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ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 2, pp. 198213. c Pleiades Publishing, Ltd., 2014.

The Dynamics of Nonholonomic Systems Consisting of a

Spherical Shell with a Moving Rigid Body Inside

Ivan A. Bizyaev1*, Alexey V. Borisov2**, and Ivan S. Mamaev1***

1Udmurt State University,ul. Universitetskaya 1, Izhevsk, 426034 Russia

2A. A. Blagonravov Mechanical Engineering Research Institute of RAS,ul. Bardina 4, Moscow, 117334 Russia

National Research Nuclear University MEPhI,Kashirskoe sh. 31, Moscow, 115409 Russia

Moscow Institute of Physics and Technology,Institutskii per. 9, Dolgoprudny, Moscow Region, 141700 Russia

Received September 4, 2013; accepted October 31, 2013

AbstractIn this paper we investigate two systems consisting of a spherical shell rolling

without slipping on a plane and a moving rigid body fixed inside the shell by means of twodifferent mechanisms. In the former case the rigid body is attached to the center of the ballon a spherical hinge. We show an isomorphism between the equations of motion for the innerbody with those for the ball moving on a smooth plane. In the latter case the rigid bodyis fixed by means of a nonholonomic hinge. Equations of motion for this system have beenobtained and new integrable cases found. A special feature of the set of tensor invariants of thissystem is that it leads to the Euler Jacobi Lie theorem, which is a new integration mechanismin nonholonomic mechanics. We also consider the problem of free motion of a bundle of twobodies connected by means of a nonholonomic hinge. For this system, integrable cases andvarious tensor invariants are found.

MSC2010 numbers: 70E18, 37J60, 37J35

DOI: 10.1134/S156035471402004X

Keywords: nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge

INTRODUCTION

This work is an extension of our cycle of investigations on the dynamics of nonholonomicsystems [18]. Such systems are of great interest for applications such as control theory and roboticswhere they are used to model the dynamics of devices involving rolling motion. For example, oneof the popular problems in robotics is investigating the dynamical control of the locomotion ofa spherical robot [9] using various driving mechanisms (pendulum-like, rotor-like etc.; for morereferences see [10]). In particular, in [1012] the control of a dynamically asymmetric ball bymeans of three balanced rotors is studied and the advantages and disadvantages of such a controlmechanism are shown.

As demonstrated in [1, 35, 1316], nonholonomic systems exhibit far more diverse behaviorsthan Hamiltonian systems. Such a diversity of behaviors (which was called the hierarchy ofdynamical behavior) is due to the presence or absence of different tensor invariants (conservationlaws), which considerably influences the dynamics of the system. We note that Hamiltonian systems,due to the existence of a tensor invariant such as the Poisson bracket, do not exhibit different typesof behavior their dynamics is always conservative. Therefore, before one studies the controlled

*E-mail: bizaev [email protected]**E-mail: [email protected]

***E-mail: [email protected]

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THE DYNAMICS OF NONHOLONOMIC SYSTEMS 199

dynamics of new nonholonomic systems, it is necessary to gain insight into their free dynamics the subject of this study.

In this paper we investigate two systems which are immediately related to robotics and areassociated with various designs of the mechanism for controlling the rolling of the ball on a plane.We use the classical model of an absolutely rough plane in which the velocity of the point of contactof the ball with the plane is zero (for a discussion of problems related to an additional dynamicalrestriction of the no-spin condition see [5, 17]).

In one of these two systems a rigid body is attached to the center of the ball on the sphericalhinge, and the center of mass of the entire system generally does not lie at the geometrical center ofthe ball. In a particular case, when a spherical pendulum is attached inside the shell, this problemwas integrated by S. A. Chaplygin [18]. The case where the inner body is dynamically symmetric wasconsidered in [7, 19, 20]. The Lagrange representation of the equations of motion of the inner body(which implies their Hamiltonian property) for this case was found by S. V. Bolotin [19]. Here wedevelop these ideas, and in the general case of an arbitrary body we write the equations of motionusing redundant coordinates and quasi-velocities. This makes it possible to represent the systemin an algebraic form convenient for analysis and to establish an isomorphism with an analogoussystem on an absolutely smooth plane, which, in turn, makes it possible to represent the equationsof motion in Hamiltonian form and to find new integrable cases.

The other system also has a rigid body inside the ball. Between the ball and the inner bodythere is a nonholonomic constraint prohibiting their relative rotations along one chosen direction

fixed in the inner body. The realization of this constraint with the aid of sharp wheels to whichthe inner body is attached was proposed by V. Vagner [21], and the kinematic mechanism realizingthis constraint between the two rigid bodies was called in [22] the nonholonomic hinge. The schemeof the spherical robot in which a nonholonomic hinge between the inner body and the sphericalshell is used for locomotion was proposed in [23] (without theoretical analysis). A special featureof the set of tensor invariants of this system is that it leads to a new integration mechanism innonholonomic mechanics the Euler Jacobi Lie theorem [24, 25], by which the integrability ofthe n-dimensional system requires the presence of an invariant measure, n 2 k first integralsand k symmetry fields.

