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ISSN 1560-3547, Regular and Chaotic Dynamics, 2011, Vol. 16, Nos. 1­2, pp. 104­116. c Pleiades Publishing, Ltd., 2011.

Hamiltonicity and Integrability of the Suslov Problem
Alexey V. Borisov* , Alexander A. Kilin** , and Ivan S. Mamaev***
Institute of Computer Science Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
Received Octob er 9, 2010; accepted Novemb er 30, 2010

Abstract--The Hamiltonian representation and integrability of the nonholonomic Suslov problem and its generalization suggested by S. A. Chaplygin are considered. This sub ject is important for understanding the qualitative features of the dynamics of this system, being in particular related to a nontrivial asymptotic behavior (i. e., to a certain scattering problem). A general approach based on studying a hierarchy in the dynamical behavior of nonholonomic systems is developed. MSC2010 numbers: 34D20, 70E40, 37J35 DOI: 10.1134/S1560354711010035 Keywords: Hamiltonian system, Poisson bracket, nonholonomic constraint, invariant measure, integrability

1. EQUATIONS OF MOTION AND THE HAMILTONIAN FORM Equations of motion, first integrals, and the invariant measure. The Suslov problem (formulated in [1]) describ es the motion of a rigid b ody with a fixed p oint sub ject to the nonholonomic constraint (, a) = 0, (1.1) where is the angular velocity in the b ody and a is the unit vector fixed in the b ody. A p ossible implementation of such a constraint with sharp-edged wheels inside a spherical shell was suggested by Wagner [19]; see Fig. 1. Suslov himself suggested an implementation of such a

Fig. 1.

constraint with a nontwisted filament that sequentially connects several Cardan joints [1]. However, the correctness of this implementation of the constraint (1.1) has not yet b een proved, although it is used even in modern studies. In addition, such a proof encounters some difficulties:
* ** ***

E-mail: borisov@rcd.ru E-mail: aka@rcd.ru E-mail: mamaev@rcd.ru

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-- all the known implementations of nonholonomic constraints based on limiting processes are associated with the presence of viscous friction in the original system [26], whereas the elastic nontwisted filament is conservative. The resulting limiting-case constraints are holonomic in this case (Wagner's implementation of the constraint (1.1) can b e considered the limiting case of the rolling motion of the wheel over a rough surface, b eing thus quite correct from this standp oint); -- it is known (see, e. g., [24]) that the rotation of the filament by a nonzero angle can b e due to changes in its shap e, rather than twisting (see Fig. 2). In this case, the elastic filament cannot implement the constraint over the interval of motion.

G. K. Suslov [1] and P. V. Voronets [2] considered also the more complex problem of the motion of two b odies connected by a nontwisted filament. For the same reasons, the formulation of this problem is also not quite correct, so that the mechanical meaning of the results is not clear as yet. The problem of the implementation of an inhomogeneous Suslov constraint ( , a) = d = 0 also remains unresolved; Suslov suggested to implement this constraint attaching a nontwisted filament to a clockwork drive [1].

Fig. 2. Illustration of a p ossible situation in which the nontwisted filament prop osed by Suslov does not imp ose a constraint.

We choose a moving coordinate system Oxy z fixed to the b ody so that Oz a and direct the Ox and Oy axes so as to have I12 = 0 for this comp onent of the inertia tensor. In this case, the equations of the constraint has the form 3 = 0, (1.2)

and the equations of motion that describ e the variations in the angular velocity with the constraint taken into account assume the form I11 1 = -2 (I13 1 + I23 2 )+ M1 , I22 2 = 1 (I13 1 + I23 2 )+ M2 , I13 1 + I23 2 = (I11 - I22 )1 2 + + M3 , (1.3)

where Mi is the comp onent of the moment of the external forces and is the Lagrange undetermined multiplier. Clearly, the last equation is used only to determine and can b e omitted. The p osition of the b odies is describ ed by the Poisson equations = â , = â , = â ,

where , , are the pro jections of the unit vectors of the motionless axes onto the coordinate system Oxy z fixed to the b ody. Let the b ody move in an axisymmetric p otential field; therefore, the unit vectors of the motionless axes can b e chosen so as to make the p otential dep endent on only one of them, U = U ( ). Then
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M =â

