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Non-holonomic dynamics and Poisson geometry

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Russian Math . Surveys 69:3 481­538 DOI 10.1070/RM2014v069n03ABEH004899

Uspekhi Mat . Nauk 69:3 87­144

Non-holonomic dynamics and Poisson geometry
A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov
Abstract. This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie­Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them. Bibliography: 95 titles. Keywords: non-holonomic systems, Poisson bracket, Chaplygin ball, Suslov system, Veselova system.

Contents 1. Introduction 1.1. Main definitions 1.2. The brackets corresponding to e(3) and rank-4 Poisson structures 1.3. The geometric formulation of the reducing multiplier method 1.4. Reducing the Poisson bracket to a canonical form 2. Motion of a rigid body on a plane 2.1. The Chaplygin ball 2.2. The conformally Hamiltonian representation 2.3. Reducing the Poisson bracket to a canonical form 2.4. Gyrostatic generalizations 3. The Chaplygin ball on a sphere: the BMF system 3.1. The Poisson bracket and a conformally Hamiltonian representation 3.2. Reducing the Poisson bracket to a canonical form 4. The Veselova system 4.1. Equivalence of the Veselova system and the Chaplygin ball problem 4.2. A gyrostatic generalization 482 484 486 488 492 495 497 498 498 501 501 502 503 504 505 507

The research of the first author was carried out in the framework of the state contract "Regular and Chaotic Dynamics" with Udmurt State University. The research of the second author was supp orted by grant no. 14-19-01303 of the Russian Science Foundation ("Dynamics and Control of Mobile Rob ototechnological Systems"). AMS 2010 Mathematics Subject Classification . Primary 70F25, 70G45; Secondary 53D17.

c 2014 Russian Academy of Sciences (DoM), London Mathematical So ciety, Turpion Ltd.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

5. A body of revolution on the plane 5.1. Hamiltonizing the reduced system 5.2. Rank-4 Poisson structure 5.3. Reducing the Poisson bracket to a canonical form 5.4. Gyrostatic generalizations 5.5. The Routh sphere 6. Dynamically symmetric ball on a surface of revolution 6.1. The Poisson bracket for a ball on a surface of revolution 6.2. Hamiltonization on a reduced submanifold 6.3. Reducing the Poisson bracket to a canonical form 6.4. Motion along a paraboloid of revolution 7. The Suslov system 7.1. Poisson structures for inertial motion 7.2. A special case of a motion in a potential field 8. Conclusion Bibliography 1. Intro duction

507 510 512 515 516 516 520 523 525 526 527 528 529 530 531 532

In this survey we consider some dynamical systems in non-holonomic mechanics which can be described using methods of Poisson geometry [21]. Without going into detail, we note that non-holonomic mechanics and non-holonomic geometry appeared at the end of the 19th and beginning of the 20th century and we currently see a continuing increase of new studies in this field (see [6], [10], [12], [16]­[19], [22]­[24], [26], [27], [36], [39]­[41], [45], [73], [78], [81]­[85]). Let us look at the difference between non-holonomic mechanics and non-holonomic geometry. In either case we deal with some differential equations generalizing the familiar Lagrange and Hamilton equations in standard classical mechanics. But in non-holonomic mechanics these equations are postulated on the basis of equations of ideal non-integrable (non-holonomic) constraints and are deduced from the D'Alembert­Lagrange principle, while in non-holonomic geometry the differential equations are deduced using Hamilton's variational principle with the constraints taken into account (although in older papers -- for instance, [86] -- `non-holonomic geometry' means in fact non-holonomic mechanics). Non-holonomic mechanics is used to describe dynamical systems that involve rolling (in which a non-holonomic constraint is used to express the absence of slipping: the point of contact has velocity zero). Non-holonomic geometry, also known as sub-Riemannian geometry (see a modern introduction to the theory in [10], [51], [52], [64], [88]­[91]), can be used in the investigation of optimal control problems (see [1]) and in the description of systems actually arising in fluid dynamics (ones in which the inertia tensor is strongly anisotropic). The latter field of study has come to be called vaconomic mechanics, and the reader can learn about it in some of the papers cited in [7]. We also mention [8], where a 1-parameter family of model systems with constraints was discovered which `links' non-holonomic geometry and classical non-holonomic mechanics. In [11] unifying vaconomic equations for non-holonomic systems were obtained that can be used in the optimal control of systems with rolling (for example, a spherical robot).


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Authors sometimes ignore the distinction between non-holonomic mechanics and non-holonomic geometry (vaconomic mechanics), and this can result in conclusions contradicting the principles of mechanics (for instance, see [43], where vaconomic equations are used in place of non-holonomic ones to describe a disc rolling on a plane). We note, however, that some tra jectories can be the same in non-holonomic and vaconomic models [59]. The equations of non-holonomic (sub-Riemannian) geometry are indeed Hamiltonian, but the corresponding Hamiltonian is degenerate with respect to the momenta, which leads to new geometric constructions (abnormal extremals and so on). The equations of non-holonomic mechanics are not Hamiltonian in the general case: they contain non-holonomic terms which cannot be removed. However, when a problem has some additional symmetry (in the mechanics context this can occur for special values of the parameters describing the dynamical and geometric properties of the system) one can obtain a representation of the system in the Hamiltonian or conformally Hamiltonian form. Depending on the parameters of the system, a hierarchy of the dynamical behaviour of non-holonomic systems arises (this was first proposed in [23]). The Poisson brackets occurring there can have a highly non-trivial non-linear structure, and they are of independent interest for the general theory of Poisson structures [21]. An alternative standpoint was taken in the survey paper [36] (see also the bibliography there), where an almost-Poissonian approach was used to describe non-holonomic systems. This involves postulating a Poisson-type bracket which, however, does not satisfy the Jacobi identity. Still, we should underscore that this approach has brought no interesting results so far. Our approach develops the idea of a hierarchy of the dynamical behaviour of non-holonomic systems and is based on an analysis of the tensor invariants of the equations of motion. In essence, it was first sketched by Kozlov [53], [55] and was systematically pursued in [23], [24], [26], and [27]. The tensor invariants can be scalar functions (first integrals), symmetry fields, Poisson structures (mostly degenerate ones), and invariant measures. For non-holonomic systems their existence is implicit (and cannot be easily derived from symmetries). We note that as long ago as the 19th century it was noted (by Hertz and Poincar´ that one cannot apply e) Hamilton's variational principle to non-holonomic systems, and it only became clear later (Chaplygin [29]; see also [58]) that a theorem analogous to Noether's can be generalized to non-holonomic systems under certain natural additional constraints. The search for and the analysis of hidden tensor invariants became possible only when powerful systems of analytic calculations were developed. The use of this approach to the problem of non-holonomic rolling of a rigid body on a plane or a sphere was shown in [23], [24], [26], and [27], where the authors introduced the notion of conformally Hamiltonian systems, which turn into Hamiltonian systems after a suitable `nice' change of time. As a result, the problem of `Hamiltonizing' non-holonomic system arises: find a necessary and sufficient number of tensor invariants which enable one to write the system in conformally Hamiltonian form. Some parts of these investigations (closely related to the existence of invariant measures and integrability) overlap with earlier results due to Veselov, Veselova, and Fedorov. From the analytic point of view the problem of Hamiltonization was treated in [12], [54], and [56], and in this case we can distinguish in it a `semilocal' aspect (in a neighbourhood of an invariant manifold) and a global aspect.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

In this survey paper, intended mainly for mathematicians, we do not dwell on deriving the equations of motion and the physical principles of non-holonomic mechanics; for this the reader can consult an excellent overview in [5]. We mainly consider classical formulations of problems which go back to Routh, Appel, Chaplygin, Suslov, and Vagner. They describe mechanical systems related to rolling of bodies without slipping (this condition results in a non-holonomic constraint), and the corresponding equations of motion are well known. We systematize the known results for these systems and present several new results related to their reduction to Poisson (Hamiltonian) form, and we also discuss phenomena obstructing such a reduction. In some cases one can only find explicitly a conformally Hamiltonian decomposition, and an understanding of what results in Poisson geometry and dynamics can be generalized to such systems is yet to come. As noted in [25] and [26], not only is non-holonomic dynamics interesting from the standpoint of applications of the well-developed machinery of Hamiltonian dynamics: it also raises a number of interesting theoretical questions in the domain where the system is very close to Hamiltonian. On this path we can see remarkable connections with the theory of reversible systems, the theory of invariant measures, and some new non-linear Poisson structures (whose symplectic leaves can display chaotic behaviour) which are of great interest to mathematics in general. 1.1. Main definitions. A dynamical system in a phase space M , dim M = n, with coordinates (x1 , . . . , xn ) = x is determined by the equations of motion xi = Xi (x), i = 1, . . . , n. (1)

This system of differential equations defines the vector field
n

X=
i=1

Xi

, xi

(2)

which is a linear operator acting in the space of smooth functions on the manifold M , and which determines the evolution of an arbitrary smooth function F (x):
n

F = X (F ) =
i=1

Xi

F . xi

For a Hamiltonian system the vector field X = P dH (3)

is defined in terms of the Hamiltonian function H (x) and the Poisson bivector (or tensor ) P (x), which defines the Poisson bracket on M by the formula
n

{f , g }P =
i,j =1

Pij (x)

f g . xi xj

In what follows we shall drop the subscript P of the bracket if this will not cause confusion.