In conclusion, we consider the problem of free motion of a bundle of two bodies connected bymeans of a nonholonomic hinge (a similar problem was discussed previously by G. K. Suslov [26]).Its interesting feature is that it can be considered to be an integrable (by the Euler Jacobi theorem)geodesic flow of the quadratic metric on the solvable Lie group.

We also note that the analysis of systems considered in this paper yields many new problemsfrom various areas of dynamical systems theory: explicit integration, topological analysis, stabilityanalysis etc.

1. A HOMOGENEOUS SPHERICAL SHELL WITH A RIGID BODY ATTACHEDTO THE GEOMETRICAL CENTER

1. Equations of motion. The Chaplygin integral. Consider a system moving on a horizontalplane and consisting of two bodies. One of them is the outer body a spherical shell such thatits tensor of inertia is spherical, IsE (E is the identity matrix) and the outer surface is a regularsphere; this body contains a cavity to whose center the inner body a spinning top is attached,such that the point of attachment coincides with the geometrical center of the outer surface of thespherical shell (see Fig. 1). We also assume that the outer body rolls without slipping on the plane.

We give a detailed derivation of the equations of motion by using quasi-coordinates in which theequations of motion take an algebraic form that is more convenient for further analysis than thatin [19]).

The position and configuration of this system is completely defined by the radius vector Rp R2of the contact point Pof the shell, by the rotation matrix of the shell QsS O(3) and the rotationmatrix of the spinning top QtS O(3).

In order to obtain equations of motion in a simpler form, we use the fact that they must beinvariant under the group of motions of the plane E(2) and (due to the dynamical symmetry ofthe shell) under the group of rotations of the outer body SO(3). In this case the projections ofvelocities, angular velocities of the bodies and the normal vector to the plane onto the moving axes

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Fig. 1. A dynamically symmetric spherical shell to whose center a spinning top is attached.

rigidly attached to the inner body (the spinning top) are magnitudes invariant under the action

of the symmetry group E(2) SO(3). Therefore we choose a coordinate system Gx1x2x3 rigidlyattached to the spinning top with the origin at its center of mass.Let V and be the velocity of the center and the angular velocity of the shell, v and the

corresponding velocities of the spinning top, the normal to the plane, and c the constant vectorfrom the center of the shell C to the center of mass of the spinning top G. The condition ofattachment of the outer body in the inner body implies that the velocity of the shell and that ofthe spinning top coincide at point C, and is expressed by a (holonomic) constraint of the form

v= V + c. (1.1)The absence of slipping at point Pcorresponds to vanishing of the velocity of the contact point ofthe shell, which is given by the (nonholonomic) constraint

V Rs = 0, (1.2)where Rs is the outer radius of the shell (see Fig. 1).

The equations governing the evolution of the momenta and angular momenta of these bodiescan be represented in the moving axes as

ms V + msV =Nc+ Np, Is + Is= RsNp ,mtv + mtv =mtgNc, It + It=Nc c,

(1.3)

where ms and mt are the mass of the shell and the mass of the spinning top, respectively, It is thetensor of inertia of the spinning top, Nc and Np are the reaction forces of the constraints (1.1)and (1.2). These equations must be supplemented by the Poisson equation governing the evolutionof the space-fixed normal vector in the moving coordinate system:

=. (1.4)Remark. Throughout, for simplicity, we set 2 = 1, where necessary.

It follows from (1.3) and (1.4) that the vertical component of the angular velocity of the shellremains constant:

z = (,) = const.

Using this integral, one can express the angular velocity of the outer body in terms of the velocityof its center V and the vector :

= R10 V + z. (1.5)


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We now express with the help of the constraints (1.1) and (1.2) the moments of the reactionforces in (1.3):

RsNp = (ms+ mt)Rs(V + V) + mtRs(vc+ vc) ,Nc c= mt(V + V) c+ mt(vc+ vc) c+ mgc,

(1.6)

where the notation vc= c is introduced.Using the relation (1.5) from the equation governing the evolution ofand the first of Eqs. (1.6),

we obtain

Is(V + V) =

(ms+ mt)R2s(V + V) + mtR2s(vc+ vc)

. (1.7)

To abbreviate some of our forthcoming formulae, we define the projection operator a:

ab= a (b a) =a2b (a,b)a.It can be shown that by virtue of the constraint equation (1.2) the following identity holds:

