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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 Bo dy Inside
Ivan A. Bizyaev1* , Alexey V. Borisov2** , and Ivan S. Mamaev1
Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia A. 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 Septemb er 4, 2013; accepted Octob er 31, 2013
1

***

2

Abstract--In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid bo dy fixed inside the shell by means of two different mechanisms. In the former case the rigid bo dy is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smo oth plane. In the latter case the rigid bo dy is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler Jacobi Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bo dies connected by means of a nonholonomic hinge. For this system, integrable cases and various 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 nonholonomic systems [18]. Such systems are of great interest for applications such as control theory and rob otics where they are used to model the dynamics of devices involving rolling motion. For example, one of the p opular problems in rob otics is investigating the dynamical control of the locomotion of a spherical rob ot [9] using various driving mechanisms (p endulum-like, rotor-like etc.; for more references see [10]). In particular, in [1012] the control of a dynamically asymmetric ball by means of three balanced rotors is studied and the advantages and disadvantages of such a control mechanism are shown. As demonstrated in [1, 35, 1316], nonholonomic systems exhibit far more diverse b ehaviors than Hamiltonian systems. Such a diversity of b ehaviors (which was called the hierarchy of dynamical b ehavior) is due to the presence or absence of different tensor invariants (conservation laws), 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 typ es of b ehavior -- their dynamics is always conservative. Therefore, b efore one studies the controlled
* ** ***

E-mail: bizaev 90@mail.ru E-mail: borisov@rcd.ru E-mail: mamaev@rcd.ru

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dynamics of new nonholonomic systems, it is necessary to gain insight into their free dynamics -- the sub ject of this study. In this pap er we investigate two systems which are immediately related to rob otics and are associated 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 p oint of contact of the ball with the plane is zero (for a discussion of problems related to an additional dynamical restriction of the no-spin condition see [5, 17]). In one of these two systems a rigid b ody is attached to the center of the ball on the spherical hinge, and the center of mass of the entire system generally does not lie at the geometrical center of the ball. In a particular case, when a spherical p endulum is attached inside the shell, this problem was integrated by S. A. Chaplygin [18]. The case where the inner b ody is dynamically symmetric was considered in [7, 19, 20]. The Lagrange representation of the equations of motion of the inner b ody (which implies their Hamiltonian prop erty) for this case was found by S. V. Bolotin [19]. Here we develop these ideas, and in the general case of an arbitrary b ody we write the equations of motion using redundant coordinates and quasi-velocities. This makes it p ossible to represent the system in an algebraic form convenient for analysis and to establish an isomorphism with an analogous system on an absolutely smooth plane, which, in turn, makes it p ossible to represent the equations of motion in Hamiltonian form and to find new integrable cases. The other system also has a rigid b ody inside the ball. Between the ball and the inner b ody there is a nonholonomic constraint prohibiting their relative rotations along one chosen direction fixed in the inner b ody. The realization of this constraint with the aid of sharp wheels to which the inner b ody is attached was prop osed by V. Vagner [21], and the kinematic mechanism realizing this constraint b etween the two rigid b odies was called in [22] the nonholonomic hinge. The scheme of the spherical rob ot in which a nonholonomic hinge b etween the inner b ody and the spherical shell is used for locomotion was prop osed in [23] (without theoretical analysis). A sp ecial feature of the set of tensor invariants of this system is that it leads to a new integration mechanism in nonholonomic mechanics -- the Euler Jacobi Lie theorem [24, 25], by which the integrability of the n-dimensional system requires the presence of an invariant measure, n - 2 - k first integrals and k symmetry fields. In conclusion, we consider the problem of free motion of a bundle of two b odies connected by means of a nonholonomic hinge (a similar problem was discussed previously by G. K. Suslov [26]). Its interesting feature is that it can b e considered to b e 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 pap er yields many new problems from various areas of dynamical systems theory: explicit integration, top ological analysis, stability analysis etc. 1. A HOMOGENEOUS SPHERICAL SHELL WITH A RIGID BODY ATTACHED TO THE GEOMETRICAL CENTER 1. Equations of motion. The Chaplygin integral. Consider a system moving on a horizontal plane and consisting of two b odies. One of them is the outer b ody -- a spherical shel l such that its tensor of inertia is spherical, Is E (E is the identity matrix) and the outer surface is a regular sphere; this b ody contains a cavity to whose center the inner b ody -- a spinning top is attached, such that the p oint of attachment coincides with the geometrical center of the outer surface of the spherical shell (see Fig. 1). We also assume that the outer b ody rolls without slipping on the plane. We give a detailed derivation of the equations of motion by using quasi-coordinates in which the equations of motion take an algebraic form that is more convenient for further analysis than that in [19]). The p osition and configuration of this system is completely defined by the radius vector Rp R2 of the contact p oint P of the shell, by the rotation matrix of the shell Qs SO(3) and the rotation matrix of the spinning top Qt SO(3). In order to obtain equations of motion in a simpler form, we use the fact that they must b e invariant under the group of motions of the plane E (2) and (due to the dynamical symmetry of the shell) under the group of rotations of the outer b ody SO(3). In this case the pro jections of velocities, angular velocities of the b odies 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 b ody (the spinning top) are magnitudes invariant under the action of the symmetry group E (2) в SO(3). Therefore we choose a coordinate system Gx1 x2 x3 rigidly attached to the spinning top with the origin at its center of mass. Let V and b e the velocity of the center and the angular velocity of the shell, v and the corresp onding velocities of the spinning top, the normal to the plane, and c the constant vector from the center of the shell C to the center of mass of the spinning top G. The condition of attachment of the outer b ody in the inner b ody implies that the velocity of the shell and that of the spinning top coincide at p oint C , and is expressed by a (holonomic) constraint of the form v = V + в c. (1.1) The absence of slipping at p oint P corresp onds to vanishing of the velocity of the contact p oint of the shell, which is given by the (nonholonomic) constraint V - Rs в = 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 b odies can b e represented in the moving axes as ms V + в ms V = Nc + Np , Is + в Is = Rs Np в , mt v + в mt v = -mt g - Nc , It + в It = -Nc в c, (1.3)