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Since the Poisson bracket is bilinear, antisymmetric, and satisfies the Jacobi identity, the coefficients of P satisfy Pij = -Pj i ,
m

P

im

Pj k +P xm

jm

Pki +P xm

km

Pij xm

= 0,

where i, j, k = 1, . . . , n. It is known [60] that these conditions can be expressed in a compact form in terms of the Schouten bracket [ · , · ] as [P, P ] = 0. We recall that the Schouten bracket [A, B ] of two bivectors A and B is the trivector whose elements have the following form in local coordinates x on M :
dim M

[A, B ]

ij k

=-
m=1

Bmk

Aij +A xm

mk

Bij + cycle(i, j, k ) . xm

(4)

The better-known Poisson bivectors are: · the constant bivector giving the canonical Poisson bracket, P= 0 -I I , 0
m

{f , g } =
i=1

f g f g - , qi pi pi qi

x = (q , p);

(5)

· the linear bivector giving a Lie­Poisson bracket,
n n

Pij =
k=1

Gk xk , ij

{xi , xj } =
k=1

Gk xk , ij

(6)

where the Gk are the structure constants of a Lie algebra. The reader can find ij a detailed guide on Poisson structures in Hamiltonian dynamics together with a comprehensive bibliography in [21]. One natural generalization of a Hamiltonian system is a conformal ly Hamiltonian system, in which the vector field in question has the representation X = g (x)P dH, (7)

where P is a Poisson tensor, and the function g (x), the reducing multiplier, is usually taken to be positive definite [6], [63]. In some cases even when we can prove theoretically that a system is Hamiltonian or conformally Hamiltonian, we cannot find an explicit representation (3) or (7). Let us therefore consider more general systems with a set of first integrals H1 (x), . . . , Hm (x) which can be used to express the vector field in terms of a Poisson tensor having the standard properties: X = g1 (x)P dH1 + · · · + gm (x)P dHm , [P, P ] = 0, {Hi , Hj }P = 0 i, j. (8)

In what follows we call such a representation a decomposition into conformal ly Hamiltonian fields (or a conformally Hamiltonian decomposition). Of course, whether we can use such a decomposition for studying the dynamics depends on the particular form of the coefficients gi in the decomposition (which are functions defined on the whole phase space).


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Remark 1. For many a priori non-Hamiltonian systems which nevertheless have a first integral H we can write the vector field X in the antisymmetric form X = P dH if we have found a suitable antisymmetric (2, 0)-tensor field P , which does not satisfy the Jacobi identity in general. Some authors call such a representation almost Hamiltonian [45], [46], [62], [87]. For example, it arises in non-holonomic mechanics. However, then we must do without all the methods of investigation developed in symplectic (Poisson) geometry and topology. For instance, we must give up identifying level sets of the integrals of motion as Lagrangian submanifolds. The aim of this paper is to represent various systems in non-holonomic mechanics in one of the possible forms (3), (7), or (8). The non-linear Poisson bivector arising on this path will usually be a deformation of the Lie­Poisson bracket of the algebra e(3). Using the theory of Poisson deformations, one can reduce different non-holonomic systems to dynamical systems on well-understood phase spaces with the canonical Poisson bracket. This makes it possible to compare different non-holonomic integrable systems with each other and to use fairly advanced methods of Poisson geometry and topology in investigating them [4], [91]. 1.2. The brackets corresp onding to e(3) and rank-4 Poisson structures. For most of the systems considered below, the phase space is diffeomorphic to the 4-manifold T S 2 . It is well known to admit a standard embedding in R6 which has a Poisson structure corresponding to the Lie algebra e(3) of the group E (3) of motions. We recall that this Lie­Poisson bracket has the form {Li , Lj }0 =
ij k

Lk ,

{Li , j }0 =

ij k k

,

{i , j }0 = 0,

(9)

where ij k is a totally antisymmetric tensor. The corresponding canonical Poisson bivector on e (3) is 0 P0 = . (10) L This definition involves the antisymmetric 3 â 3 matrices 0 3 - 2 0 L3 0 1 , 0 = -3 L = -L3 2 -1 0 L2 -L1 which are used to define the standard isomorphism z1 0 z3 0 z = z2 Z = -z3 z3 z2 -z1 -z2 z1 0 -L2 L1 , 0

between the two 3-dimensional Lie algebras (R3 , a â b) with cross product and (so(3), [a, b]) with matrix commutator. We recall that the Poisson bracket (9) has two Casimir functions, C1 = ( , ), C2 = ( , L), P0 dCi = 0, i = 1, 2,

and that the symplectic leaves of the canonical Poisson bivector P0 on the level set C2 = 0 are symplectomorphic to the cotangent bundle T S 2 of the 2-sphere.


Non-holonomic dynamics and Poisson geometry

487

Remark 2. Almost all the Poisson brackets found below (including non-linear ones) can be reduced to the Lie­Poisson bracket corresponding to e(3). For a dynamical system that is integrable by the Euler­Jacobi last-multiplier theorem, the description of its behaviour in a neighbourhood of a compact integral manifold 2 = {x M : H1 (x) = h1 , . . . , H h
n-2

(x) = hn

-2

},

h k C,

is based on the classical results on dynamical systems on a 2-torus due to Poincar´ e, Siegel, and Kolmogorov (see, for instance, [5], [12], [57]). We recall that if a connected component of a level set of integrals of motion is diffeomorphic to a 2-torus and if we have a dynamical system with an invariant measure on this torus, then by Kolmogorov's theorem [49] we can introduce coordinate variables 1 , 2 (mod 2 ) in which the system assumes the form c1 c2 1 = , 2 = , (11) where is a smooth positive function on the torus and the c1,2 are some constants. By making the change of time dt d one can linearize the equations of motion so that the motion will take place along a straight-line winding on the torus, although it will not necessarily be uniform [53]. In a neighbourhood of a non-singular torus, after the change of time -1 dt d the angle variables 1 , 2 can be supplemented by the canonical action variables I1 , I2 [12], so that the vector field (11) is conformally Hamiltonian: X = g1 P dH1 , g1 = -1 , of rank 4 0 1 . 0 0

with respect to the canonical Poincar´ bivector e 0 01 0 00 P = -1 0 0 0 -1 0

However, so far in the general case no effective algorithm has been obtained for finding the Poisson structure and the conformally Hamiltonian representation of the field X in terms of the original physical variables corresponding to this structure. Among such effective methods is Chaplygin's reducing multiplier method , which enables one to effectively find the Poisson bracket on a completely reduced phase space M /G [24]. In the case of a partially reduced space M /G1 we shall use the method in [81]­[83] for finding Poisson brackets. From the point of view of mathematics, the central idea in this method is to identify common level surfaces of the integrals of motion and the Lagrangian foliation by symplectic leaves of the required Poisson structure. Indeed, if a vector field X on an n-dimensional phase space (1) has n - 2 integrals of motion H1 , . . . , Hn-2 and an integral invariant (for example, an invariant measure), then this vector field is integrable in quadratures by the Euler­Jacobi theorem. We shall assume that level sets of the integrals are Lagrangian surfaces in symplectic leaves of some Poisson bivector P , just as in the case of Liouville-integrable


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Hamiltonian systems. Since the dimension of a Lagrangian submanifold is half the dimension of the symplectic leaf, we assume that rank P = 4 almost everywhere. Examples of rank-2 brackets can be found in [37] and [66]. Remark 3. Points in a Poisson manifold at which rank P is maximal (and equal to 4 in our case) are said to be regular, and points where rank P is less than its maximum value are said to be singular. From the point of view of dynamics, singular points usually coincide with (stable) equilibria, and lower-dimensional symplectic leaves (thin orbits) and additional Casimir functions appear at these points. In most of the examples below, the completely or partially reduced phase space has dimension 6, dim M /G = 6, with local coordinates (1 , 2 , 3 , M1 , M2 , M3 ). The vector field X on this manifold is integrable in the Euler­Jacobi sense and has four integrals of motion H1 , . . . , H4 and an invariant measure. To construct the required bivector P we must solve the system of equations [P, P ] = 0, {Hk , Hm } = 0, k , m = 1, . . . , 4, (12)

where [ · , · ] is the Schouten bracket. Here the main problem lies with the fact that a priori the system of equation (12) has infinitely many solutions [79]. Thus, we confine ourselves to bivectors P which are linear in M1 , M2 , and M3 :
3

Pij =
k=1

ck ( )Mk + dij ( ), ij

(13)

where the ck and dij are unknown functions of . This restricts the space in which ij solutions of (12) should be sought and enables us to find particular solutions. We remark that in many cases in our search for a Poisson structure, among the first integrals H1 , . . . , H4 we can choose a pair of Casimir functions on physical grounds. One example is 2 and (M , ) in the problem of the Chaplygin ball [22]. Here we do not use this approach, and by analogy with the search for a bi-Hamiltonian structure in the problem of the Lagrange top [80], we assume that all the integrals H1 , . . . , H4 are on equal terms. 1.3. The geometric formulation of the reducing multiplier metho d. Generalized Chaplygin systems. We recall that, in accordance with [24], a generalized Chaplygin system is a mechanical system with two degrees of freedom whose equations of motion can be written as d L dt q1 - L = q2 S , q1 d L dt q2 - L = -q1 S, q2 (14)

S = a1 (q )q1 + a2 (q )q2 + b(q ), where L is a function of the generalized coordinates q = (q1 , q2 ) and the velocities q = (q1 , q2 ); we shall also call L the Lagrangian of the system. It can easily be verified that this system has an energy integral of the usual form: E=
i

L qi - L. qi

(15)


