straume (2009), Springer, 159–172.

17. IBRAGIMOV, N. H. A practical course in differential equations and mathematical modelling.Higher Education Press, Beijing (P. R. China), 2009. World Scientific, Singapore, 2009.

18. KHABIROV, S. V. Lie-Backlund group methods in mathematical physics.Doctor of Sciencethesis. Institute of Mathemtics and Mechanics, Sverdlovsk, 1991. (Russian)

19. LIE, S. Vorlesungen uber Differentialgleichungen mit bekannteninfinitesimalen Transforma-tionen. (Bearbeited und herausgegeben von Dr. G. Scheffers), B. G.Teubner, Leipzig, 1891.

20. LIE, S. Theorie der Transformationsgruppen,Vol. III. (Bearbeitet unter Mitwirkung von F.Engel), B. G. Teubner, Leipzig, 1893.

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21. MAHOMED, F. M., AND LEACH, P. G. L. Lie algebras associated with scalar second-orderordinary differential equations.J. Math. Phys. 30, No. 12(1989), 2770–2777.

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Index

Admitted, 51group, 51operator, 62

Algebraic equation, 108Approximate, 135

Cauchy problem, 137, 143conservation law, 173equality, 135exponential map, 146group generator, 142integration, 162invariant, 166invariant solution, 168Lie algebra, 156Lie equations, 144solution, 165symmetry group, 151transformation, 140, 144transformation group, 140travelling waves, 170

Canonical parameter, 25, 32Canonical variables, 36, 94Cauchy problem, 137

approximate, 137Commutator, 75, 155

relations, 80table, 76, 155

Conformal group, 20Conformal transformation, 17Continuous group, 7

finite, 7infinite, 7

Crystallography, 19

Deformation of symmetry, 153Derived algebra, 82

Determining equation, 67, 152for algebraic equation, 108for approximate symmetries, 152for deformations, 153

Differential algebra, 52Differential function, 52

spaceA , 52Differential variables, 52Dilation, 8, 37

Equation, 131approximate Lie, 144determining, 152differential, 131Korteweg-de Vries (KdV), 170Lie, 131nonlinear wave, 154perturbed, 153perturbed KdV, 170unperturbed, 153van der Pol, 162

Euclidean group, 14in IRn, 19

Exponential map, 23, 29, 133approximate, 146

Frame of differential equations, 54

Galilean group, 4Galois group, 114, 116Generator, 26Group, v, 6, 129

r-parameter, 7admitted, v, 51continuous, 10dilation, 8discontinuous, 10

183

184 Index

Euclidean, 14, 19finite continuous, 7Galilean, 4Galois, 116general linear, 7global, 13infinite continuous, 7local, 12Lorentz, 15mixed, 10, 72multi-parameter, 102of isometric motions, 14one-parameter, 6Poincare, 15projective, 13, 15prolonged, 55, 56rotation, 12transformation, 129two-parameter, 7

Group generator, 26

Ideal, 82Induced group, 43, 114Infinitesimal, 25

generator, 26symmetry, 62transformation, 25

Infinitesimal transformation, 25Invariance, vInvariant, 166

approximate, 166Invariant equations, 41Invariant manifold, 41

regular, 44singular, 44

Inversion, 16

Jacobi identity, 78

Kelvin’s transformation, 21

Lie algebra, 75, 80, 155Abelian, 83admitted, 80approximate, 156approximate Abelian, 161dimension, 80solvable, 83structure constants, 80

Lie algebras, 80isomorphic, 84similar (equivalent), 86

Lie equations, 23, 27, 131approximate, 144

Lie symmetriesof algebraic equations, 108

Linear span, 80Local group, 12

r-parameter, 101one-parameter, 24

Lorentz group, 15, 21Lorentz transformation, 15Lorentz transformations, 21

Maclaurin expansion, 42Matrix group, 19Minkowski space-time, 21

One-parameter group, 6Orbit (G-orbit), 10, 27

Poincare group, 15, 21Projective group, 13, 15

canonical variables, 38on line, 13on plane, 15special, 16, 63

Prolongation, 3, 51, 55kth-order, 67first, 55second, 55

Prolongation formula, 57first, 57, 60general, 58second, 58, 60third, 58

Prolonged group, 55, 56Pseudoscalar product, 89

Rotation, 4Rotation group, 12

generator, 27prolonged generator, 57

Shallow-water equations, 39Similar groups, 8Solution, 137

approximately invariant, 166of Cauchy problem, 137

Solvable Lie algebra, 83Spiral transformation group, 48Structure constants, 80Subalgebra, 81Subgroup, 8Summation convention, 26Symmetry, 152

approximate, 152exact, 153

Index 185

stable, 153Symmetry group, v, 51, 67

Galois representation of, 107Symmetry Lie algebra, 80

Total differentiation, 52Transformation, v, 3, 5, 129

approximate, 140conformal, 17Galilean, 4group, 129infinitesimal, 25Lorentz, 15

point, 3projective, 11

Transformation group, v, 6, 129approximate, 140

Translation, 3group, 6, 14

Tschirnhausen’s transformation, 109Two-parameter group, 7

Van der Pol equation, 162approximate solution, 165exact symmetry, 163symmetries, 162

Transformation Groups and Lie Algebras

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