U , and we arrive at the following closed system for the vectors and : I11 1 = -2 (I13 1 + I23 2 )+ 2 U U - 3 , 3 2 U U - 1 , I22 2 = 1 (I13 1 + I23 2 )+ 3 1 3 1 = -3 2 , 2 = 3 1 , 3 = 1 2 - 2 1 . 1 2 2 (I11 1 + I22 2 )+ U ( ), 2

(1.4)

Equations (1.4) admit an energy integral and a geometric integral, E= 2 = 1. (1.5)

System (1.4) has also an invariant measure, (I13 1 + I23 2 )-1 d1 d2 d , (1.6) which is, however, singular. This prevents applying the geometric version of the Euler­Jacobi theorem formulated in [3], although makes it p ossible to obtain explicit quadratures if an additional energy integral is present. Hamiltonian representation. Although system (1.4) is not holonomic, it can b e written in a generalized Hamiltonian form with a degenerate Poisson bracket. As this is done, the corresp onding Poisson structure proves to contain singularities at a prop erly chosen p otential U ( ), which, in turn, results in effects absent in "standard" Hamiltonian systems. We first consider the case of U = 0, where the equations for 1 and 2 can b e separated and the following statement is valid: Prop osition 1. The equations describing the variation of 1 and 2 at U = 0 can be represented in a Hamiltonian form with a Lie­Poisson bracket on a corresponding two-dimensional solvable algebra. Proof. The Lie­Poisson bracket of the two-dimensional solvable algebra has the form {X, Y } = Y. We make the substitution X = I13 I22 2 - I23 I11 1 , It can easily b e shown that X = {X, H0 } = Y 2 , Y = {Y, H0 } = - 1 XY , I11 I22
2

(1.7) (1.8)

Y = I13 1 + I23 2 .

2 1 I 2 I22 + I23 I11 1 E= X2 + Y H0 = 13 I11 I22 2 I11 I22

(1.9)

.

Fig. 3.

Recall that the symplectic sheets of the bracket (1.7) are constructed as follows: two twodimensional sheets are the upp er and lower half-planes (Y > 0 and Y < 0), and each p oint of
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the straight line Y = 0 is a zero-dimensional symplectic sheet. The tra jectories of the system (1.9) in the plane X, Y are arcs of the ellipse H0 = const (see Fig. 3) and fixed p oints in the straight line Y = 0. It can easily b e seen that equations (1.9) have an invariant measure dX d Y whose density, = 1 , is singular in the straight line Y = 0. Y Remark 1. At first glance, the prop osition 1 contradicts the statements of [3] that no invariant measure exists and, therefore, the Suslov problem is not Hamiltonian. Generally, according to [4], the existence of an invariant measure of the Euler­Poincar´ equations on a Lie algebra requires that e this algebra b e unimodular. The considered two-dimensional solvable Lie algebra is not unimodular. The seeming contradiction can b e accounted for by the fact that the measure of equations (1.9) is singular, which nevertheless does not prohibit writing a generalized Hamiltonian representation and introducing canonical (symplectic) variables separately in either half-plane, e. g., q = ln Y , p = X . To the point, the Hamiltonian H0 in these variables coincides with the Hamiltonian of the simplest Toda chain. The Poisson structure (1.7) has not b een successfully extended to the full system (1.4) in an explicit form. Equations (1.9) can b e explicitly integrated in p olar coordinates in the plane X , Y : X= 2h0 I11 I22 cos ,
(t+t0 )