Non-holonomic dynamics and Poisson geometry

489

Remark 4. The ordinary Chaplygin system is given by a special choice of the function S (where b(q ) = 0 in any case) [32]. A slightly different generalization of Chaplygin systems was proposed in [28], [48], and [76]. It is known that if there exists an invariant measure with density depending only on the coordinates, then the system can be represented in the conformally Hamiltonian form [24] (for b(q ) = 0 this was proved by Chaplygin himself [32]). To show this we take the Legendre transform for the original system (14), Pi = L , qi H=
i

Pi qi - L

qi Pi

,

so that the equations of motion (14) can now be written as qi = H H H H H , P1 = - + S, P2 = - - S, Pi q1 P2 q2 P1 S = a1 (q )q1 + a2 (q )q2 + b(q ) = A1 (q )P1 + A2 (q )P2 + B (q ),

(16)

where H coincides with the energy integral (15) expressed in terms of the new variables. Assume now that the system has an invariant measure with density depending only on the coordinates: µ = N (q ) dP1 dP2 dq1 dq2 . Then Liouville's equation for N (q ) is taken to the form q1 1 N 1 N - A2 ( q ) + q 2 + A1 (q ) N q1 N q2 = 0, (17)

and since N depends only on the coordinates, each expression in parentheses must vanish: 1 N 1 N - A2 (q ) = 0, + A1 (q ) = 0. (18) N q1 N q2 The condition for the local solvability of (18) has the form A2 A1 + = 0. q1 q2 Making the change of variables Pi = pi , N (q ) i = 1, 2,

we obtain the following relations for the derivatives: H H =N , Pi pi H H 1 N = + qi qi N qi H H p1 + p p1 p2
2

,

where H (q , p) = H q , P (q , p) is the Hamiltonian expressed in the new variables.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Substituting the above relations in (16) and using (18), we get that H , pi H H p1 = N (q ) - + N (q )B (q ) , q1 p2 qi = N (q ) p2 = N (q ) - H H - N (q )B (q ) . q2 p1

Thus, we have arrived at the following result. Theorem 1. If the system (16) has an invariant measure of the form (17) , then it can be represented in the conformal ly Hamiltonian form qi = N (q ){qi , H }, with the Poisson bracket defined by {qi , pj } = ij , {qi , qj } = 0, {p1 , p2 } = N (q )B (q ). pi = N (q ){pi , H }, i = 1, 2,

Proof . The proof reduces to a direct verification of the Jacobi identity. The Chaplygin system on T S 2 . Assume that the system is described by a pair of 3-dimensional vectors M , and the equations of motion have the form H H +â , M = (M - S ) â M =â H , M (19)

where the `Hamiltonian' H (M , ) is an arbitrary function (quadratic in M ) and S (M , ) is a function linear in M : S = K ( ), M = K1 ( )M1 + K2 ( )M2 + K3 ( )M3 . A direct verification shows that (19) always has three integrals of motion: F1 = 2 = const, F2 = (M , ) = const, F3 = const.

Without loss of generality we can set 2 = 1, so that equations (18) will describe a dynamical system on a family of 4-manifolds Mc4 = {M , | 2 = 1, (M , ) = c}, each of which is diffeomorphic to T S 2 . Letting x = ( , M ) denote the full system of variables, we can represent (19) in the standard antisymmetric form x=P with respect to the Poisson bivector P0 = 0 0 - S ( x) M 0 0 ,
0

dH dx


Non-holonomic dynamics and Poisson geometry

491

where 0 M = M3 -M 2 -M 3 0 M1 M2 -M 1 0 0 and = 3 - 2 -3 0 1 2 - 1 . 0

Here the first term is the standard Poisson structure corresponding to the algebra e(3). In addition, P0 satisfies the equations P
0

F2 F1 = 0 and P0 = 0. x x

As above, we assume that (19) has an invariant measure whose density depends only on : µ = ( ) dM d . (20) Then Liouville's equation for the vector field V (M , ) given by (19) assumes the form H div V = , â K - â = 0. M Since the Hamiltonian is non-degenerate with respect to M , we therefore obtain the vector equation 1 - K â = 0, (21) which can be used to prove the following result. Prop osition 1. If the function ( ) satisfies (21) , then the tensor P = satisfies the Jacobi identity . In this way we finally obtain the following theorem. Theorem 2. If the system (19) has an invariant measure (20) whose density depends only on , then it can be represented in the conformal ly Hamiltonian form x = ( )P (x) where P (x) =
-1

1 P ( )

0

H , x

P0 (x) is a Poisson structure of rank 4 with the Casimir functions F1 =
2

and

F2 = (M , ).

We can solve (21) with respect to the vector K as follows: K = f ( ) + 1 ,

where f ( ) is an arbitrary function. Thus, we have obtained in a natural way a special class of Poisson structures on R6 (M , ) that can be expressed as follows: P= 1 0 - M 1 + f ( )( , M ) 2 0 0 0 . (22)

This type of Poisson structure is discussed in [84] from the standpoint of deformations of the canonical Poisson bracket.


492

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Remark 5. Adding a term of the form ( ) 0 0 0

to the bivector (22), where ( ) can be an arbitrary function, we also preserve the Jacobi identity. 1.4. Reducing the Poisson bracket to a canonical form. Setting g = -1 , one can express the Poisson structure (22) in the following form, which is more convenient for analysis: P =g 0 + M g - f · , M 0 0 0 . (23)

We now look more closely at the class of such Poisson structures. First of all, we point out that these structures are parametrized by two arbitrary functions g ( ) > 0 and f ( ). Hence we denote the corresponding Poisson structure by Pg,f . All these Pg,f have the same Casimir functions (M , ) and ( , ). For simplicity we confine ourselves to the physical case 2 = ( , ) = 1, that is, we restrict all the ob jects to the 5-dimensional (Poisson) manifold S 2 ( ) â R3 (M ). It is one of our aims to find a canonical form to which we can reduce these Poisson structures. First we note that the symplectic leaves of Pg,f are diffeomorphic to the cotangent bundle T S 2 of the sphere. We can conclude from the explicit expression (13) for the Poisson structure that the symplectic structure on a leaf T S 2 must be the sum of the canonical form dp dq and some magnetic component, that is, a closed 2-form magn on the sphere. By Moser's theorem [65], all such forms magn are parametrized, up to a symplectomorphism, by a single quantity, their integral over the sphere: S 2 magn . Thus, for each Poisson structure we have a 1-parameter family of symplectic leaves whose type is described by a single parameter. This observation prompts the conjecture that by `rearranging' the symplectic leaves when necessary, and applying some symplectomorphism to each separate leaf, we can transform any structure Pg,f into any other structure Pg,f . Remark 6. On the zero level (M , ) = 0 the Poisson structure (23) is reduced to the canonical form corresponding to the Lie algebra e(3) by the simple substitution [24] (M , ) g
-1

( )M , .

Remark 7. The whole range of these problems is also closely related to Novikov's papers [70], [71]. We start by describing a class of natural transformations that preserve the form of the Poisson tensor Pg,f but change the parameters g and f . For example, consider transformations of the form (M , ) (M , ), M = A( )M , (24)

where A( ) is a linear operator on R3 whose components depend on .


Non-holonomic dynamics and Poisson geometry

493

Prop osition 2. For each point S 2 consider the orthogonal decomposition M = M + M , where M = (M , ) is the projection of M on and M = M - M is the projection of M on the plane orthogonal to . Let M = ( )M + cM + M â h( ), where c is a constant , ( ) > 0 is an arbitrary scalar function , and h( ) is some vector-valued function of . Then (24) transforms the Poisson structure Pg,f into a Poisson structure Pg,f of similar form whose parameters are defined as fol lows : g = g , f= 2 f+ -1 c c g - , g + h 1 + g 2 rot , g c g . (25)

Proof . The proof consists in a direct verification. We confine ourselves to a comment about the geometric meaning of the transformation M M used in this result. Let us pick an orthonormal basis e1 , e2 , e3 related to in R3 (M ). Namely, e1 and e2 are two orthonormal vectors in the tangent plane to the unit sphere at the point , and e3 is the unit normal to the sphere at this point, so that e3 = . In this basis the operator A has the matrix 0a A = 0 b , 00c where , a, and b depend on and c is a constant. This is just the general form of a transformation A satisfying our requirements. Indeed, the Casimir function (M , ) must be taken into itself up to a multiplicative constant, which is just c. Then the plane given by the equation (M , ) = 0 must be taken into itself, while in the orthogonal direction the transformation must be the dilation with coefficient c, where c must be independent of . These conditions completely determine the last line of A. Furthermore, the relations {Mi , j } = -g · ij k k can be formally rewritten in vector form as {M , } = -g M â . Since they keep this form, we obtain the condition g A( )M â = g M â , which means precisely that the operator acts as multiplication by some number (depending on ) on the tangent plane to the sphere. There are no restrictions on the elements a and b; they are determined by the vector-valued function h (it has three components, but only two of them are essential, because we do not change anything by adding to h a vector proportional to ). The following factor is useful to take into account: the set of transformations described in Proposition 2 forms a group (an infinite-dimensional group, of course, since its parameters involve the arbitrary functions and h). We can easily verify that making two transformations with parameters (1 , c1 , h1 ) and (2 , c2 , h2 ) one after the other is equivalent to the transformation with the parameters (1 2 , c1 c2 , h1 2 + h2 c1 ). This defines a group law which is simply a reproduction of matrix multiplication: 2 0 h2 c2 1 0 h1 c1 = 1 2 0 h1 2 + h2 c c1 c2
1

.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

This group acts in a natural way on the class of Poisson structures under consideration, or equivalently, on the space of parameters g , f . The above rules (25) are precisely the formulae for this action. Denoting the action formally by (g , f ) = (,c,h) (g , f ) and making two successive transformations, we can easily see that this action satisfies the standard rule: namely, if (g , f ) = then
(1 ,c1 ,h1 )

~~ (g , f ) and (g , f ) = ~~

(2 ,c2 ,h2 )

(g , f ),

~~ (g , f ) = ~~

(1 2 ,c1 c2 ,h1 2 +h2 c1 )

(g , f ).