(V + V) = V + V.This allows us to rewrite equation (1.7) as

V + V =(vc+ vc), = mtR2s

I0+ (ms+ mt)R2s. (1.8)

Since the normal to the plane is fixed in space, it follows from (1.8) that the vector parallelto the plane

K=V + vc, (K,) = 0 (1.9)

is also fixed in space, i.e., its evolution in the moving axes Gx1x2x3 is given by the equation

K=K.The vectorKexpressed in the fixed axes coincides with the vector integral found by S. A. Chaplygin

for such systems [18].Using the second of Eqs. (1.6) and Eq. (1.8), we represent the evolution of the vector as

It + It= mt

E (vc+ vc) c + mtg c. (1.10)

This equation, along with the Poisson equation (1.4), forms a system that is closed relative tothe vectors and . We rewrite it in a form more convenient for further analysis. To do this, weuse the identity

(vc+ vc) c=

c + c

and transfer a part of the terms in (1.10) from the right-hand side to the left-hand side. We finallyobtain the following result.

Proposition 1. The equations of motion governing the evolution of the vectors and form aclosed(reduced) system and can be represented as

J + J= mt(vc+ vc,) c+ mtg c, = ,J= It+ (1 )mtc= It+ (1 )mtc2E (1 )mtc c.

(1.11)

This system of equations possesses a geometrical integral, an area integral and an energy integral:

F0= (,) = 1, F1= (J,) = const, E=1

2(, J) +

1

2mt(vc,)

2 + mt(c,).


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For a complete description of the dynamics of this system we choose the axes of a fixed coordinatesystemOxyz in such a way that Ox K, and let and denote the unit vectors of the fixed axesexpressed in a moving coordinate system; then

= , = , K=|K|. (1.12)The unit vectors , and completely define the rotation of the spinning top. Using (1.9), wefind the velocity of the contact point in the fixed coordinate system in the form

x= (V,) =|K| (vc,), y= (V,) =(vc, ).Given , and , we find the angular velocity of the shell, , from (1.5). After that we also findthe rotation of the shell via the Poisson equations.

2. A ball with a displaced center on a smooth plane. We now consider an obviouslyHamiltonian (conservative) system governing the sliding motion of a dynamically asymmetric ballwith a displaced center on an absolutely smooth plane.

In the case of absolute motion, this system has five degrees of freedom, but since in this case thereaction of the plane is perpendicular to it, two projections of the momentum of the system ontothis plane are preserved. Choosing a coordinate system Gx1x2x3 rigidly attached to the body withorigin at the center of mass (thereby excluding its horizontal uniform rectilinear displacement), forthe motion in a potential field U() we obtain the Lagrange function

L=1

2 (, I) +1

2m(, r )2 U(), (1.13)where I is the tensor of inertia of the body relative to the center of mass, m is the mass of thebody, is the angular velocity in projections onto the axes attached to the body, is the normalvector to the plane in the same axes, and r is the vector from the point of contact to the center ofmass of the body (see Fig. 2). For the gravitational field U() =mg(a,).

Fig. 2. A ball with a displaced center of mass on a smooth plane.

As seen from Fig. 2, the vector r for the ball is expressed in terms of the constant vector ofdisplacement of the center of mass a and the vector by the formula

r=R a,where R is the radius of the balls shell.

The equations of motion can be represented in the form of the Lagrange equations in the quasi-

velocities (they are sometimes called the Euler Poincare equations); in this case they are writtenin vector form as follows:

L

= L

+ L

.

Substituting the Lagrange function (1.13) and simplifying, we obtain the closed system

I + I= m(va+ va,)a + mga, = , (1.14)where the notation va= ais introduced and the last equation expresses the condition that thenormal vector is fixed in space.


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A complete description of rotations of the ball is also achieved by adding the Poisson equationsgoverning the evolution of the fixed unit vectors and parallel to the plane

= , = . (1.15)3. Isomorphism, Hamiltonian property and integrable cases.Comparing the equations

of motion of the two systems described above, we obtain the following result.

Theorem 1. The reduced system of equations (1.11)(1.12) governing the motion of a spinningtop attached to the shell rolling without slipping on a plane becomes equivalent to the system ofequations (1.14)(1.15) in the problem of the dynamics of an unbalanced dynamically asymmetricball on an absolutely smooth plane upon the change of the parameters

mt = m, c=a

, J= I.

Hence, the equations of motion of the initial nonholonomic reduced system (1.11)(1.12) turnout to be Hamiltonian. We recall that the Hamiltonian formalism in the redundant coordinatesand quasi-velocities is described by the Poincare Chetaev equations [27]. In this case the Legendretransformation in the notation of the nonholonomic system has the form

M= L

=J + mt(, c

)c

,

H= (M,) L|M=12

(JAM, AM) +1

2mt(AM, c )2 + mtg(c,),

A= (J + mt(c ) (c ))1,wheredenotes the tensor product.