where ms and mt are the mass of the shell and the mass of the spinning top, resp ectively, It is the tensor of inertia of the spinning top, Nc and Np are the reaction forces of the constraints (1.1) and (1.2). These equations must b e supplemented by the Poisson equation governing the evolution of the space-fixed normal vector in the moving coordinate system: = в . Remark. Throughout, for simplicity, we set 2 = 1, where necessary. It follows from (1.3) and (1.4) that the vertical comp onent of the angular velocity of the shell remains constant: z = (, ) = const. Using this integral, one can express the angular velocity of the outer b ody in terms of the velocity of its center V and the vector : =R
-1 0

(1.4)

в V +z .
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We now express with the help of the constraints (1.1) and (1.2) the moments of the reaction forces in (1.3): Rs Np в = (ms + mt )Rs (V + в V ) в + mt Rs (vc + в vc ) в , -Nc в c = mt (V + в V ) в c + mt (vc + в vc ) в c + mg в c, (1.6)

where the notation vc = в c is introduced. Using the relation (1.5) from the equation governing the evolution of and the first of Eqs. (1.6), we obtain Is (V + в V ) = - в
2 2 (ms + mt )Rs (V + в V )+ mt Rs (vc + в vc ) в .

(1.7)

To abbreviate some of our forthcoming formulae, we define the pro jection op erator a: ab = a в (b в a) = a2 b - (a, b)a. It can b e shown that by virtue of the constraint equation (1.2) the following identity holds: (V + в V ) = V + в V . This allows us to rewrite equation (1.7) as V + в V = - (vc + в vc ), =
2 mt Rs . 2 I0 +(ms + mt )Rs

(1.8)

Since the normal to the plane is fixed in space, it follows from (1.8) that the vector parallel to the plane K = V + vc , (K , ) = 0 (1.9)

is also fixed in space, i.e., its evolution in the moving axes Gx1 x2 x3 is given by the equation K = K в . The vector K expressed 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 + в It = mt E - (vc + в vc ) в c + mt g в c. (1.10)

This equation, along with the Poisson equation (1.4), forms a system that is closed relative to the vectors and . We rewrite it in a form more convenient for further analysis. To do this, we use the identity (vc + в vc ) в c = - c + в c and transfer a part of the terms in (1.10) from the right-hand side to the left-hand side. We finally obtain the following result. Prop osition 1. The equations of motion governing the evolution of the vectors and form a closed (reduced) system and can be represented as J + в J = mt (vc + в vc , ) в c + mt g в c,
2

= в ,

J = It +(1 - )mt c = It +(1 - )mt c E - (1 - )mt c c. 1 1 E = (, J)+ mt (vc , )2 + mt (c, ). 2 2
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This system of equations possesses a geometrical integral, an area integral and an energy integral: F0 = ( , ) = 1, F1 = (J, ) = const,
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For a complete description of the dynamics of this system we choose the axes of a fixed coordinate system Oxy z in such a way that Ox K , and let and denote the unit vectors of the fixed axes expressed in a moving coordinate system; then = в , = в , K = |K |. (1.12)