Y= =

2h0 sin , 2h0 , I11 I22

= arctg e- 2

,

where H0 = h0 is the value of the integration constant. Thus, the limiting value of the angular velocity of the b ody at t ± is determined by the relationships 2h0 I11 I22 I23 2h0 I11 I22 I13 2 = ± 2 . 1 = 2 2I , 2 I13 I22 + I23 11 I13 I22 + I23 I11 Remark 2. Substituting the solution (t) into the Poisson equation = â in the system (1.4) yields a linear nonautonomous system, which cannot b e generally resolved in quadratures. Its solution can b e found in the time complex plane [27]. Wagner [19] reduced the problem of integrating these equations to a Riccati-typ e equation using the Darb oux method [29]. Now we consider the Hamiltonization of system (1.4) with the presence of a p otential U ( ). Virtually, this issue has not b een studied in the general case. Under the additional restriction that the vector a is directed along the principal axis of the inertia tensor (i. e., if we put I13 = I23 = 0), system (1.4) has a standard invariant measure d2 d3 and can, up on the substitution of time 3 dt = d on the level surface 2 = 1, b e represented as the following natural Lagrangian system [5]: L d L = 0, i = 1, 2, - d i i (1.10) 1 2 2 I22 (1 ) + I11 (2 ) - U ± (1 , 2 ), L= 2 , U ± (1 , 2 ) = U ( ) =±1- 2 - 2 , and the signs ± corresp ond to the upp er and
3 1 2

where i =

di d

lower hemispheres. Clearly, up on the Legendre transformation and with the restriction 2 = 1 imp osed on the sphere, we obtain Hamiltonian equations with the energy integral (1.5) as the Hamiltonian. Note that b oth the substitution of time and the restriction imp osed on the sphere have singularities at the p oints 3 = 0; therefore, more strictly, equations (1.10) describ e two different systems (corresp onding to the upp er, 3 > 0, and the lower, 3 < 0, hemisphere), which are defined
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2 2 on an op en (noncompact) subset of the plane (1 , 2 ), sp ecified by the inequality 1 + 2 < 1. If the function U ( ) is even with resp ect to 3 , these systems are identical.

Borisov and Dudoladov [6] raised the question of hamiltonizing the system (1.4) without a time substitution and found a corresp onding Poisson structure of rank 4 for the p otential U (3 ) = µ3 ; however, the Hamiltonian in this case is the function 2 . Remark 3. The Poisson structure was found in [6] by analyzing the Kovalevskaya exp onents, which are related to the bifurcation of the general solution on the complex plane of time and to the Kovalevskaya­Lyapunov method (Painlev´ test) for the analysis of integrability. The Kovalevskaya e exp onents are defined for quasi-homogeneous systems, to which the Suslov problem with a uniform field of gravity b elongs. As shown in [6], as a Poisson structure (a tensor invariant) is present, the Kovalevskaya exp onent in the general-p osition situation are paired, except for the exp onents corresp onding to the Casimir functions. Thus, an analysis of the Kovalevskaya exp onents can b e considered (for quasi-homogeneous systems) a test for p ossibilities for Hamiltonization. The result obtained in [6] can b e generalized as follows: Theorem 1. If, in the system (1.4), I13 = I23 = 0 (i. e., the vector a is directed along the principal inertia axis) and U ( ) = µ3 + V (1 , 2 ), the equations of motion can be represented in the Hamiltonian form i = {i , H }, i = {i , H }, H= 12 2 2 ( + 2 + 3 ) 21 (1.11)

with the Poisson bracket (only nonzero ones are indicated )
- {1 , 2 } = I111 µ, - {1 , 3 } = -I111 - {2 , 1 } = -I221 µ,

V , 2

- {2 ,3 } = I221

V , 1

{1 , 3 } = -2 ,

{2 , 3 } = 1 .