Calculating the Poisson brackets (see [27, 2, Chapter 1]), we finally obtainTheorem 2. The equations of motion (1.11)(1.12) in the variablesM and are representedin Hamiltonian form

Mi={Mi, H}, i={i, H}, i ={i, H}, i ={i, H}, i= 1, 2, 3, (1.16)with the Lie Poisson bracket of the following form:

{Mi, Mj}=ijkMk, {Mi, j}=ijkk, {Mi, j}=ijkk, {Mi, j}=ijkk,where the remaining brackets are equal to zero.

The result on the Hamiltonian property of the original nonholonomic system for variousparameter values, is completely nonobvious, since the nonholonomic systems associated with rollingare, as a rule, represented in Hamiltonian or conformally Hamiltonian form only for special valuesof geometrical and dynamical parameters [1, 35].

Analogous problems in the dynamics of a general system of material points inside a sphericalshell rolling on an (absolutely rough) plane are investigated in the paper [19], in which it is shownthat in the case of a dynamically symmetric top the equations of motion are Hamiltonian.

For (1.11) and the Euler Poisson equations to be integrable, we need an additional integral.Below we show the special cases where it can be found.

The Euler case. In the case c = 0 the system of equations (1.11) possesses an additionalintegralF2 = M

2.The Lagrange case. For I

1= I

2, c1 = 0 and c2= 0 there is an additional integral F2= M3.

The integrability of this case is established in [7].

The particular Hess integral.Forc2= 0 andc3

I12 I1

1 c1

I13 I1

2 = 0 there exists

the invariant Hess relation F2= 1I1

I12 I1

1

3I3

I13 I1

2 = 0. We note that the Hess

case was established by finding an isomorphism. The Hess case for a ball with a displaced centerof mass on a smooth plane is found in [28].


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2. A HOMOGENEOUS SPHERICAL SHELL WITH A NONHOLONOMIC HINGE INSIDE

1. Equations of motion.G. K. Suslov (see [26]) considered a system consisting of two bodiesof which each rotates about a fixed point and which are connected with each other in such a waythat the (nonholonomic) constraint is satisfied (Fig. 3)

3

+ 3

= 0,

where and are the angular velocity vectors of the bodies, which are assumed to be given in the

moving coordinate systems rigidly attached to each of the bodies. He assumed that this constraintcan be realized by means of a long torsion-free thread. Such a realization is incorrect, since it iswell known that the rotation of the thread through a nonzero angle can arise not due to torsionbut due to a change in shape [6, 29]. A correct (from the theoretical point of view) realization ofthe Suslov constraint was proposed by V. Vagner [21]. Later an analogous realization, called by theauthors thenonholonomic hinge, was also pointed out in [22].

Fig. 3

In this paper we consider another problem of a spherically symmetric shell rolling on anabsolutely rough plane with a rigid body moving inside the shell and connected with it by meansof sharp wheels in such a manner that relative rotations about the vector e fixed in the inner bodyare excluded (Fig. 4):

( , e) = 0, (2.1)where and are the angular velocities of the shell and the inner body, respectively. In order toprohibit relative rotations of the bodies only along one direction, the points of contact of the wheels

with the inner surface of the shell must lie on one straight line passing through the center of thesphere C (Fig. 4). The arising constraint (2.1) is completely equivalent to the Suslov constraint.Furthermore, we shall assume that the centers of mass of the shell and the body coincide and areat the geometrical center of the sphere C.

We choose a moving coordinate system Cx1x2x3 rigidly attached to the inner body in such away that the axis Cx3e. Then the constraint equations become

f0= 3 3= 0 (the Suslov constraint),f=V R= 0 (no-slip constraint at pointP), (2.2)

whereV is the velocity of the center of mass of the system and is the normal vector to the plane.As in the previous section, we shall assume that the tensor of inertia of the shell,IsE, is spherical.

Moreover, we restrict ourselves to the case where the vector e coincides with the direction of one

of the principal axes of inertia of the inner body. The kinetic energy of the entire system can berepresented as

T =1

2

mV2 + Is

2 + (, I)

,

where m is the mass of the entire system, I= diag(I1

, I2

, I3

) is the tensor of inertia of the innerbody (the axes Cx1x2x3 are assumed to be the principal axes of inertia).