For a direct verification it is convenient to write (25) as g = g , f= 2 g f + g - , c - g - , g + h g2 , rot c g ,

after which the verification is straightforward. From the standpoint of group theory it is natural to ask: what structure do the orbits of this action have ? In other words, we would like to know what Poisson structures can be transformed one into another by these transformations. The answer is quite simple: the action described has a unique orbit, that is, all the Poisson structures in the class under consideration are equivalent. In particular, we have the following result. Theorem 3. Each Poisson structure Pg,f of the form (23) on the level set 2 = 1 is isomorphic to the standard Poisson structure P1,0 corresponding to the Lie algebra e(3) . Proof . It is sufficient to choose the parameters (, c, h) of the transformation in (25) so that g = 1 and f = 0. The first condition immediately gives us the function , namely, = g -1 . The second condition now takes the much simpler form 2 f+ c or equivalently, 2 f + + , - c + ( , rot h) = 0, 1 -1 + , c c 1 + ( , rot h) = 0 c

where the constant c and the vector-valued function h are unknown. Then we can write the equation as ( , rot h) = F ( ) + c, (26) where F ( ) is some prescribed function, and the above condition need hold the unit sphere 2 = 1. Conditions for the solvability of such an equation known. From the point of view of differential geometry this simply means are looking for an antiderivative of a 2-form of type (F + c) d on the unit where d is the standard area form. We can find it if and only if S 2 (F + c) which can always be attained by picking a suitable c. only on are well that we sphere, d = 0,


Non-holonomic dynamics and Poisson geometry

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Remark 8. Similarly, we can reduce the bracket for Pg,f to a standard form on the whole of e (3), that is, without imposing the additional constraint 2 = 1, but then we must extend the class of transformations: we will have to assume that c depends on 2 . Since 2 is a Casimir function, in all the transformations we can treat c( 2 ) as a constant as before, so the formulae will not change much. The conditions for the solvability of the equation ( , rot h) = F ( ) + c( 2 ) remain the same, but now we must verify them on spheres of arbitrary radii. As previously, we can fulfill these conditions because the constants we shall select can now depend on the radius 2 . For example, in the problem mentioned above of a Chaplygin ball on a plane the function F ( ) in (26) has the form F ( ) = - (D D -1 - ( , A )) .

-1

3/2

In this case solutions of equation (26) with the unknowns c and h can be expressed in terms of complete and incomplete elliptic integrals. Thus, although it is not difficult to prove theoretically that the bracket for Pg,f can be reduced to a bracket on e(3), the resulting transformation can be very unwieldy and transcendental. An analogous reduction of the bracket for (23) to a bracket on e (3) was given in [84] and [85], where it was also noted that if we allow transformations which are singular on S 2 , then the solution of (26) can be explicitly expressed in terms of elementary functions. 2. Motion of a rigid b o dy on a plane In what follows we shall consider various cases of a rigid body rolling along a given surface under the action of external forces. We assume that during its motion the body makes contact with the surface only at a single point and rolls without slipping. It is known that such constraints are non-holonomic, and the corresponding equations of motion can be obtained using various approaches (see, for instance, [2], [3], [23], [30], [50], [68], [69], [75], [94]). In this section we give only the definitions and equations needed for the case of a rigid body rolling on a plane. We introduce two coordinate systems (see Fig. 1): · the fixed reference frame OX Y Z , with origin O located at some point in the plane and axis OZ orthogonal to the plane; · the body frame C xy z , with origin C at the centre of mass of the body and axes parallel to the principal axes of inertia. Let , , be the pro jections of the unit vectors of the coordinate axes of OX Y Z on the axes of the body frame C xy z , and let R = (R1 , R2 , R3 ) be the coordinates of the body's centre of mass in the fixed reference frame OX Y Z . Then for the orthogonal matrix 1 1 1 Q = 2 2 2 SO(3); (27) 3 3 3 the pair (R, Q) R3 SO(3) uniquely determines the position of the body. Let v and be the velocity of the centre of mass and the angular velocity of the body, which we shall consider in the body frame C xy z . Because of the absence of


496

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Figure 1

slipping at the point of contact, this point has velocity zero: v + â r = 0, (28)

where r is the radius vector from the centre of mass to the point of contact, also taken in the body frame. If the body is everywhere convex and its surface has equation f(r) = 0, then r can be uniquely expressed in terms of the vector using the inverse of the Gauss map: f(r) =- . (29) |f(r)| (This equation expresses the fact that the normal to the surface of the body is also normal to the plane.) In what follows we assume that r = r( ) is some given function. Using the equations for the dynamics of a rigid body in the body frame C xy z , and eliminating the support reaction using the constraint (28), we obtain a complete system of equations describing the evolution of the system: I + mr â ( â r) + mr â ( â r) = I â + m( , r) â r + MF , = â , = â , = â , R1 = (, v ), R2 = ( , v ), R3 = ( , v ),

(30)

where m is the mass of the body, I = diag(I1 , I2 , I3 ) is the inertia tensor with respect to the centre of mass, and MF is the moment of the external forces acting on the body. In what follows we assume that the external forces are potential forces with potential U depending only on , so that MF = â U .

In this case the system admits the symmetry group E (2) consisting of the translations parallel to the plane and the rotations about the vertical direction. Hence we can split off a 6-dimensional reduced system which describes the evolution of the


Non-holonomic dynamics and Poisson geometry

497

vectors and . (The components of these vectors with respect to the axes of the body frame are invariants of the action of E (2) on the phase space of the system.) Next we write the resulting equations in a more convenient form by introducing the angular momentum of the body with respect to the point of contact: M = I + mr â ( â r). (31)

Solving this equation for , we obtain equations on the reduced 6-dimensional phase space M with the coordinates x = ( , M ): U M = M â + mr â ( â r) + â , = â , AM + mr â (Ar â AM ) -1 , A = I + (r, r) ; = 1 - m(r, Ar)

(32)

these will be studied in what follows. The vector field X corresponding to the dynamical equations (32) has two integrals, 1 H = (M , ) + U ( ) and C = ( , ) = 1. (33) 2 The given system displays various types of dynamical behaviour, depending on the parameters and the relevant conservation laws: from the simplest regular behaviour of integrable systems when there exists a `complete set of conservation laws', to intricate chaotic behaviour which is characteristic for dissipative systems when there there are no invariants. For the possible spectrum of various patterns of behaviour of non-holonomic dynamical systems in the presence of various tensor invariants the term `hierarchy of dynamics' was proposed in [23] and [27]. In the special cases below, the vector field X has additional first integrals and an invariant measure. 2.1. The Chaplygin ball. We consider a rolling motion of a dynamically asymmetric balanced rigid body with spherical surface of radius R, the so-called Chaplygin ball [31]. We recall that a ball is said to be balanced if its centre of mass coincides with its geometric centre, and it is dynamically asymmetric if the three moments of inertia are distinct. In this case we obtain from (29) that r = -R , and in the absence of external forces the equations of motion (32) on the reduced 6-dimensional phase space take the form M = M â , = â , (34) where M = (I + dE) - d( , ) , d = mb2 . (35) The vector field X corresponding to equations (34) has four integrals of motion H1 = (M , ), and an invariant measure µ=g
-1

H2 = (M , M ),

H3 = ( , ),

H4 = ( , M )

(36)

( ) d dM ,

g ( ) =

1 - d( , A ) ,

(37)


498

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

where a1 A=0 0 0 a2 0 0 (I1 + d)- = (I + dE)-1 = 0 0 a3 0
1

0 (I2 + d) 0

-1

0 0 (I3 + d)


-1

.

Thus, X is integrable in quadratures by the Euler­Jacobi theorem. 2.2. The conformally Hamiltonian representation. By [22] and [24] the rank-4 Poisson bivector Pg = g ( ) 0 d 0 (M , A ) - 0 M g ( ) 0 (38)

for the Chaplygin ball involves the same 3 â 3 matrices 0 3 - 2 0 M3 0 1 , 0 = -3 M = -M3 2 - 1 0 M2 -M1

-M 2 M1 0

as the canonical Poisson bivector P0 in (10), and the function g g ( ) defines the density of the invariant measure (37). The corresponding Poisson bracket is {Mi , Mj }g =
ij k

g Mk -

d(M , A ) g

k

,

{Mi , j }g =

ij k

g k ,

{i , j }g = 0,

(39) where ij k is a totally antisymmetric tensor. Using this Poisson bracket on the completely reduced phase space, one can write the original equations of motion (34) as dxk Xk = g dt
-1

{H, xk }g ,

where H =

H1 , 2

(40)

H1 is the integral of motion (36), and as usual, x = (1 , 2 , 3 , M1 , M2 , M3 ). 2.3. Reducing the Poisson bracket to a canonical form. 2.3.1. The nature of the Poisson bracket (39) can be explained using a deformation of the canonical Poisson bracket on the cotangent bundle and the moment map (see also § 1.4). Indeed, if g (q ) is a function on some configuration space Q with coordinates q = (q1 , . . . , qn ), then replacing the conjugate variables p = (p1 , . . . , pn ) in the Liouville canonical 1-form and in the corresponding symplectic 2-form = p1 dq1 + · · · + pn dqn according to the rule pk g (q ) pk , we obtain the forms g = g (q )(p1 dq1 + · · · + pn dqn ) and g = dg . (41) and = d,


Non-holonomic dynamics and Poisson geometry

499

The corresponding Poisson bracket, {qi , qj }g = 0, {qi , pj }g = g ij , {pi , pj }g = pi j g - pj i g , (42)

where k = / qk , is a simple deformation of the original canonical Poisson bracket {qi , qj } = 0, {qi , pj } = ij , {pi , pj } = 0.