The unit vectors , and completely define the rotation of the spinning top. Using (1.9), we find the velocity of the contact p oint in the fixed coordinate system in the form x = (V , ) = |K |- (vc , ), y = (V , в ) = (vc , в ). Given , and , we find the angular velocity of the shell, , from (1.5). After that we also find the rotation of the shell via the Poisson equations. 2. A ball with a displaced center on a smooth plane. We now consider an obviously Hamiltonian (conservative) system governing the sliding motion of a dynamically asymmetric ball with 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 the reaction of the plane is p erp endicular to it, two pro jections of the momentum of the system onto this plane are preserved. Choosing a coordinate system Gx1 x2 x3 rigidly attached to the b ody with origin at the center of mass (thereby excluding its horizontal uniform rectilinear displacement), for the motion in a p otential field U ( ) we obtain the Lagrange function 1 1 (1.13) L = ( , I)+ m( , r в )2 - U ( ), 2 2 where I is the tensor of inertia of the b ody relative to the center of mass, m is the mass of the body, is the angular velocity in pro jections onto the axes attached to the b ody, is the normal vector to the plane in the same axes, and r is the vector from the p oint of contact to the center of mass of the b ody (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 of displacement of the center of mass a and the vector by the formula r = -R - a , where R is the radius of the ball's shell. The equations of motion can b e represented in the form of the Lagrange equations in the quasivelocities (they are sometimes called the Euler Poincarґ equations); in this case they are written e in vector form as follows: L L · L в+ в . = Substituting the Lagrange function (1.13) and simplifying, we obtain the closed system I + в I = m(va + в va , ) в a + mg в a, = в , (1.14) where the notation va = в a is introduced and the last equation expresses the condition that the normal vector is fixed in space.
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A complete description of rotations of the ball is also achieved by adding the Poisson equations governing the evolution of the fixed unit vectors and parallel to the plane = в , = в . (1.15)

3. Isomorphism, Hamiltonian prop erty and integrable cases. Comparing the equations of motion of the two systems describ ed ab ove, we obtain the following result. Theorem 1. The reduced system of equations (1.11)(1.12) governing the motion of a spinning top attached to the shel l rol ling without slipping on a plane becomes equivalent to the system of equations (1.14)(1.15) in the problem of the dynamics of an unbalanced dynamical ly asymmetric bal l on an absolutely smooth plane upon the change of the parameters a mt = m, c = , J = I. Hence, the equations of motion of the initial nonholonomic reduced system (1.11)(1.12) turn out to b e Hamiltonian. We recall that the Hamiltonian formalism in the redundant coordinates and quasi-velocities is describ ed by the Poincarґ Chetaev equations [27]. In this case the Legendre e transformation in the notation of the nonholonomic system has the form M= H = (M , ) - L| L = J + mt ( , c в )c в , 1 1 2 M = (JAM , AM )+ mt (AM , c в ) + mt g (c, ), 2 2 A = (J + mt (c в ) (c в ))-1 ,

where denotes the tensor product. Calculating the Poisson brackets (see [27, § 2, Chapter 1]), we finally obtain Theorem 2. The equations of motion (1.11)(1.12) in the variables M and are represented in Hamiltonian form Mi = {Mi ,H }, i = {i ,H }, {Mi ,Mj } = -ij k Mk , i = {i ,H }, i = {i ,H }, i = 1, 2, 3, (1.16)

with the Lie Poisson bracket of the fol lowing form: {Mi ,j } = -ij k k , {Mi ,j } = -ij k k , {Mi ,j } = -ij k k , where the remaining brackets are equal to zero. The result on the Hamiltonian prop erty of the original nonholonomic system for various parameter values, is completely nonobvious, since the nonholonomic systems associated with rolling are, as a rule, represented in Hamiltonian or conformally Hamiltonian form only for sp ecial values of geometrical and dynamical parameters [1, 35]. Analogous problems in the dynamics of a general system of material p oints inside a spherical shell rolling on an (absolutely rough) plane are investigated in the pap er [19], in which it is shown that in the case of a dynamically symmetric top the equations of motion are Hamiltonian. For (1.11) and the Euler Poisson equations to b e integrable, we need an additional integral. Below we show the sp ecial cases where it can b e found. The Euler case. In the case c = 0 the system of equations (1.11) p ossesses an additional integral F2 = M 2 . The Lagrange case. For I1 = I2 , 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. For c2 = 0 and c3
- - I2 1 - I1 1 c1 - - I3 1 - I2 1 = 0 there exists

- - - - the invariant Hess relation F2 = 1 I1 I2 1 - I1 1 ± 3 I3 I3 1 - I2 1 = 0. We note that the Hess case was established by finding an isomorphism. The Hess case for a ball with a displaced center of 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 b odies of which each rotates ab out a fixed p oint and which are connected with each other in such a way that the (nonholonomic) constraint is satisfied (Fig. 3) 3 +3 = 0, where and are the angular velocity vectors of the b odies, which are assumed to b e given in the moving coordinate systems rigidly attached to each of the b odies. He assumed that this constraint can b e realized by means of a long torsion-free thread. Such a realization is incorrect, since it is well known that the rotation of the thread through a nonzero angle can arise not due to torsion but due to a change in shap e [6, 29]. A correct (from the theoretical p oint of view) realization of the Suslov constraint was prop osed by V. Vagner [21]. Later an analogous realization, called by the authors the nonholonomic hinge, was also p ointed out in [22].