Using the formalism of [4], the equations of motion in the moving coordinate system rigidlyattached to the inner body can be written explicitly as

mV + m V =, Is + Is = R0e, I + I= 0e, (2.3)


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Fig. 4. A dynamically symmetric spherical shell on a plane with a nonholonomic hinge inside.

where = (1, 2, 3), 0 are the undetermined multipliers, e= (0, 0, 1). Adding the Poisson

equations for evolution of the normal , eliminating the undetermined multipliers , 0 with thehelp of the constraint equations (2.2) and simplifying, we obtain the following closed system:

(I) = (I) 0e, I= (I) + 0e, =,I= (Is+ mR2s)E mR2s, 0 =Is(Is+ mR2s)(12(I1 I2) + I3(21 12))Is(Is+ mR2s+ I3) + mR2sI323 . (2.4)Thus, we see that for the angular momentum vector of the system relative to the point of contact

M= I + I, (2.5)the equations of motion have the form of the Poisson equations

M=M.

Hence, the angular momentum Mis constant in the fixed axes. This is an analog of the Chaplyginintegral for this system.

2. First integrals and invariant measure.In the general case, the system of equations (2.4)possesses the following integrals:

F0= 2, F1 = 3 3, F2= M2, F3= (M,),

F4 = I1(I1 I3)21+ I2(I2 I3)22,(2.6)

where the physical constants of the integrals F0 and F1 are fixed,

F0= 1, F1= 0.

On the level set ofF1= 0 (i.e., on the constraint f0= 0) Eqs. (2.4) also admit the energy integral

E=

1

2 (,I) +12(, I). (2.7)In addition, the system (2.4) preserves an invariant measure = d3 d3 d3with density

=

Is(Is+ mR

2s+ I3) + mR

2sI3

2

3. (2.8)

3. Absolute dynamics.For a complete description of rotations of the inner body we proceedin a standard way. We add the Poisson equations governing the evolution of the fixed unit vectors and parallel to the plane:

= , = .


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We express the velocity of the center of the ball, V, from the constraint equation (2.2) and therelation (2.5):

V = Rs

Is+ mR2s

(M I) .

If we choose a fixed coordinate system in such a way that (M,) = 0, that is,

M=

M

+M

, M

, M

= const,

then the projections of the velocity of the center of mass onto the fixed axes take the form

x

Rs= (I,)

Is+ mR2s

, y

Rs=

(I,) MIs+ mR

2s

.

As we can see, for the reduced system (2.4) to be integrable by the Euler Jacobi theorem, weneed an additional integral, and in the general case, when we restrict ourselves to the common levelsurface of the integrals (2.6) and (2.7), we obtain a nonintegrable three-dimensional flow.

The simplest integrable (by the Euler Jacobi theorem) caseI1

= I2

= I3

is considered in [22].We note that F4 degenerates in this case, but the additional integrals 1= const and 2= constarise instead.

4. Zero angular momentum (M2

= 0). Obviously, M2

= 0 implies that each component ofthe angular momentum vector is zero:

M1 = 0, M2= 0, M3= 0.

Remark. The case M= 0 is particularly important from the viewpoint of control theory, sincein practice the controlled motion of a system usually starts and ends in the state of rest.

In this case it is convenient to rewrite the equations in the variables M, , 1

and 2

byeliminating and

3, using the relation (2.5) and the constraint

3

3= 0:

3

= IsM3+ mR

2s3

(M,) I1

11 I222

2 .

Setting M 0, we obtain a closed system of equations in the formI1

1= (I3 I2)32, I22=(I3 I1)31,1=3(2+ 2), 2 = 3(1+ 1), 3= 11 22,

= mR2s

2 (I

111+ I222).

(2.9)

This system possesses the invariant measure

1 d1

d2

d3,

and the integrals F0, F4 and E. Hence,the system(2.9) is integrable by the Euler Jacobi theorem.

5. The dynamically symmetric case (I1

= I2

=I

3). In the case of dynamical symmetry

the system possesses an additional symmetry field and therefore turns out to be integrable by theEuler Jacobi Lie theorem [24]. Performing a reduction for this symmetry field, we obtain a systemintegrable by the Euler Jacobi theorem, with two-dimensional invariant manifolds, although theinitial three-dimensional manifold formed by the integrals (2.6) and (2.7) is not foliated into afamily of two-dimensional submanifolds in this case either.

The symmetry fieldu induced by the invariance of the system under rotations of the axis ofdynamical symmetry has the form

u= 1

2

2

1

+ 1

2

2

1

+ 1

2 2

1. (2.10)


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According to the Lie theorem, to reduce the order with the help of this field, it is necessary tochoose the integrals of this field (i.e., functions yi such thatu(yi) = 0) as variables of the reducedsystem. In this case, on the level set of the first

2 = 1, 3

3= 0, (M,) =M,

it is more convenient to take the following functions as variables:

3, K1=

11+ 221 23

, K2=

12

211 23

, K3=

1M2

2M11 23

, K4 =

M3

M31 23

.