These questions were also discussed in [72]. If we now identify Q with the 2-sphere S 2 embedded in R3 so that qi = i , i = 1, 2, 3, then the standard moment map T S 2 (p, ) (M , ) so(3) given by the cross product M =âp takes the Poisson bracket (42) to the Poisson bracket for (23),
3

R

3

(43)

{Mi , Mj }g = {Mi , j }g =

ij k

g ( )Mk + g ( ) k ,

k m=1

Mm m g ( ) ,

(44)

ij k

{i , j }g = 0,

where m = / m as before. If the function in (37) is taken for g ( ), then the Poisson bracket (44) will coincide with the bracket (39) appearing in the analysis of the non-holonomic Chaplygin ball. 2.3.2. Since the Poisson bracket (39) is a deformation of the canonical Lie­Poisson bracket of a special form (see [81]), it can be reduced to this canonical bracket in various ways (see § 1.4). For example, the change of variables proposed in [83], although it contains a singularity for c = 0, can be expressed in terms of elementary functions: b1 2 c1 L1 = g -1 M1 - 1+ 3 +2 2, ( , ) 1 + 2 b2 2 c2 (45) L2 = g -1 M2 - 1+ 3 +2 2, ( , ) 1 + 2 2 b3 2 + 2 L3 = g -1 M3 - 1- 1 , ( , ) where b = ( , M ), c = ( , L),
2 2 2 2 = 1 + 2 - d( , )(a1 1 + a2 2 ),

and it reduces the Poisson bracket (39) to the canonical Lie­Poisson bracket (9) on the algebra e (3). Here b and c are the numerical values of the Casimir operators that correspond to interrelated symplectic leaves.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Applying the Poisson map M1 = g L1 - c1 b1 2 + 1+ 3 , 2 2 1 + 2 ( , ) 2 b2 c2 3 + M2 = g L2 - 2 1+ , 2 1 + 2 ( , ) 2 2 + 2 b3 1- 1 M3 = g L3 + ( , )

(46)

inverse to (45) to the original integrals system on e (3) with integrals of motion Lie­Poisson bracket (9). Furthermore, setting c = 0, we obtain leaves of the Poisson bracket { · , · }g in ( ( , ) = 1 ,

of motion (36), we obtain an integrable in involution with respect to the canonical the Poisson map (45) between symplectic 39) and the symplectic leaf ( , L) = 0

of the Lie­Poisson bracket { · , · } in (9), which is symplectomorphic to the cotangent bundle T S 2 of the unit sphere. In this case the original integrals of motion (36) take the form H1 = g 2 (L, Ag L) + 2bg ( , AL) - a3 3 L3 2 2 b2 (a1 1 + a2 2 ) 1 - d(a1 1 + a2 2 ) + , 2 2bg 3 L3 2 ( 2 + 2 ) H2 = g 2 (L, L) + + b2 1 + 3 1 2 2 .

(47)

Remark 9. Using the sphero-conical coordinate system u1 , u2 on the sphere, we now obtain the following expressions for the integrals of motion on T S 2 : H1 = ap
2 u1 2 u1

+ 2bpu1 pu2 + cp + 2Bpu1 p
u2

2 u2

+ dpu1 + ep + Dp
u1

u2

+ f,
u2

H2 = Ap

+ Cp

2 u2

(48)

+ Ep

+ F.

Here the pu1,2 are the momenta conjugate to the coordinate variables u1,2 , and the coefficients of the polynomials in (48) are analytic functions of the u1,2 . According to [61], Hamiltonian systems with such integrals admit a partial separation of variables in the Hamilton­Jacobi equations. In this case the separated variables s1,2 are the roots of the equation (B - bs)2 - (A - as)(C - cs) = 0, which in our case are s1,2 = d(1 + u1,2 )u
-1 1,2

.

As in the Clebsch case [61], these variables evolve on the Prym variety of an algebraic curve which is not hyperelliptic, in contrast to the separation of variables in the Chaplygin case [20], [31].


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2.4. Gyrostatic generalizations. Following [24], we study the changes in the Poisson structure (38) when a uniformly rotating balanced rotor is added to the body. The corresponding system can be interpreted as a non-holonomic gyrostat. The effect of a gyrostat can also be obtained by making multiply-connected cavities in the body which are completely filled with an ideal incompressible fluid. In this case the equations of motion (32) are modified as follows: M = (M + S ) â + mr â ( â r) + MF , = â , (49)

where S is the constant 3-dimensional gyrostatic moment vector. For the Chaplygin ball on the plane this vector can be arbitrary, and the integrals of motion (36) change as follows: H1 = (M , ), H2 = (M + S, M + S ), H3 = ( , ), H4 = ( , M + S ).

The Poisson bivector corresponding to (49) and the new integrals of motion have the form 0 00 Psg = g - dg -1 (M , A ) , M+S 0 where 0 S = -S3 S2 S3 0 -S1 -S2 S1 . 0

In this case the vector field remains conformally Hamiltonian: X=g
-1

P

sg

dH.

The corresponding Poisson bracket was found in [24]. 3. The Chaplygin ball on a sphere: the BMF system Following [16] and [24], we now study a dynamically asymmetric balanced rigid body with spherical surface of radius b which rolls on the surface of a fixed sphere of radius a without slipping. In this case the symmetry group SO(3) consists of the rotations of the fixed base on which the body rolls. In the body frame connected with the principal axes of the moving body the equations of motion of the reduced system depend on the ratio = a/(a + b) of the radii of the spheres: M = M â , = â , (50)

where, as before, the angular momentum M with respect of the point of contact is given by (35). For each the equations of motion (50) have the three integrals of motion H1 = (M , ), H2 = (M , M ), H3 = ( , ) (51) and the invariant measure µ=g
-1

( ) d dM ,

g ( ) =

1 - d( , A ) ,

(52)


502

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

where a1 A=0 0 0 a2 0 0 (I1 + d)- = (I + dE)-1 = 0 0 a3 0
1

0 (I2 + d) 0

-1

0 0 (I3 + d)


-1

.

For = ±1 a further integral of motion b1 0 H4 = ( , BM ), B = 0 b2 00

exists: 0 0 = tr A b3

-1

E + ( - 1)A

-1

,

(53)

where E is the identity matrix. The value = 1 corresponds to the limiting case when the support surface is flat, that is, we obtain the problem of the Chaplygin ball considered above (see § 2.4). For = -1 the integrability of the equations of motion (50) was proved in [16], and a conformally Hamiltonian representation and explicit quadratures for H4 = 0 were found in [17], so following [82], we shall call this system the Borisov­Mamaev­ Fedorov system or the BMF-system for brevity. 3.1. The Poisson bracket and a conformally Hamiltonian representation. For = -1 we have a more complicated Poisson bivector, whose structure is nevertheless similar to that of Pg in (38). Indeed, we can easily verify that the integrals of motion (51) and (53) are in involution with respect to the Poisson bracket defined by a Poisson bivector which is linear in M : Pb = g 0 - M +g
-1

(2d( , ) - tr B)

0 0

0 ,

(54)

and we have [Pb , Pb ] = 0. The matrix depends only on : = where
2 1 C = 2 3

( , )E - C -

B b , 2d



1 1

1 2 2 3

2

2

1 3 2 3 , 2 3

0 b = -b3 3 b2 2



b3 3 0 -b1 1

-b2 2 b1 1 , 0

and the remaining two matrices M and have entries of the form Mij = -ij ij = - where k = C2 + bk ( , M ), C2 = ( , BM ).
k

k k - ( , )bk Mk +

b2 M k 2d

k

,

ij k bk k (bi + bj )k + (bk - bi )(bk - bj )Mk k , (b1 + b2 )(b2 + b3 )(b1 + b3 )