(1.12)

The proof is simply in checking the equations and the Jacobi identity. If the function V (1 , 2 ) has no singularities, the equations can b e represented in the Hamiltonian form (1.11) everywhere in the phase space. The rank of the Poisson structure (1.12) is 4 at µ = 0 an 2 at µ = 0. Thus, it app ears at first glance that, if µ = 0, the system (1.11) is integrable for an arbitrary p otential V (1 , 2 ), since it can b e restricted to a two-dimensional symplectic sheet, thus b eing reduced to a system with one degree of freedom. This is nevertheless not the case; indeed, if we set I11 = I22 in system (1.10) and choose the p otential of some nonintegrable natural system in the plane, the b ehavior of this system will b e chaotic. This is due to the fact that global Casimir functions do not necessarily exist for complex Poisson structures, since various generalizations of the Darb oux theorems for degenerate Poisson structures are local. The resultant situation is analogous to that related to the existence of first integrals: while any system is locally integrable according to the rectification theorem, the existence of global first integrals is an exception. Note also that, while the Casimir functions for semisimple Lie algebras (parametrizing orbits of the coadjoint representation) are well known and describ ed in numerous textb ooks on Lie algebras, in the general case, (e. g., for general solvable Lie algebras) global Casimir functions may not exist. An interesting question emerges in this context: can an example of a rank-two Poisson structure be given, such that, for an arbitrary (analytical) Hamilton function, the corresponding Hamiltonian system be not integrable? In other words, the symplectic sheets of such a Poisson system must b e chaotically emb edded in the phase space. In the general case, the issue can b e formulated as follows: obstacles to the existence of global analytical Casimir functions for analytical Poisson structures of an arbitrary rank should b e studied. The similar question of the existence of analytical integrals for Hamiltonian systems was e, raised by H. Poincar´ who noted the basic dynamical effects that hamp er their existence. An idea of the state of the art in this problem can b e formed using an excellent b ook by V. V. Kozlov [4].
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Now let the p otential U ( ) b e arbitrary. We write it as U± ( ) = 3 + V± (1 , 2 ), V± (1 ,2 ) = U ( )
3 =±



1-

2 1

-

2 2



2 2 1 - 1 - 2 ,

(1.13)

where the upp er sign corresp onds to 3 > 0 and the lower sign to 3 < 0. According to Theorem 1, we find that the equations of motion can b e represented in a Hamiltonian form in two different (noncompact) regions of the phase space, 3 > 0 and 3 < 0. Since a radical app ears in the 2 2 potential U± , the system is defined only in the region 1 + 2 1, and a singularity is present 2 + 2 = 1. in the Poisson bracket (1.12) at 1 2 Note that such a Hamiltonian representation in different op en regions of the phase space does not contradict the existence of such singularities in the system on the whole that are not typical of Hamiltonian systems with a Poisson bracket without singularities. In particular, Tatarinov [8] notes p otentials U = U (1 , 2 ) for which system (1.4) is integrable and the level surface of the first integrals is diffeomorphic to an oriented two-dimensional manifold of kind g 2 (i. e., to a sphere with two or more handles). At the same time, in a canonical Hamiltonian case, according to the Liouville­Arnold theorem, only tori are p ossible (i. e., g = 1). This seeming contradiction (which was considered by Ya. V. Tatarinov to b e a real difference of the integrable nonholonomic systems from the Hamiltonian ones) is related to the singularities of the Poisson system and inapplicability of the canonical version of the Liouville­Arnold theorem to the system on the whole. In particular, for the systems considered in [8], the vector field vanishes at the common level surface of the first integrals. Note also a relationship b etween the Hamiltonicity based on the substitution of time 3 dt = d (see (1.10)) and the Hamiltonicity based on the representation (1.13). We use the following well-known Prop osition 2. Let a Hamiltonian system be specified in the canonical form q= H , p p=- H . q

Upon the substitution of time, (q, p) dt = d , these equations of motion become equivalent, at the energy level H = h, to the equations of motion of the Hamiltonian system H dq = , d p dp H =- , d q H= 1 H (q, p) - h . (q, p)
1 2

Applying this statement to system (1.11) at the level H = 1 2 = 2 a Hamiltonian system exactly corresp onding to (1.10).