In terms of the new variables the equations of motion become

3=K2

1 2

3, K1= K2

K1 2

3

, K2= K1 K

1 23

, K3= K1K4

1 23

, K4= K1K31 2

3

,

K= 3(Is+ mR2s)(I31 23K5 I1(Is+ I3)K1) + IsI3(1 23)K4I1

2 .

(2.11)

Equations (2.11) possess the integrals

F1= K21 + K22 , F2=K23+ K24 ,E= 12

(K21 + K2

2) +1

2

(K3 I1K2)2Is+ mR

2s

+1

2

(Is+ I323

)(I1

3K1+ K4)2

2

+3

2

K5

1 23

I1

K1

(I

1(Is+ mR

2s)3K1 I3(1 23)K4)

+Is+ mR

2s+ I3(1 23)

22

K2

5 2I1K1K5

1 23

+ I21

(1 + 23

)K21

,

(2.12)

where is defined above (2.8). The integrals of the reduced system are related to the originalintegrals (2.6)(2.7) as follows:

F1 = F4I1

(I1 I

3)

, F2 = F2 (M,)2, E=E.The system (2.11) also preserves the invariant measure

= 1

d3 dK1 . . . d K 4.

Thus, the reduced system(2.11) is integrable by the Euler Jacobi theorem.

Remark. In a similar problem (the Chaplygin ball with a cavity filled with a liquid [2]) the systemobtained by reduction with the help of the symmetry field turns out to be nonintegrable and exhibitschaotic behavior.

As is well known, the integrable Hamiltonian systems are, as a rule, bi-Hamiltonian. Moreover, inmany cases it is possible to find their explicit form [30] for systems with quadratic (in the velocities)first integrals of a bi-Hamiltonian representation. It turns out that a generalization of this resultholds for the reduced system (2.11).

Theorem 3. Equations (2.11) can be represented in the conformally bi-Hamiltonian form

x= 1J1H1

x =1J2

H2

x , x= (3, K1, K2, K3, K4),

H1= Is+ mR

2s

I1

E 12I

1

F2, H2= 12I

1

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relative to the consistent (nonlinear) Poisson structures J1 and J2 of rank four (rankJ1=rankJ2= 4). The nonzero brackets corresponding to the Poisson tensorJ1 have the form

{K1, K4}1 = K21 2

3

, {K2, K4}1= K11 2

3

, {K3, 3}1 =

1 23

,

{K3, K4}1=

=1

(Is+ mR2s+ I3)(mR

2sI1

1 23

K1 (Is+ mR2s)M) +(Is+ I3)(Is+ mR

2s)3K4

1 23

,

and forJ2 they have the form

{K1, K3}2= I1K41 2

3

, {K1, K4}2 = I1K31 2

3

, {K2, 3}2=

1 23

,

{K1, K2}2= 1

3(Is+ mR2s)

(Is+ I3)I1K1

1 23

I3

M

IsI3

1 2

3K4

.

The proof is a straightforward calculation of the equations and the Jacobi identity.

Remark. We recall that the consistency of the Poisson structures J1 and J2 means that theirlinear combination J1+ J2, = const, also defines the Poisson structure (i.e., it satisfies theJacobi identity).

The Casimir function of the first bracket isF1 and that of the second bracket isF2:J1

F1x

= 0, J2F2x

= 0.

Thus, a bundle of Poisson structures J1+ J2 has naturally arisen in the system underconsideration. They are linear in the velocities but nonlinear in 3.

As an illustration we also show how the canonical variables are given for the Poisson bracket J1on the symplectic leaf

F1 = c:

K1= c cos q1, K2 = c sin q1, 3= cos q2,

K3 =mR2sI1(Is+ mR2s+ I3) c

2sin q1sin

2 q2 p2, K4=p1

1 +mR2s

Is

Mcos q2,

where q1[0, 2), q2[0, ] are the angle variables and p1, p2 are the corresponding momenta.

3. A NONHOLONOMIC BUNDLE OF TWO BODIES(THE GENERALIZED SUSLOV PROBLEM)

1. Equations of motion.In conclusion, we consider the problem of free motion of a balancednonholonomic bundle of two bodies considered in the previous section (see Fig. 4). In this case theorigin of the center of mass system (which executes a uniform and rectilinear motion) coincides withthe geometrical center C of the shell. As above, we write the equations of motion in the movingcoordinate system C x1x2x3 rigidly attached to the inner body, so that C x3e, and direct the axesCx1 and C x2in such a way that one component of the inertia tensor of the body vanishes: I12= 0.In this case, the constraint equation and the inertia tensor of the inner body take the form

f0= 3 3= 0,

I=

I1

0 I13

0 I2

I23

I13

I23

I3

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The equations of motion in quasi-velocities with the undetermined multiplier 0 have the form

d

dt

T

+ T

=0

f0

,

d

dt

T

+ T

=0

f0

, (3.1)

where Tis the kinetic energy of the entire system in the center of mass system:

T=1

2

Is2 +

1

2

(, I). (3.2)

Using the constraint, we explicitly obtain a system that is closed relative to ,

Is= Is 0e, I= I + 0e,

0=( I1(I ),e)

(I1e+ I1s e,e),

(3.3)

where e= (0, 0, 1).