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503

This bivector is the unique solution of equations (12) in the class of Poisson bivectors linear in M : the proof consists in solving equations (12) directly while taking the ansatz of the most general form (13) for a bivector P which is linear in the Mi . Using this Poisson structure, we can write the original equations of motion (50) in the form of a decomposition into conformally Hamiltonian fields: dxk - - Xk = g1 1 {H1 , xk }b + g2 1 {H2 , xk }b , dt where g1 ( ) = and s( ) = 4d2 ( , )( , B ) - 2d (E tr B - B) , B + det B. (56) Thus, in contrast to the ordinary Chaplygin ball, the original vector field is not conformally Hamiltonian with respect to a Poisson structure linear in M when the energy integral H1 is taken as the Hamiltonian. Nevertheless, we can obtain a representation (17) by changing the Hamiltonian. Prop osition 3. The vector field X corresponding to the system (50) with = -1 is conformal ly Hamiltonian with respect to the Poisson structure (54) : X= with the Hamiltonian H = (2d( , ) - tr B)H1 + 2H2 . Proof . The proof consists in the straightforward construction of the equations of motion with the help of the Poisson tensor Pb and the integrals of motion H1,2 and H3 . 3.2. Reducing the Poisson bracket to a canonical form. As in the case of the Chaplygin ball on the plane, we also know the change of variables L1 = b1 3 h( ) 1 2 + 2 ) 1 (b1 2 M1 - b2 1 M2 ) + 1 M3 + 2 + 2 s( )b1 b2 (1 2 1 2 c1 , 2 2 1 + 2 1 b2 3 h( ) 2 (b1 2 M1 - b2 1 M2 ) + 2 M3 + 2 2 2 2 s( )b1 b2 (1 + 2 ) 1 + 2 c2 2 2, 1 + 2 1 2 2 3 (b1 2 M1 - b2 1 M2 ) + 3 M3 - bh( ) , s( )b1 b2 (1 + 2 ) b = (B , M ), c = ( , L), g ( )s( ) d
-1

(55)

g ( ) s( ) , (2d( , ) - tr B)d

g2 ( ) =

g ( )s( ) , 2d

Pb dH

g( ) +

L2 =

g( ) +

L3 =

g( )


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

which takes Pb to the canonical bivector P0 in (10). Here s( ) is the function defined in (56),
2 2 2 1 = 2d2 2db1 1 + 2db2 (2 + 3 ) - b1 b2 , 2 2 2 2 = -2d1 2db1 (1 + 3 ) + 2db2 2 - b1 b2 ,

3 = 4d2 1 2 3 (b1 - b2 ), and
2 2 2 1 = -2d1 3 2d 1 (b1 b2 - b1 b3 + b2 b3 ) + b2 (b1 1 + b3 3 ) - b1 b2 b3 , 2 2 2 2 = -2d2 3 2d 1 (b1 b3 + b1 b2 - b2 b3 ) + b1 (b2 2 + b3 3 ) - b1 b2 b3 , 2 2 2 2 2 2 2 3 = 4d2 b3 3 (b2 1 + b1 2 ) + 2db1 b2 (1 + 2 ) 2d(1 + 2 ) - b3 .

We do not present here the explicit expression for h( ), which is a fairly cumbersome solution of the differential equations {Li , Lj }b = ij k Lk . We note only that such a solution does exist and is not unique. (The change of variables is also singular for c = 0.) Using this change of variables for c = ( , L) = 0, we obtain, as in the case of the Chaplygin ball, a dynamical system on the 2-sphere's cotangent bundle T S 2 endowed with the canonical Poisson bracket. 4. The Veselova system Let us consider the motion of a rigid body about a fixed point in the absence of external forces but in the presence of the non-holonomic constraint ( , ) = 0. (57)

That is, the pro jection of the angular velocity on the direction of the vector vanishes, which means that there is no rotation about [92], [93]. The equations of motion have the form M = M â + , = â , (58)

where M is the angular momentum, = AM is the angular velocity of the body, is the unit vector on the fixed axis in the body frame, and A = I-1 is a non-singular diagonal matrix (the inverse of the inertia tensor): a1 0 0 A = 0 a2 0 . 0 0 a3 Differentiating the constraint equation (57) and using (58), we find the Lagrange multiplier (AM â M , A ) . = (A , ) The vector field X has the four first integrals (on the level surface defined by the constraint) H1 = (M , AM ), H2 = (M , M ) - ( , )-1 ( , M )2 , H3 = ( , ), H4 = ( , )


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505

and has the invariant measure µ = g d dM , g ( ) = ( , A )
1/2

.

(59)

As shown in [24], this system is equivalent to the problem of the Chaplygin ball rolling on the plane without spinning. The integrals of motion are in involution with respect to the Poisson bracket: {Mi , Mj }ag ij k = ak g ai aj
ij k 3 -1

Mk + ,

k m=1

am Mm m g

-1

,

{Mi , j }ag =

g

-1 -1 ai k

{i , j }ag = 0,

provided that (57) is satisfied. 4.1. Equivalence of the Veselova system and the Chaplygin ball problem. One can verify that (58) remains integrable also in the case of an inhomogeneous constraint [38] (see also [13] and [83]), that is, when we get ( , ) = b, b R, (60)

instead of (57). It is convenient to introduce in place of M a vector K which is constant in the space, using the formula K=M- ((E - A)M , ) , ( , ) ( , K ) = ( , ) = b, (61)

and then the equations of motion (58) for the Veselova system assume the form K = K â , = â , (62)

where = AM . Since the Jacobian of this coordinate transformation is proportional to the square of the density of the invariant measure in the original Veselova problem, ( , M ) = g 2 ( ) = ( , A ), ( , K ) the transformation is invertible (at points where the invariant measure is defined), and hence there exists a well-defined change of time enabling us to integrate the equations of motion [93]. In the new variables the invariant measure has the form µ=g
-1

d dK,

g ( ) = ( , A )

1/2

,

(63)

and the integrals of motion are H1 = (K, Ag K ), where Ag = A - g
-2

H2 = (K, K ),

H3 = ( , ),

H4 = ( , K ),

(64)

(E - A)( )(E - A).


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

These integrals of motion are in involution with respect to the following Poisson bracket, for which ( , ) and ( , K ) are Casimir functions:
3

{Ki , Kj }v = {Ki , j }v =

ij k

g Kk + g k ,

ij k



k m=1

Km m g -

b , g

(65)

ij k

{i , j }v = 0.

Using this Poisson structure, we can rewrite the equations of motion (62) in a conformally Hamiltonian form as d xi = g dt
-1

( ){H, xi }v ,

H=

H1 , 2

i = 1, . . . , 6.

(66)

Prop osition 4. The Poisson bracket { · , · }v in (65) and the invariant measure (63) for the Veselova system coincide with the Poisson bracket { · , · }g in (39) and the invariant measure (37) for the Chaplygin bal l , provided that ( , ) = 1 and the parameters of the two problems are related by ai = 1 - dai . (67)

The integrals of motion Hi of the Veselova system (see (64) ) can be expressed in terms of the integrals of motion Hi of the Chaplygin system (see (51) ): H1 = H2 - dH1 , H2 = H2 , H3 = H3 , H4 = H4 . (68)

Since the sets of first integrals of the two systems coincide (after a suitable change of parameters), they define the same foliation of the phase space by invariant tori. Remark 10. If ( , ) = 1, then the connection (67) between the parameters of the two problems has the form ai = ( , )-1 - dai and H1 = ( , )-1 H2 - dH1 . As a consequence, the Poisson bracket { · , · }v for the Veselova system can be reduced to the canonical Poisson bracket on the Lie algebra e (3) by means of the transformation L1 = g L2 = g L3 = g
-1

-1

-1

b1 1+ ( , ) b2 K2 - 1+ ( , ) b3 K3 - 1- ( , ) K1 -

2 3 c1 +2 2, 1 + 2 2 3 c2 +2 2, 1 + 2 2 2 1 + 2 2 2 , = a1 1 + a2 2 ,

(69)

which coincides with (45) up to the change of parameters (67). It was shown in [13] that this analogy (on the level of the corresponding Poisson structures) between the conformally Hamiltonian representations for the Veselova system and the Chaplygin ball problem survives the addition of a potential field of external forces with potential U ( ). In that case both systems will be non-integrable in general.


Non-holonomic dynamics and Poisson geometry

507

4.2. A gyrostatic generalization. Now let us consider the case when a rotor with gyroscopic moment k is added to the rigid body. Direct calculations show that the system remains conformally Hamiltonian, x=g
-1

Pk ( x)

H , x

with respect to the Poisson bivector Pk of the more general form Pk (x) = g where 0 Kk = K3 + k3 -K2 - k2 S= - -K3 - k3 0 K1 + k1 K2 + k2 -K 1 - k 1 , 0 + ( ), 0 Kk - gS 0 0 0 , (70)

g + f ( ) , K

x = ( , K ) is the full set of variables, and the function g depends only on . Then the Jacobi identity certainly holds, and the Casimir functions are F1 =
2

and F2 = (K + k , ).
-1

When the gyrostat is added, (61) must be modified. We shall take K = A , where we find the coefficient from the condition (K + k , ) = ( , ). Then we get that K=A
-1

+

- (A

-1

- E) + k , , (AK - K - k , ) (A , ) ,

= A(K - S ),

S=

where S coincides with the corresponding function in (70) if g is given by g= Then the Hamiltonian has the form H= 1 (AK - K - k , )2 (AK , K ) + . 2 (A , ) (A , ) .

5. A b o dy of revolution on the plane Starting with [3], [30], [42], [50], and [75], many authors have been concerned with a body of revolution rolling along the plane (see, for instance, the bibliographies in [23] and [33]). Here we shall mainly follow [23].


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We recall that for a body of revolution its surface and the central ellipsoid of inertia are two coaxial surfaces of revolution. Mathematically, this means that we impose the following constraints: I1 = I2 = I3 , r1 = f1 (3 )1 , r2 = f1 (3 )2 , r3 = f2 (3 ).
1,2

(71) (3 )

Since the surface is described by a single equation f(r) = 0, the functions f in (71) satisfy the equation
2 1 - 3 df1 (3 ) df2 (3 ) = f1 (3 ) - , d3 3 d3

which describes the meridional section of the surface of revolution. From the physical standpoint this is simply a consequence of the fact that the pro jection of the velocity of the motion of the vector r on the normal is zero: (r, ) = 0.