with choosing = 3 yields

All the known integrable generalizations of the Suslov problem that introduce certain additional p otential terms corresp ond to the cases of separation of variables in system (1.10) in either Cartesian or parab olic coordinates [8, 21]. The Suslov problem with a gyrostat. A gyrostatic generalization of equations (1.4) [9, 10] is p ossible. If the gyrostatic moment is k = (k1 , k2 , k3 ), the equations of motion acquire the form I11 1 = -2 (I13 1 + I23 2 + k3 )+ 2 U U - 3 , 3 2 U U - 1 , I22 2 = 1 (I13 1 + I23 2 + k3 )+ 3 1 3 2 = 3 1 , 3 = 1 2 - 2 1 ; 1 = -3 2 ,

in this case, the form of the first integrals (1.5) remains unchanged.
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Up on the substitution of time, 3 dt = d , and under the restriction on the sphere U ± (1 ,2 ) = U ( ) =±1- 2 - 2 , we arrive at the equations
3 1 2

I22 1 = -

U ± (I13 2 - I23 1 + k3 ) ± 2 , 2 2 1 1 - 1 - 2

(I13 2 - I23 1 + k3 ) U ± 1 . I11 2 = - 2 2 2 1 - 1 - 2

(1.14)

Prop osition. At I13 = I23 = 0, equations (1.14) can be written in a Hamiltonian form if we set p1 = I22 1 , p2 = I11 1 : dpi di = {i , H }, = {pi , H }, d d 1 p2 p2 1 H= + 2 + U ± (1 , 2 ), 2 I22 I11 where the Poisson bracket is {i , pj } = ij , {p1 , p2 } = ± k3
2 2 1 - 1 - 2

.

Clearly, the equations can in this case b e also represented in the Lagrangian form (1.10) is, for example, we assume
2 1 L = (I22 12 + I11 2 ) - U ± (1 , 2 ) k3 1 arctg 2

2
2 2 1 - 1 - 2

.

Multidimensional analogs of the Suslov problem were studied in [11­14], mainly from the standp oint of integrability and the analytical prop erties of the general solution. Seemingly, the Hamiltonicity of these analogs have never b een considered. 2. KOVALEVSKAYA EXPONENTS AND THE ADDITIONAL INTEGRAL Kovalevskaya exponents. We write equations (1.4) at U = 0 in the form x = v (x), (2.1) where the notation x = (1 , 2 , 1 , 2 , 3 ) is introduced and v (x) are the right-hand sides of equations (1.4). Since the equations are homogeneous (or, in the general case, quasi-homogeneous), the equations admit a partial solution of the form xµ = cµ t
-µ

,

µ = 1,... , 5, µ = 1,... , 5,

(2.2) (2.3)

where the constant coefficient cµ satisfy the algebraic system of equations vµ (c1 , ... ,c5 ) = -µ cµ , and can generally b e complex. In the case considered, the partial solution (2.2) has the form µ = 1,... , 5, µ = 1, -I11 I22 I23 ± i I11 I22 I13 I21 I11 I22 I13 ± i I11 I22 I23 I11 , c2 = , c1 = 2 2 2 2 I13 I22 + I23 I11 I13 I22 + I23 I11 c3 = c4 = c5 = 0.

(2.4)

For the solution (2.4), we calculate the Kovalevskaya exp onents [17], which are the eigennumb ers v of the matrix x (c)+ , = diag(1 ,... ,5 ), and obtain 1 = -1,
4, 5 2 2 (I11 I23 - I22 I13

=1±

2 = 2, 3 = 1, ± 2i I11 I22 I13 I23 )(I11 - I22 )I11 I22
2 I13 I22

(2.5) .
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+

2 I23 I11

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According to the Kovalevskaya­Lyapunov method, for system (1.4) to have an additional integral of motion, we have to assume that the exp onents (2.5) b e integer real numb ers (the more strict Kovalevskaya­Lyapunov test requires also their nonnegativity; see [17] for further details). This condition is satisfied under the following restrictions on the inertia moments of the system: 1) I13 = 0, 2) I
23

I11 = I22 + I22 = I11

= 0,

2 I23 k I22 I2 + 13 k I11

2

4, 5 = 1 ± k, 4, 5 = 1 ± k,

k Z, (2.6) k Z.