The vector of the total angular momentum relative to the center of mass

M=Is + I (3.4)

preserves its magnitude and orientation in space:M=M.

Thus, the system of equations (3.3) possesses two general integrals:

F0 = 3 3, F1= M2.On the level set ofF0= 0, the energy T is also preserved.

Throughout this section we shall assume

3 = 3.

Consider in more detail some special cases of this system.

Remark. We note that to determine the absolute dynamics, it is also necessary to additionallyintegrate the Poisson equations for the direction cosines:

= , = , = .

2. The total angular momentum of the system is zero (M= 0). In this case, usingEq. (3.4) and the constraint

3=

3, we eliminate the variables and

3 from the equations of

motion and obtain the following system on the plane:

1

= (I131+ I232)(Is+ I3)

2(det I + IsI1I2)(A01+ A12),

2=

(I13

1

+ I23

2

)

(Is+ I3)2(det I + IsI1I2)(A

21+ A02),

A0= I13I23

Is(Is+ TrI) I212 I223+ (I1+ I2)I3

,

Ai = (Is+ I3)(Is+ Ii)

I2i3 (Is+ I3)Ii I2i3I213+ I223 (Is+ I3)Ii.

These equations obviously preserve a singular invariant measure

d1

d2

I13

1

+ I23

2

.


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210 BIZYAEV et al.

3. The vector e is directed along the principal axis of the tensor of inertia (I13= 0,I23 = 0). In this case the equations of motion (3.3) can be represented as

1= 3(2 2), 2=3(1 1), 1=b23, 2= a13,3=

1

2

21

1 + c c(a b)12

(1 ab)(1 + c) ,

a=

I3

I1

I2

, b=

I3

I2

I1

, c=

I3

Is .

(3.5)

Remark. Under physical restrictions on the moments of inertia (Ii+ Ij Ik) the inequality

1 ab 0 holds. Note that equality is achieved only for I3

= I1

+ I2

(i.e., when the inner body is

flat). Under this condition no singularity arises in the equation, since the quantity ab1ab

= I

1I

2

I3

remains bounded.

Equations (3.5) preserve the standard invariant measure = d1

d2

d1

d2

d3

and possessanother additional integral

F2 =a2

1+ b2

2. (3.6)

Remark. This integral coincides with the last of the integrals (2.6) of the previous system, i.e.,

its existence is determined by the specificity of the Suslov constraint and the mass distribution ofthe body (more precisely, by the condition that the vector e is directed along the principal axis ofinertia).

The presence of four first integrals and an invariant measure allows the conclusion that

the system of equations (3.5) is integrable by quadratures by the Euler Jacobi theorem, andits integral invariant manifolds are two-dimensional.

In order to integrate the system by quadratures, it is necessary to make the change of time3

dtdt and solve the resulting linear system for the variables 1

, 2

, 1

and 2

.

It turns out that equations (3.5) preserve another tensor invariant, a Poisson structure, and can

therefore be represented in Hamiltonian form.Proposition 2. The equations of motion (3.5) can be represented in Hamiltonian form

x= JH

x, x= (1, 2, 1, 2, 3),

H=1

2

21

+ 22

1 + c +

1

2

c((1 a)21

+ (1 b)22

)

(1 ab)(1 + c) +1

223,

(3.7)

with the degenerate Lie Poisson bracket of rank two (rankJ= 2)

{1

, 3}=

2

2, {

2,

3}=

1

1,

{1, 3}=b2, {2, 3}= a1. (3.8)

The proof is a straightforward calculation of the equations and the Jacobi identity.At first sight, the fact that the invariant manifolds of the system under consideration are two-dimensional contradicts this proposition. Indeed, the symplectic leaves of the Poisson structure (3.8)are two-dimensional and must intersect with the level surface of the Hamiltonian (3.7). Theseintersections must be one-dimensional invariant manifolds. The solution of this contradiction isthat the bracket (3.8) admits only two globally defined Casimir functions

C1=F2, C2 = 2

1+ 2

2+ (1 ab)(21+ 22) 2(1 a)11 2(1 b)22,while the third function does not exist (it is defined only locally). This is due to the fact that inthe general case the two-dimensional symplectic leaf is embedded in R5 ={x} in a fairly complex


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way (for example, ifab >0 and

abis an irrational number, this leaf is the product of the straightline R1 by the everywhere dense winding of the two-dimensional torus), therefore, in the generalcase its intersection with the surface H= const turns out to be a nonclosed fairly complex orbit

(for example, ifab >0 and

ab is an irrational number, and moreover, for sufficiently large valuesofH=H0, we obtain everywhere a dense winding of the two-dimensional torus).