Figure 2. The meridional section of a body of revolution.

In addition, we assume that the potential U of the external forces depends only on 3 , that is, on the angle between the axis of revolution and the vertical direction. In the case of motion in a gravitational field this implies that the centre of mass of the body lies on the axis of revolution. In the present case the symmetry group G = E (2) â SO(2) of the general system (30) consists of two subgroups. The subgroup E (2) of motions of the plane was mentioned above (see § 1); the second subgroup SO(2) is formed by the rotations of the body about the axis of revolution C z (Fig. 2), which are sometimes called the internal symmetries of the body [35], [36]. Thus, the vector field X of the reduced system (32) is invariant under the action 1 2 3 M1 M2 M3 cos 1 - sin sin 1 + cos 3 cos M1 - sin sin M1 + cos M3 2 2 M2 M2

(72)


Non-holonomic dynamics and Poisson geometry

509

of SO(2). This action is not free, because it has the two fixed points 0 0 ±1 e± = . 0 0 M3

(73)

These points correspond to two families of relative equilibria (parametrized by M3 ) when the body rotates about its axis, which is directed vertically. Remark 11. In the terminology of some authors (see [48], for instance), if the symmetry group of a system has fixed points, then this is not a Chaplygin system. Remark 12. The points (73) can occur on the same integral manifold. In this case it was conjectured in [34] that there can exist obstructions to `Hamiltonizing' the non-holonomic system, a non-Hamiltonian monodromy. As a concrete example of such a system the authors of [34] point out the problem of a rolling elongated balanced ellipsoid of revolution. As shown in [14], there is no non-Hamiltonian monodromy in that problem, because its Liouville foliation is equivalent to the foliation of the Hamiltonian system describing an ellipsoid rolling on an absolutely smooth plane. Furthermore, it was shown in [14] how the monodromy can be used to `Hamiltonize' the system for a suitable choice of Casimir functions. The generator of the action (72) of SO(2), which is called the symmetric field XS , has the form XS = 1 - 2 + M1 - M2 (74) 2 1 M2 M1 and, as we show below, it appears in a decomposition of X of the type (8): X = P dH + XS . It is known (see, for instance, [3], [23], or [30]) that in this case we can always find a pair of integrals of motion for the vector field X corresponding to (32) which are linear in the momenta: Jk = v
(k) 1

(3 )(1 M1 + 2 M2 ) + v
(k) (k)

(k) 2

(3 )M3 ,

k = 1, 2.

(75)

In general the coefficients v1 and v2 are real analytic but not necessarily algebraic functions of 3 , and they satisfy the following system of differential equations:
g 2 ((v2 ) (k)

+v

(k ) 1

)

m

2 = mf1 f2 (1 - 3 )f1 + 3 f2 (v 2 + I1 f1 (1 - 3 )(v

(k) 2 f1

-v

(k) 1 f2

)

+v
g 2 ( v1 ) m (k)

(k ) 1

2 (3 - 1) +

(k) (k) 1 f2 - v2 f1 (k ) 3 v2 f 1 , (k) 2 f1

) - 3 v

(k ) 1 f2

(76)
(k) 1 f2

= mf

2 1

2 (1 - 3 )f1 + 3 f2 (v (k ) 2

-v

)

2 + I3 f1 v

- (v

(k ) 1 f2

-v

(k) 2 f1

)f2 .


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov
1,2

This system is obtained by equating to zero the coefficients of M equations Jk = X (Jk ) = 0, k = 1, 2.

and M3 in the

Here and throughout this section we set ( , ) = 1 and drop the indication of the dependence of functions on 3 , that is, we use the notation F ( 3 ) = F , dF (3 ) =F d3


for all functions F of 3 . Apart from the four integrals of motion H , C , and J vector field (32) also has the invariant measure =g
-1

1,2

, in the present case the

( ) d dM ,

g ( ) =

I1 I3 + m(r, Ir) .

(77)

Thus , the system (32) for a body of revolution is integrable in quadratures by the Euler­Jacobi theorem. Example 1. Consider a round disc with centre of mass shifted along the axis of dynamical symmetry, and let f1 = R 1-
2 3

,

f2 = a,

where R is the radius of the disc (coin) and a is the displacement of the centre of mass. If a = 0, then the functions v
(k) 1

= L(k) (b- , 3 ),

v

(k ) 2

= c L(k) (b+ , 3 ) - 3 L(k) (b- , 3 ) ,

k = 1, 2, (78)

where b = 1 g 2 - 4mI3 R2 , 2g 2 c=- I1
1/2

g 2 - 4mI3 R2 + g I1 , 2mg R2

can be expressed in terms of the Legendre functions of the first and second kind L(1,2) (bk , 3 ), which are defined in the standard way in terms of a hypergeometric function [23]. For a = 0 solutions of (76) are combinations of Whittaker functions (see [8]). 5.1. Hamiltonizing the reduced system. Following [23], we reduce the system with respect to the action (72) of the symmetry group SO(2), taking as new variables the following functions, which are invariant under the symmetry field: 3 , K1 = (M , r) , f1 K2 = g 3 , K3 = (1 M2 - 2 M1 ) , 2 1 - 3 (79)

where =

3 1 - 3 . I1 + m(r, r)


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511

Then we get that 3 = K3 , K1 = -K3 g K2 = -K3 g K3 = -
22 I1 f1 -1 -1

I

3

1-

f2 f1



K2 , (80)

mf1 (f1 - f2 )K1 , 2 2 22 2 f2 f1 (1 - 3 ) + 3 f2 (mf1 K1 + I3 K2 )

2 (1 - 3 )
-1

2 2 2 + 3 f1 I1 K1 + f1 f1 (1 - 3 ) + 23 f2 K1 K2 g 2 + mf1 g 2 f2 (1 - 3 )(3 f1 I1 - f2 I3 )K1 K2 -

U (3 ) , 3

and the energy integral takes the form H= I3 mf 1 2 2 2 ) K1 - mf 2 K2 + I 2I1 (1 - 3 1 1
2 2

K1 -

K2 g mf1 f2

2

12 + K3 + U (3 ). (81) 2

The vector field X of the reduced system (80) is Hamiltonian with respect to the rank-2 Poisson structure: X = P dH, 000 0 0 0 P = 0 0 0 1 -I3 g -1 1 - (f2 /f1 ) K2 , -mg -1 f1 (f1 - f2 )K1 0

rank P = 2

(82)

(here and below, asterisks denote terms which can be determined from the antisymmetry condition). To find Casimir functions of the bracket for (82) we find linear integrals of the system (80), and to do this we divide the second and third equations by the first, obtaining the two non-autonomous linear equations dK1 = -g d3
-1

I

3

1-

f2 f1



K2 ,

dK2 = -mg d3

-1

f1 (f1 - f2 )K1 .

(83)

They were obtained by Chaplygin [30] using different notation (see the discussion in [23]). This system of equations always has a general solution of the form Ki = c1 1 (3 ) + c2 2 (3 ), i = 1, 2, (84)

where the 1,2 are the fundamental solutions of equations (83), and the integration constants c1,2 are first integrals of the system and can be expressed in terms of the first integrals (75). Solving the equations (84) for c1,2 , we find these integrals of motion, and they are Casimir functions of the Poisson bivector P .


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

5.2. Rank-4 Poisson structure. In contrast to the case of the Chaplygin ball, in the present case we have explicit expressions for the mechanical energy H of the system and the geometric integral C (see (33)), but the two integrals of motion J1,2 linear in the momenta (see (75)) are defined only implicitly. Nevertheless, taking the ansatz (13) linear in the Mi and using the equations (76) defining the functions (k) v1,2 , we can obtain solutions of (12). In contrast to the Lagrange top with similar symmetries, in our case equations (12) have only two and not three solutions P (k) linear in the Mi . These solutions can provisionally be decomposed into a `vertical' and a `horizontal' part: P where P
(k) (k ) ( = Pk) + P ,

k = 1, 2,

(85)

and P are compatible rank-2 Poisson bivectors, [P
(k)

, P ] = 0,

which satisfy (12), and and are some arbitrary functions of 2 /1 and 3 , respectively. (k ) The first component depends on the functions v1,2 in the definition of the integrals of motion Jk in (75), which are linear in the momenta: P where 1 2 v2 2 + 2 (k) 1 2 v1 (k) 2 = 1 v2 - 2 + 2 (k) 1 2 v1 0 1 M 1 + 2 M 0 2 2 1 + 2 M = 0 k = exp + I1 v 1 v
(k) 2 1g (k)

= k

0 -

, M

rank P

(k )

= 2,

k = 1, 2,

(86)



(k)

(k) 2 2 v2 -2 2 2 1 + 2 v (k) 1 (k) , 1 2 v2 -2 1 2 (k) 1 + 2 v 1 0 0 (k ) 2 v2 -M 2 (k ) v1 , M1 0
2 1 2 (1 - 3 )f2 + 3 f2 (f1 v (k) 2 (k) 2

m mf

- f2 v

(k) 1

) .

(k ) 1 f1

2 (1 - 3 )f1 - 3 f (2)

1

+ I3 v

2 (f1 f2 - f1 ) d

3

The Poisson bivectors P P
(k)

(1)

and P
(k )

have in common three Casimir functions P
(k)

dC = 0,

P

d3 = 0,

d(1 M2 - 2 M1 ) = 0

along with two individual Casimir functions P
(1)

dJ1 = 0,

P

(2)

dJ2 = 0,


Non-holonomic dynamics and Poisson geometry

513

and acting on M3 these bivectors generate the symmetry field P
(k)

dM3 = XS .