2

We will consider only the first case in (2.6), since the second case can easily b e obtained from the first one by p ermuting the subscripts 1 and 2. The algebraic first integral. The condition (2.6) is necessary for the existence of an additional integral of motion. We now prove that these conditions are sufficient for odd k. Theorem 2. If the conditions (2.6) are satisfied at I13 = 0 for odd k = 2n +1, system (1.4) has an additional algebraic integral of motion of degree k = 2n +1 in momenta, Fn = 1 1 f where fn = (f
(n) 1 (n) 1

+ 2 2 f

(n) 2

+ I23 2 3 f

(n) 3

,

(2.7)

,f

(n) 2

,f

(n) 3

)=

n-1 i=0

A-1 U fn , k , ,

2 2 22 U = diag(1 , 2 , I23 2 ), Ak = A0 - 2kC 1 1 0 -1 I11 - I11 0 0 A0 = 0 - 1 1 , C = 0 I1 0 I22 22 2 I23 I2 1 -1 - I22 0 0 I23 22

and fn is an eigenvector of the matrix An with zero eigenvalue, i. e., An fn = 0. Proof. To prove this theorem, we use the algorithm develop ed in [15, 16]; it needs to b e modernize with our aim in view. We make the substitution of the variables and time
2 22 u2 = 2 , u3 = I23 2 , 2 3 s2 = , s3 = , I23 1 I23 2 d 2 = 1 2 I23 . dt In the new variables, the equations of motion assume the form 2 u1 = 1 , 1 , s1 = I23 2

(2.8)

u1 =

2 , I11

u2 =

2 , I22

s = U-1 A0 s,

(2.9)

where s = (s1 , s2 , s3 ) and the primes denote differentiation with resp ect to the new time . We seek the integral of system (2.9) in the form
3

Fn =
i=1

ui si f

(n) i

(u).

(2.10)

From the conservation condition for the integral (2.10), we obtain the following equation for the vector function fn : 2UD fn = A0 fn , where the differential op erator D = -
1 J11 u
1

(2.11)

+

1 J22 u

2

is introduced.
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An obvious solution of equation (2.11) a the constant vector f0 such that A0 f0 = 0, and the existence condition for this vector, det A0 = 0, coincides with the condition (2.6) for k = 1; thus, the theorem has b een proved for n = 0 (k = 1). To prove the theorem for an arbitrary n, we note that the substitution f1 = A-1 Uf 0 brings the equations for f1 to a form similar to (2.11): 2UD f1 = A1 f1 , A1 = A0 - 2C. (2.13)
1

(2.12)

The solution of equation (2.13) is the eigenvector of the matrix A1 with zero eigenvalue. By carrying out the substitution (2.12) sequentially (n - 1) times for matrices A0 , A1 ,... , An-1 , we obtain again an equation of the form (2.11), whose solution can b e found from the equation An fn = 0 and whose condition of existence exactly coincides with the condition (2.6) at odd k = 2n +1. The question of existence of an additional algebraic integral at even k remains op en. Absolute dynamics. As already noted ab ove, motion in absolute space (i. e., the motion of the b ody relative to motionless axes) is a transition from one steady rotation (at t -) to another (at t -). The rotational axis in a coordinate system fixed to the b ody changes then its direction to an opp osite one, while, in a motionless coordinate system, the rotational axis turns by a certain angle. Thus, the issue of absolute motion in the Suslov problem can b e interpreted as a scattering problem. For the particular case of I13 = 0 (or, accordingly, I23 = 0), an explicit analytical expression for the turning angle of the axis was recently obtained [18]. Is was shown that the angle b etween the axes of limiting steady rotations is energyindep endent and, at I13 = 0, is determined by the relationship cos k= = cos 2 I22 (I11 - I22 I23 k k ch , 2 2d ) I11 - I22 , d= . I22