Fig. 3

In this pap er we consider another problem of a spherically symmetric shell rolling on an absolutely rough plane with a rigid b ody moving inside the shell and connected with it by means of sharp wheels in such a manner that relative rotations ab out the vector e fixed in the inner b ody are excluded (Fig. 4): ( - , e) = 0, (2.1) where and are the angular velocities of the shell and the inner b ody, resp ectively. In order to prohibit relative rotations of the b odies only along one direction, the p oints of contact of the wheels with the inner surface of the shell must lie on one straight line passing through the center of the sphere 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 b ody coincide and are at the geometrical center of the sphere C . We choose a moving coordinate system Cx1 x2 x3 rigidly attached to the inner b ody in such a way that the axis Cx3 e. Then the constraint equations b ecome f0 = 3 - 3 = 0 (the Suslov constraint), f = V - в R = 0 (no-slip constraint at point P ), (2.2)

where V 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, Is E, 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 b ody. The kinetic energy of the entire system can b e represented as 1 T = mV 2 + Is 2 +(, I) , 2 where m is the mass of the entire system, I = diag(I1 ,I2 ,I3 ) is the tensor of inertia of the inner b ody (the axes Cx1 x2 x3 are assumed to b e the principal axes of inertia). Using the formalism of [4], the equations of motion in the moving coordinate system rigidly attached to the inner b ody can b e written explicitly as (2.3) mV + m в V = , Is + Is в = в R - 0 e, I + в I = 0 e,
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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 the help of the constraint equations (2.2) and simplifying, we obtain the following closed system: (I)· = (I) в - 0 e,
2 2 I = (Is + mRs )E - mRs ,

I = (I ) в + 0 e, Is (Is +
2 mRs

= в ,

0 = -

)(1 2 (I1 - I2 )+ I3 (2 1 - 1 2 )) (2.4) . 2 2 2 Is (Is + mRs + I3 )+ mRs I3 3 (2.5)

Thus, we see that for the angular momentum vector of the system relative to the p oint of contact M = I + I, the equations of motion have the form of the Poisson equations M = M в . Hence, the angular momentum M is constant in the fixed axes. This is an analog of the Chaplygin integral for this system. 2. First integrals and invariant measure. In the general case, the system of equations (2.4) p ossesses the following integrals: F0 = 2 , F1 = 3 - 3 ,
2 1

F2 = M 2 ,

F3 = (M , ),

2 F4 = I1 (I1 - I3 ) + I2 (I2 - I3 )2 ,

(2.6)

where the physical constants of the integrals F0 and F1 are fixed, F0 = 1, F1 = 0. On the level set of F1 = 0 (i.e., on the constraint f0 = 0) Eqs. (2.4) also admit the energy integral E= 1 1 (, I)+ (, I). 2 2 (2.7)

In addition, the system (2.4) preserves an invariant measure = d3 d3 d3 with density =
2 2 2 Is (Is + mRs + I3 )+ mRs I3 3 .

(2.8)

3. Absolute dynamics. For a complete description of rotations of the inner b ody we proceed in a standard way. We add the Poisson equations governing the evolution of the fixed unit vectors and parallel to the plane: = в , = в .
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We express the velocity of the center of the ball, V , from the constraint equation (2.2) and the relation (2.5): V= Rs (M - I) в . 2 Is + mRs M ,M = const, y (I, ) - M = . 2 Rs Is + mRs

If we choose a fixed coordinate system in such a way that (M , ) = 0, that is, M = M + M , (I, ) x =- , 2 Rs Is + mRs then the pro jections of the velocity of the center of mass onto the fixed axes take the form

As we can see, for the reduced system (2.4) to b e integrable by the Euler Jacobi theorem, we need an additional integral, and in the general case, when we restrict ourselves to the common level surface of the integrals (2.6) and (2.7), we obtain a nonintegrable three-dimensional flow. The simplest integrable (by the Euler Jacobi theorem) case I1 = I2 = I3 is considered in [22]. We note that F4 degenerates in this case, but the additional integrals 1 = const and 2 = const arise instead. 4. Zero angular momentum (M 2 = 0). Obviously, M 2 = 0 implies that each comp onent of the angular momentum vector is zero: M1 = 0, M2 = 0, M3 = 0. Remark. The case M = 0 is particularly imp ortant from the viewp oint of control theory, since in 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 by eliminating and 3 , using the relation (2.5) and the constraint 3 - 3 = 0: 3 =
2 Is M3 + mRs 3 (M , ) - I1 1 1 - I2 2 2 . 2