Remark. In the case ab >0 the Casimir function of the bracket (3.8) can be represented as a

multiple-valued function as follows:C3= (a

2

1 b2

2)cos() 212

ab sin(),

where the angle variable is defined from the equation

tan

2

ab

=

(1 b)2 (1 ab)

2

(1 a)1 (1 ab)

1

.

The Lie Poisson bracket (3.8) corresponds to a solvable algebra. According to the classificationof [31], this is the algebra Aspq

5,17 with p= q= 0. A change of variables that leads to the canonical

form with a >0 and b >0 can be represented as

1=

a(y1+ y2) + 2(1

a)y3

2a(1 ab) ,

2=

b(y2

y1) + 2(1

b)y4

2b(1 ab),

1=

y3

a,

2=

y4

b,

{3

, y1}=y2, {3, y2}= y1, {3, y3}=

ab y4, {3, y4}=

ab y3.

Usually one considers a generalization of the Suslov problem on semi-simple algebras. However,as we see, real problems lead to systems with a quadratic Hamiltonian on the solvable Lie algebra.

We show two more special cases where the system of equations (3.5) possesses additional tensorinvariants under some additional restrictions on the moments of inertia of the inner body.

1) I3

= I1

+ I2

, i.e., the inner body is a flat plate perpendicular to the vector e. In this casea= b = 1 and the system (3.5) possesses four quadratic integrals

C1=2

1+ 2

2, C2= 2

1+ 2

2 11 22, C3 = 12+ 21 12,

H=12 23+ 12

I1

21+

I2

22+

Is(

2

1+

2

2)Is+ I3,

where the first three integrals are the globally defined Casimir functions of the bracket (3.8).

In this case the system is superintegrable and all trajectories turn out to be closed. We also

note that at any rational

abthe system (3.5) is superintegrable, although the additional integralis much more complex.

2)I1

= I2=I

3,i.e., the inner body is dynamically symmetric aboute. Under this condition a = b

and a pair of commuting symmetry fields arises

u1=2

1

+ 1

2

2

1

+ 1

2

,

u2= 1 2I1I3

1 1

+2 2I1I3

2 2 1

1 2

2

,

[u1,u2] = 0.After straightforward simplifications the integrals can be represented as

F1= 21+ 22, F2 = 21+ 22 2I1I3

(11+ 22), H=1

2

IsIs+ I3

(21+ 2

2) +1

223. (3.9)

If the existence of the fieldu1 is obvious and is induced by the invariance under rotations aboutthe axis of dynamical symmetry, then the appearance of the fieldu2 is sufficiently nonobvious.


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212 BIZYAEV et al.

Moreover, as opposed tou1, the symmetry fieldu2 does not preserve the first integrals (3.9)(i.e.,u2( Fi)= 0,u2( H)= 0). This allows one to use it for Hamiltonization of the system with thehelp of the Poisson structure of rank two as follows:

v = JFF

x, JF =

1

u2(F)

v u2,where any of the integrals (3.9) can be chosen as the function F, and v is the initial vector field of

the system (3.5). In the components the tensor JFcan be represented as JF =viui2vjui2

k

uk2

Fxk

.Remark. The system (3.5) a consequence of homogeneity is invariant under the extensionsii, ii with a simultaneous change of time, dtdt. They correspond to the vectorfield

u3= 1 1

+ 2

2

+ 1

1

+ 2

2

+ 3

3

.

This vector field commutes with the fieldsu1 andu2. With the initial fieldv its commutator isequal to

[u3,v] =v.ACKNOWLEDGMENTS

The work of A. V. Borisov was carried out within the framework of the state assignment tothe Udmurt State University Regular and Chaotic Dynamics. The work of I. S. Mamaev wassupported by the RFBR grants 13-01-12462-ofi m. The work of I. A. Bizyaev was supported by theGrant of the President of the Russian Federation for Support of Young Doctors of Science MD-2324.2013.1, and by the Grant of the President of the Russian Federation for Support of LeadingScientific Schools NSh-2964.2014.1.

REFERENCES1. Borisov, A. V. and Mamaev, I. S., The Rolling of a Rigid Body on a Plane and Sphere: Hierarchy

of Dynamics, Regul. Chaotic Dyn., 2002, vol.7, no. 2, pp. 177200.

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