(2.14)

As can b e seen from (2.14), at odd k (i. e., with the ab ove-noted hand side vanishes and the rotational axis in b oth the absolute and the system changes its direction to an opp osite one, i. e., = ± . The rotational-axis turning angle in the general case, I23 , I13 = 0 remains

integral present), the rightfixed-to-the-b ody coordinate question of the value of the op en as yet.

3. THE ROLLING OF A DYNAMICALLY ASYMMETRIC SPHERE OVER A PLANE WITH THE SUSLOV CONSTRAINT IMPOSED In his unpublished work, S. A Chaplygin considered a modification of the problem (which he had brilliantly solved b efore) of the motion of a dynamically asymmetric equilibrated sphere over a plane, as a nontwisted filament is attached to this sphere. The year of writing this pap er is not exactly known; however, the study was obviously done b efore 1903, b ecause S. A. Chaplygin later passed to problems of fluid mechanics. In the case at hand, two nonintegrable (nonholonomic) constraints are present: first, the no-slip condition at the p oint of contact b etween the sphere and the plane and, second, the vanishing pro jection of the angular velocity to some axis fixed to the b ody. Although the implementation of the constraint suggested by Chaplygin (to all app earances, indep endently of Suslov) is also not quite correct, the problem is highly instructive in terms of explicit integration and sp ecial techniques of asymptotic analysis. In addition, Borisov and Mamaev [20] suggested a correct implementation of these two constraint based on a "compact" version of Chaplygin's problem of the sphere (spherical susp ension) and implementing the Suslov constraint with the use of thin wheels, similarly to Wagner's implementation [19], see Fig. 4. Regrettably, Chaplygin did not make progress in solving this problem, so that the question of analyzing it still remains unresolved. In the general case, the problem does not seem to b e integrable and to have an (even singular) invariant measure. We note here only a straightforward observation
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Fig. 4. A p ossible implementation of the system where the dynamics of Chaplygin's sphere is sub ject to Suslov's constraint.

that makes it p ossible to reduce the analysis of this system to a scattering problem and formulate the op en questions more accurately. The equations of motion of the system with the presence of the two ab ove-noted constraints in a coordinate system fixed to the sphere are -(I13 1 + I23 2 )2 1 , I = (3.1) 2 (I13 1 + I23 2 )1 = â , where is the vector normal to the plane, = (1 , I is the tensor of inertia with resp ect to the contact p 2 2 I11 + mR2 (2 + 3 ) I= -mR2 1 2 I22 2 , 0) is the angular velocity of the sphere, oint, 2 -mR 1 2 , 2 ( 2 + 2 ) + mR 1 3

m and R are the mass and radius of the sphere, and Iij are the comp onents of the inertia tensor relative to the center of mass; the coordinate axes are chosen so that I12 = 0. Equations (3.1) have obviously the integrals of motion 1 (I, ), energy, 2 ( , ) = 1, geometric integral, H= (3.2)

where = (1 , 2 ). For Eulerian­Jacobian integrability, one more additional integral and an invariant measure are lacking. However, neither an invariant measure (even singular), nor the first integrals noted in the preceding section cannot b e directly generalized to system (3.1), in which, in contrast to the preceding case, the equations for the angular velocity cannot b e separated. The common level of the first integrals (3.2), Mh = , | H = h, 2 = 1 , is generally a three-dimensional manifold, which is pro jected onto the plane of angular velocities (1 ,2 ) within the region b ounded by the ellipses (see Fig. 5):
2 2 1 : 2h = I11 1 + I22 2 , 2 2 2 : 2h = (I11 + mR2 )1 +(I22 + mR2 )2 .