Setting M 0, we obtain a closed system of equations in the form I1 1 = (I3 - I2 )3 2 , I2 2 = -(I3 - I1 )3 1 , 1 = -3 (2 + 2 ), 2 = 3 (1 + 1 ), 3 = 1 1 - 2 2 , mR2 = 2 s (I1 1 1 + I2 2 2 ). This system p ossesses the invariant measure
-1

(2.9)

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 = I3 ). In the case of dynamical symmetry the system p ossesses an additional symmetry field and therefore turns out to b e integrable by the Euler Jacobi Lie theorem [24]. Performing a reduction for this symmetry field, we obtain a system integrable by the Euler Jacobi theorem, with two-dimensional invariant manifolds, although the initial three-dimensional manifold formed by the integrals (2.6) and (2.7) is not foliated into a family of two-dimensional submanifolds in this case either. The symmetry field u induced by the invariance of the system under rotations of the axis of dynamical symmetry has the form u=
1

- 2 + 1 - 2 + 1 - 2 . 2 1 2 1 2 1
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According to the Lie theorem, to reduce the order with the help of this field, it is necessary to choose the integrals of this field (i.e., functions yi such that u(yi ) = 0) as variables of the reduced system. In this case, on the level set of the first 2 = 1, 1 1 + 2
2 1 - 3

3 - 3 = 0, 1 2 - 2
2 1 - 3

(M , ) = M , 1 M2 - 2 M1
2 1 - 3

it is more convenient to take the following functions as variables: 3 , K1 =
2

,

K2 =

1

,

K3 =

,

K4 =

M3 - M 3
2 1 - 3

.

In terms of the new variables the equations of motion b ecome 3 = K2
2 1 - 3 ,

K2 K K1 = - , 2 1 - 3
2 3 (Is + mRs )(I3

K2 =

K1 K 1-
2 3

,

K1 K4 K3 = , 2 1 - 3

K1 K3 K4 = - , 2 1 - 3

K=

2 2 1 - 3 K5 - I1 (Is + I3 )K1 )+ Is I3 (1 - 3 )K4 . I1 2

(2.11) Equations (2.11) p ossess the integrals
2 2 F1 = K1 + K2 , 2 2 F2 = K3 + K4 , 2 1 (K3 - I1 K2 )2 1 (Is + I3 3 )(I1 3 K1 + K4 )2 12 2 (K1 + K2 )+ + 2 2 2 Is + mRs 2 2

E= +

3 2

K5 1-
2 3

- I1 K1

2 2 (I1 (Is + mRs )3 K1 - I3 (1 - 3 )K4 )

(2.12)

+

2 2 Is + mRs + I3 (1 - 3 ) 22

2 K5 -

2I1 K1 K5
2 1 - 3

2 2 2 + I1 (1 + 3 )K1 ,

where is defined ab ove (2.8). The integrals of the reduced system are related to the original integrals (2.6)(2.7) as follows: F1 = F4 , I1 (I1 - I3 ) F2 = F2 - (M , )2 , E = E.

The system (2.11) also preserves the invariant measure = 1 d3 dK1 ... dK4 .

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 system obtained by reduction with the help of the symmetry field turns out to b e nonintegrable and exhibits chaotic b ehavior. As is well known, the integrable Hamiltonian systems are, as a rule, bi-Hamiltonian. Moreover, in many cases it is p ossible 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 result holds for the reduced system (2.11). Theorem 3. Equations (2.11) can be represented in the conformal ly bi-Hamiltonian form x=
-1

H1 H2 = -1 J2 , x = (3 ,K1 ,K2 ,K3 ,K4 ), x x 2 I + mRs 1 1 E- F2 , H2 = F1 , H1 = s I1 2I1 2I1 J1
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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 tensor J1 have the form {K1 ,K4 }1 = {K3 ,K4 }1 = =
-1 2 2 (Is + mRs + I3 )(mRs I1 2 2 1 - 3 K1 - (Is + mRs )M )+ 2 (Is + I3 )(Is + mRs )3 K4 2 1 - 3

K2 1-
2 3

,

{K2 ,K4 }1 = -

K1
2 1 - 3

,

{K3 ,3 }1 =

2 1 - 3 ,

,

and for J2 they have the form {K1 ,K3 }2 = - {K1 ,K2 }2 = -
1

I1 K4 1-
2 3

,

{K1 ,K4 }2 =

I1 K3
2 1 - 3

,

{K2 ,3 }2 = - - Is I3

2 1 - 3 ,

2 3 (Is + mRs )

(Is + I3 )I1 K1
2 1 - 3

- I3 M

2 1 - 3 K4 .

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 their linear combination J1 + J2 , = const, also defines the Poisson structure (i.e., it satisfies the Jacobi identity). The Casimir function of the first bracket is F1 and that of the second bracket is F2 : J
1

F1 = 0, x

J2

F2 = 0. x

Thus, a bundle of Poisson structures J1 + J2 has naturally arisen in the system under consideration. 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 J1 on the symplectic leaf F1 = c: K1 = c cos q1 ,
2 2 K3 = -mRs I1 (Is + mRs + I3 )

K2 = c sin q1 ,

3 = cos q2 , K4 = p1 - 1+
2 mRs Is

c sin q1 sin2 q2 - p2 , 2

M cos q2 ,

where q1 [0, 2 ), q2 [0, ] are the angle variables and p1 ,p2 are the corresp onding 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 balanced nonholonomic bundle of two b odies considered in the previous section (see Fig. 4). In this case the origin of the center of mass system (which executes a uniform and rectilinear motion) coincides with the geometrical center C of the shell. As ab ove, we write the equations of motion in the moving coordinate system Cx1 x2 x3 rigidly attached to the inner b ody, so that Cx3 e, and direct the axes Cx1 and Cx2 in such a way that one comp onent of the inertia tensor of the b ody vanishes: I12 = 0. In this case, the constraint equation and the inertia tensor of the inner b ody take the form f0 = 3 - 3 = 0, 0 I13 I1 I = 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 +в T f0 = 0 , d dt T +в T f0 = 0 , (3.1)

where T is the kinetic energy of the entire system in the center of mass system: 1 1 T = Is 2 + (, I). 2 2 Using the constraint, we explicitly obtain a system that is closed relative to , Is = Is в - 0 e, 0 = where e = (0, 0, 1). The vector of the total angular momentum relative to the center of mass M = Is + I preserves its magnitude and orientation in space: M = M в . Thus, the system of equations (3.3) p ossesses two general integrals: F0 = 3 - 3 , Throughout this section we shall assume 3 = 3 . Consider in more detail some sp ecial cases of this system. Remark. We note that to determine the absolute dynamics, it is also necessary to additionally integrate the Poisson equations for the direction cosines: = в , = в , = в . F1 = M 2 . (3.4) I = I в + 0 e,
-1

(3.2)

( в - I (I в ), e) , - (I-1 e + Is 1 e, e)

(3.3)

On the level set of F0 = 0, the energy T is also preserved.

2. The total angular momentum of the system is zero (M = 0). In this case, using Eq. (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 = - (I13 1 + I23 2 ) (A0 1 + A1 2 ), (Is + I3 )2 (det I + Is I1 I2 ) (I13 1 + I23 2 ) (A2 1 + A0 2 ), 2 = (Is + I3 )2 (det I + Is I1 I2 )

2 2 A0 = I13 I23 Is (Is +TrI) - I12 - I23 +(I1 + I2 )I3 , 2 2 Ai = (Is + I3 )(Is + Ii ) Ii2 - (Is + I3 )Ii - Ii2 I13 + I23 - (Is + I3 )Ii . 3 3

These equations obviously preserve a singular invariant measure d1 d2 . I13 1 + I23 2

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13

3. The vector e is directed along the principal axis of the tensor of inertia (I I23 = 0). In this case the equations of motion (3.3) can b e represented as 1 = 3 (2 - 2 ), 2 = - 3 - 3 = 1 2 1+ I - I1 , a= 3 I2 (1 - 1 ), c(a 2 1 - c (1 - I - I2 b= 3 I1 1 = -b2 3 , - b)1 2 , ab)(1 + c) I , c = 3. Is 2 = a1 3 ,

= 0,

(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 b ody is I -I a-b flat). Under this condition no singularity arises in the equation, since the quantity 1-ab = 1I 2 3 remains b ounded. Equations (3.5) preserve the standard invariant measure = d1 d2 d1 d2 d3 and p ossess another additional integral
2 2 F2 = a1 + b2 .

(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 sp ecificity of the Suslov constraint and the mass distribution of the b ody (more precisely, by the condition that the vector e is directed along the principal axis of inertia). 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, and its integral invariant manifolds are two-dimensional. In order to integrate the system by quadratures, it is necessary to make the change of time 3 dt dt 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 b e represented in Hamiltonian form. Prop osition 2. The equations of motion (3.5) can be represented in Hamiltonian form H , x = (1 , 2 ,1 ,2 ,3 ), x 2 2 1 2 +2 1 c((1 - a)1 +(1 - b)2 ) 1 2 1 2 + + 3 , H= 2 1+ c 2 (1 - ab)(1 + c) 2 x=J 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)

(3.7)