According to (3.1), since I is p ositively definite, under the condition I13 1 + I23 2 = 0,
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Fig. 5. The region of p ossible motions at a fixed energy.

Fig. 6. The characteristic form of the pro jections of the tra jectories onto the plane (1 , 2 ) at I I22 = 1.5, I13 = 0.7, I23 = 1.2, m = 10, R = 1, h = 10.

11

= 1,

the angular velocities remain constant, 1 , 2 = const, (as known, the angular velocity in this case is constant in the motionless coordinate system). Equation (3.3) (as can b e seen from Fig. 5) determines, inside Mh , a pair of invariant two-dimensional submanifolds N u , N s , which are diffeomorphic to the sphere. By calculating the derivative of the quantity I13 1 + I23 2 in the neighb orhood of these manifolds, it can b e shown that the submanifold N u is unstable, while N s is stable. We now show the validity of the following Prop osition 3. Any trajectory of system (2.14) goes to N s at t + and to N
u

at t -.

Proof. We make the linear substitution of variables (1.8) and pass to p olar coordinates in the plane (X, Y ): X = cos , Y = sin . Then the manifold N u is determined by the relationship = and N s by the relationship = 0. The derivative of the angle can b e represented as =- (A11 X 2 +2A12 XY + A22 Y 2 ) Y = -k(X, Y )Y, 2 2 (I11 I23 + I22 I13 )B (X 2 + Y 2 )

2 2 2 2 2 B = I11 I22 + d I11 1 + 3 )+ I22 (2 + 3 ) + d2 3 , 2 2 2 2 2 A11 = I13 I22 + I11 I23 + (I13 + I23 )3 +(2 I23 + I13 1 )2 d, 2 2 22 22 2 A22 = I11 I22 (I13 I22 + I11 I23 )+ (I11 I23 + I13 I22 )3 +(-I13 I22 2 + I11 I23 1 )2 d,

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2 A12 = (2 I23 + I13 1 )(-I13 I22 2 + I11 I23 1 )+ I13 I23 (-I22 + I11 )1 d,

115

where d = mR2 . According to these relationships, everywhere in the region of p ossible motions in the plane (X, Y ), the function k(X, Y ) > 0; therefore, the function (t) monotonically decreases in the upp er half-plane (Y > 0) and monotonically increases in the lower half-plane (Y < 0). It can b e shown in addition that the function (t) approaches arbitrarily closely the value + = 0 at t + and the value - = at t -. The characteristic form of the pro jections of the tra jectories of the system onto the plane (1 , 2 ) is shown in Fig. 6. It can easily b e understood that the invariable manifolds N u and N s are filled with p eriodic tra jectories, for which the vector is constant and the vector circumscrib es circles on the sphere, around the axis sp ecified by the vector a = (I23 , -I13 , 0) (parallel to ). Thus, for any tra jectory of the system (t) = (t), (t) , the limits cos + = lim and cos - = lim
t- t+

(t),

a |a|

(t),

a |a|

, where + , - are certain constants. S : -

Remark 4. It app ears at first glance that the one-dimensional scattering mapping
+

could naturally b e defined. Nevertheless, as numerical simulations show, except the angle, the "phase" of scattering is also imp ortant, and the correctly defined scattering mapping must b e two-dimensional. We are grateful to A. Maciejewski and M. Przybylska for useful comments made during our rep eated discussions at the seminars of the Institute of Computer Science in Izhevsk and to Yu. N. Fedorov for helpful discussions. This work was done as part of the Federal Program "Scientists and Science Teachers of Innovative Russia" for 2009­2013 (GK 02.740.11.0195). The work of A. A. Kilin was subsidized by the grants of the President of the Russian Federation for the Supp ort of Young Russian Scientists -- Candidates of Science (MK-6376.2008.1). The work is supp orted by the Grant of The Government of the Russian Federation for the supp ort of the scientific research pro ject implemented under the sup ervision of leading scientists at Russian institutions of higher education (11.G34.31.0039). REFERENCES
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