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 twodimensional contradicts this prop osition. 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). These intersections must b e one-dimensional invariant manifolds. The solution of this contradiction is that the bracket (3.8) admits only two globally defined Casimir functions C1 = F2 ,
2 2 C2 = 1 + 2 +(1 - ab)(2 +2 ) - 2(1 - a)1 1 - 2(1 - b)2 2 , 1 2

while the third function does not exist (it is defined only locally). This is due to the fact that in the general case the two-dimensional symplectic leaf is emb edded in R5 = {x} in a fairly complex
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THE DYNAMICS OF NONHOLONOMIC SYSTEMS

way (for example, if ab > 0 and ab is an irrational numb er, this leaf is the product of the straight line R1 by the everywhere dense winding of the two-dimensional torus), therefore, in the general case its intersection with the surface H = const turns out to b e a nonclosed fairly complex orbit (for example, if ab > 0 and ab is an irrational numb er, and moreover, for sufficiently large values of H = 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 b e represented as a multiple-valued function as follows: 2 2 C3 = (a1 - b2 )cos( ) - 21 2 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 solvable algebra. According to the classification change of variables that leads to the canonical y y 3 , 2 = 4 , a b - ab y3 .

211

The Lie Poisson bracket (3.8) corresp onds to a of [31], this is the algebra Aspq with p = q = 0. A 5,17 form with a > 0 and b > 0 can b e represented as a(y1 + y2 )+2(1 - a)y3 b(y2 , 2 = 1 = 2 a(1 - ab) {3 ,y1 } = -y2 , {3 ,y2 } = y1 ,

- y1 )+2(1 - b)y4 , 1 = 2 b(1 - ab) {3 ,y3 } = ab y4 , {3 ,y4 } =

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 sp ecial cases where the system of equations (3.5) p ossesses additional tensor invariants under some additional restrictions on the moments of inertia of the inner b ody. 1) I3 = I1 + I2 , i.e., the inner body is a flat plate perpendicular to the vector e. In this case a = b = 1 and the system (3.5) p ossesses four quadratic integrals
2 2 C1 = 1 + 2 ,

C2 = 2 +2 - 1 1 - 2 2 , 1 2 H= 12 1 + 23 2
2 I1 1

C3 = 1 2 +2 1 - 1 2 ,

+

2 I2 2

+ Is (2 +2 ) 1 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 sup erintegrable and all tra jectories turn out to b e closed. We also note that at any rational ab the system (3.5) is sup erintegrable, although the additional integral is much more complex. 2) I1 = I2 = I3 , i.e., the inner body is dynamical ly symmetric about e. Under this condition a = b and a pair of commuting symmetry fields arises u1 = - u2 = 1 - 2I1 I3
1 2

+1 - 2 + 1 , 1 2 1 2 2I + 2 - 1 2 - 1 - 2 , 1 I3 2 1 2 [u1 , u2 ] = 0. 2I1 (1 1 +2 2 ), I3 1 Is 12 (2 +2 )+ 3 . 1 2 2 Is + I3 2

After straightforward simplifications the integrals can b e represented as
2 2 F1 = 1 + 2 ,

F2 = 2 +2 - 1 2

H=

(3.9)

If the existence of the field u1 is obvious and is induced by the invariance under rotations ab out the axis of dynamical symmetry, then the app earance of the field u2 is sufficiently nonobvious.
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Moreover, as opp osed to u1 , the symmetry field u2 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 the help of the Poisson structure of rank two as follows: v = JF F , x JF = 1 v u2 , u2 (F )
v i ui -v j u 2 F uk x 2
k

where any of the integrals (3.9) can b e chosen as the function F , and v is the initial vector field of the system (3.5). In the comp onents the tensor JF can b e represented as JF =

х

i 2

.

k

Remark. The system (3.5) -- a consequence of homogeneity -- is invariant under the extensions i i , i i with a simultaneous change of time, dt dt. They corresp ond to the vector field +2 + 1 + 2 + 3 . u3 = 1 1 2 1 2 3 This vector field commutes with the fields u1 and u2 . With the initial field v its commutator is equal to [u3 , v ] = v . ACKNOWLEDGMENTS The work of A. V. Borisov was carried out within the framework of the state assignment to the Udmurt State University "Regular and Chaotic Dynamics". The work of I. S. Mamaev was supp orted by the RFBR grants 13-01-12462-ofi m. The work of I. A. Bizyaev was supp orted by the Grant of the President of the Russian Federation for Supp ort of Young Doctors of Science MD2324.2013.1, and by the Grant of the President of the Russian Federation for Supp ort of Leading Scientific Schools NSh-2964.2014.1. REFERENCES
1. Borisov, A. V. and Mamaev, I. S., The Rolling of a Rigid Bo dy on a Plane and Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177200. 2. Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490496; see also: Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 1, pp. 103111 (Russian). 3. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., Rolling of a ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201220. 4. Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443490; see also: Rus. J. Nonlin. Dyn., 2008, vol. 4, no. 3, pp. 223280 (Russian). 5. Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Bo dy Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277328. 6. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 12, pp. 104116. 7. Borisov, A. V. and Mamaev, I. S., Two Non-Holonomic Integrable Problems Tracing Back to Chaplygin, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 191198. 8. Bizyaev, I. A. and Tsiganov, A. V., On the Routh Sphere, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 3, pp. 569583 (Russian). 9. Shen, J., Schneider, D. A., and Blo ch, A. M., Controllability and Motion Planning of a Multibody Chaplygin's Sphere and Chaplygin's Top, Internat. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905945. 10. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., How To Control Chaplygin's Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 34, pp. 258-272; see also: Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 2, pp. 289307 (Russian). 11. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., How To Control the Chaplygin Ball Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 12, pp. 144158; see also: Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 1, pp. 5976 (Russian).
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12. Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Mo del and Motion Planning for a Class of Spherical Rolling Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (ICRA, 1418 May, 2012), pp. 32263231. 13. Zenkov, D. V. and Bloch, A. M., Invariant Measures of Nonholonomic Flows with Internal Degrees of Freedom, Nonlinearity, 2003, vol. 16, no. 5, pp. 17931807. 14. Blo ch, A. M., Krishnaprasad, P. S., Marsden, J. E., and Murray, R. M., Nonholonomic Mechanical Systems with Symmetry, Arch. Rational Mech. Anal., 1996, vol. 136, no. 1, pp. 2199. 15. Fernandez, O. E., Mestdag, T., and Blo ch, A. M., A Generalization of Chaplygin's Reducibility Theorem, Regul. Chaotic Dyn., 2009, vol. 14, no. 6, pp. 635655. 16. Ohsawa, T., Fernandez, O. E., Blo ch, A. M., and Zenkov, D. V., Nonholonomic Hamilton Jacobi Theory via Chaplygin Hamiltonization, J. Geom. Phys., 2011, vol. 61, no. 8, pp. 12631291. 17. Borisov, A. V., Mamaev, I. S., and Treschev, D. V., Rolling of a Rigid Bo dy without Slipping and Spinning: Kinematics and Dynamics, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 4, pp. 783797 (Russian). 18. Chaplygin, S. A., On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls, in Col lected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 2656 (Russian). 19. Bolotin, S. V. and Popova, T. V., On the Motion of a Mechanical System Inside a Rolling Ball, Regul. Chaotic Dyn., 2013, vol. 18, nos. 12, pp. 159165. 20. Pivovarova, E. N. and Ivanova, T. B., Stability Analysis of Perio dic Solutions in the Problem of the Rolling of a Ball with a Pendulum, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 4, pp. 146155 (Russian). 21. Vagner, V. V., A Geometric Interpretation of Nonholonomic Dynamical Systems, Tr. semin. po vectorn. i tenzorn. anal., 1941, no. 5, pp. 301327 (Russian). 22. Kharlamov, A. P. and Kharlamov, M. P., A Nonholonomic Hinge, Mekh. Tverd. Tela, 1995, vol. 27, no. 1, pp. 17 (Russian). 23. Halme, A., Schonberg, T., and Wang, Y., Motion Control of a Spherical Mobile Robot, in Proc. of the 4th IEEE Internat. Workshop on Advanced Motion Control (Mie, Japan, 1996): Vol. 1, pp. 259264. 24. Kozlov, V. V., The Euler Jacobi Lie Integrability Theorem, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 329343. 25. Kozlov, V. V., Notes on Integrable Systems, Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 3, pp. 459478 (Russian). 26. Suslov, G. K., Theoretical Mechanics, Moscow: Gostekhizdat, 1946 (Russian). 27. Borisov, A. V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Moscow: R&C Dynamics, ICS, 2005 (Russian). 28. Burov, A. A., On Partial Integrals of the Equations of Motion of a Rigid Bo dy on a Frictionless Horizontal Plane, in Research Problems of Stability and Stabilisation of Motion, V. V. Rumyantsev, V. S. Sergeev, S. Ya. Stepanov, A. S. Sumbatov (Eds.), Moscow: Computing Centre of the USSR Acad. Sci., 1985, pp. 118121. 29. Fuller, F. B., The Writhing Number of a Space Curve, Proc. Natl. Acad. Sci. USA, 1971, vol. 68, pp. 815 819. 30. Tsiganov, A. V., On Natural Poisson Bivectors on the Sphere, J. Phys. A, 2011, vol. 44, no. 10, 105203, 21 pp. 31. Patera, J., Sharp, R. T., Winternitz, P., and Zassenhaus, H., Invariants of Real Low Dimension Lie Algebras, J. Math. Phys., 1976, vol. 17, no. 6, pp. 986994.

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