They can be regarded as the `vertical' component of the full solution (85), a component which vanishes upon reduction with respect to the remaining SO(2)-symmetry. The second component of the solution (85) has the form P = where = 1 2 3 2 2 1 + 2 2 2 3 2 2 1 + 2 -2 2 3 - 21 2 1 + 2 1 2 3 -2 2 1 + 2 1 0 , 0 0 -m 2 , m 1 0


0 -

, M

rank P = 2,

(87)

0 M = =

-M 3 +

3 (1 M1 + 2 M2 ) + m 2 2 1 + 2 0

2 2 f2 f1 (I3 f2 (1 M1 + 2 M2 ) - (1 + 2 )I1 f1 M3 ) f2 - 1- , 2 f1 g f1 1 3 2 2 = 2 (1 M1 + 2 M2 ) m(r, )f1 + I3 (f1 + f1 f2 ) + M3 m(r, )f1 f2 g 2 2 + 3 f1 - (1 + 2 )f1 I1 f1 .

The rank-2 Poisson bivector P has the Casimir functions P dC = 0, P dJ1 = 0, P dJ2 = 0, P d 2 = 0. 1

We note that this Poisson structure depends only on the shape of the body of revolution (that is, only on the functions f1,2 ), and it can be regarded as the `horizontal' component of the full solution of (12), a component which is preserved upon reduction with respect to the remaining symmetry XS of the system and is taken to the bracket for (82). Formally speaking, the bivectors P (k) are defined locally, since M and M have singularities for 1,2 = 0, 3 = ±1, M1,2 = 0. The singularity in the entry 1 M1 + 2 M 2 v 2 2 1 + 2 v
(k) 2 (k) 1

of the matrix M can be removed by taking the integrals of motion such that (k) v2 = 0 for 3 = 1 or for 3 = -1. On the other hand, using the freedom we have


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

in choosing the function , we can multiply the bivector P by =1-
2 3

and thereby obtain a general solution without singularities. Thus, it follows from the foregoing that each bivector P (k) in (85) has two Casimir functions on the regular part of the Poisson manifold: P
(k )

dC = 0,

P

(k)

dJk = 0,

k = 1, 2.

Furthermore, the vanishing of the Schouten bracket is equivalent to the existence of the linear integrals of motion J1,2 in (75): [P
(k )

,P

(k)

]=0



Jk = X (Jk ) = 0.

The resulting Poisson structure on the partially reduced phase space enables us to construct a conformally Hamiltonian decomposition of the original vector field X . Prop osition 5. For a body of revolution on a plane the vector field X corresponding to (32) is a linear combination of a Hamiltonian vector field and the symmetry field XS in (74) : X=
-1

P

(k )

dH + k XS ,

k =

u1 (1 M1 + 2 M2 ) + u2 M3 , g2

(88)

where the coefficients u1 = 1
(k) (1 - )v1 (k ) + I 3 ( 3 v1 2 3 2 mf1 (1 - 3 ) -1

k (v

(k) 1 f2

-v

(k ) 2 f1

)+v

(k ) 1

(3 f1 - f2 )

+

-1

k v
(k ) 1 f2

(k ) 2

),
(k) 2 f1

u2 =

1 v
(k ) 1

mf2

-1

k (v

-v

)+v

(k ) 1

(3 f1 - f2 ) - I1 v

(k ) 1

(

-1

k + 1)

depend on the functions v

(k) 1,2

in the definition of the integrals of motion Jk in (75) .

We now use our freedom in choosing the functions and in the solution (85). With = const and =-
2 (1 - 3 ) f1 v 2 3 (k ) 2

- f2 v

(1 - )(3 f1 - f2 ) f

(k) 2 m + I1 1 (k) (k) 1 v2 - f2 v1

2 (1 - 3 )(v

(k) 2 1) 2 3

+ I3 (v

m + I1 (1 - )v

(k ) 2 2) (k ) (k) 1 + I 3 v2

k v
(k) 1

, (89)

the vector field X takes the form X= where = 3 v
2 (1 - 3 )(f1 v (k) 2 (k ) 1 -1

P

(k)

dH + Jk XS ,

(90)

-v

(k) 2 (k ) 2 1)

- f2 v

(k ) 2 1)

2 m + I1 (1 - 3 )(v

+ I3 (v

(k) 2 2)

.


Non-holonomic dynamics and Poisson geometry

515

Prop osition 6. For a body of revolution on a plane the vector field X corresponding to (32) is conformal ly Hamiltonian , X=
-1

P

(k)

dH,

with respect to a rank- 4 Poisson bivector if one of the integrals of motion linear in the momenta has value zero : Jk = 0, k = 1, 2.

Thus, after a change of time X can be reduced to a Hamiltonian vector field on the zero level set of one of the Casimir functions of P (k) . 5.3. Reducing the Poisson bracket to a canonical form. As in the case of the Chaplygin ball, we consider changes of variables of the form L1 = L2 = 1 2 1 + 1 +
2 1

2 2

1 2

3

v v

(k) 1

( 1 M1 + 2 M 2 ) v
(k ) 2 k

+ b + b

k

+ -

2 (1 M2 - 2 M1 ) + c 1 (1 M2 - 2 M1 ) + c

1

, , (91)

3

(k) 1

( 1 M1 + 2 M 2 ) v
(k ) 2 k

2 2

k

2

M3 1 + k k v2 L3 = - (k ) k k v2 where

(k)

b,

b = Jk ,

c = (L, ),

k = and µk = 1 - mg
-2

v (v

(k) 1 µk (k ) 2 2 ) k

d

3

(92)

2 22 (1 - 3 )(I1 + mf2 )f1 + mg

-2

3 2 m3 f2 - I1 (1 - 3 )f2 + I1 3 f2 f1 .

It can easily be verified that if = const, then the Poisson maps k : ( , M ) ( , L) of the form (91) take the Poisson bracket { · , · } corresponding to the bivectors P (k) in (85) to the canonical Lie-Poisson bracket (9) on the Lie algebra e (3). We underscore that in the general case these maps are well defined only on the regular part of the Poisson manifolds under consideration, and are not defined at the singular points e± , at which rank P (k) = 2. For c = ( , L) = 0 we have Poisson maps which identify our partially reduced 6-dimensional phase space with the 2-sphere's cotangent bundle T S 2 endowed with the canonical Poisson bracket. For the Poisson bivector P (1) with the Casimir function J1 in (75), P
(1)

dJ1 = 0,


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

the integrals of motion H and J2 assume the following form after a change of variables: H = AL2 + B (L1 2 - L2 1 )2 + bC L3 + b2 D, 3 J2 = 1 (v
(1) (2) 1 v2 - (1) v1

(93) b, (94)

v

(2) (1) 1 v2

)

(L3 + b1 ) +

v v

(2) 2 (1) 2

where A, B , C , and D are functions of 3 . 5.4. Gyrostatic generalizations. We recall that when we add a rotor which rotates with a constant angular velocity to a body rolling on the plane, the equations of motion (32) transform into M = (M + S ) â + mr â ( â r) + MF , = â ,

where S is the constant 3-dimensional gyrostatic moment vector. For a Chaplygin ball on the plane this vector can be arbitrary, but for a body of revolution on the plane integrability is preserved if and only if the gyrostatic moment is directed along the symmetry axis: S = (0, 0, s). (95)

Here the integrals of motion (75), which are linear in the momenta, simply `shift' by a term linear in s: Jk
(k) (s)

= Jk + s

v

(k) 1

d3 ,

k = 1, 2,

where the v1 are functions of 3 determining the integrals of motion Jk in (75) for the body without a rotor. Note that already in this case, even for the disc in Example 1, we obtain non-algebraic integrals of motion, because we obtain non-algebraic expressions on taking the integral L(1,2) (b- , 3 ) d3 of the Legendre functions L(1,2) . In this case the solutions of (12) are simply `shifts' of the solutions already obtained: 00 ( Ps k) = P (k) - , (96) 0S that is, the rank-2 bivector P is shifted. As before, the vector field X=
-1

P

(k) s

dH + k XS ,

k = 1, 2,

will be a linear combination of a Hamiltonian field and a symmetry field with the same coefficients k as for the body of revolution without a rotor. 5.5. The Routh sphere. For example, let us consider the Routh sphere rolling on the plane [9], [23], [32], [33], [44], [74]. In this case the centre of mass of the sphere does not lie at the geometric centre of the sphere, and the axis joining the centre of mass and the geometric centre is the axis of dynamical symmetry, that is, the two principal moments of inertia with respect to the axes orthogonal to


Non-holonomic dynamics and Poisson geometry

517

it are equal. The sphere rolls on a horizontal plane under the action of a constant vertical gravitational force. The shape of the body of revolution here is described by the functions f1 = R and f2 = R3 + a, namely, r = (R1 , R2 , R3 + a), where R is the radius of the sphere and a is the distance from the geometric centre to the centre of mass. Substituting the functions f1,2 in equations (76), we easily find the integrals of motion J1,2 which are linear in the momenta (see (75)). The first of them, J1 = (M , r), (97)

is the Jellet integral [47]; see also § 243 in Routh's book [75]. The second integral of motion, J2 = g ( )3 , (98) was found by Routh [75] in 1884 and later rediscovered by Chaplygin [30]. Recall that 3 is the third component of the angular velocity vector and can be expressed in terms of the angular momentum using (32), while g g ( ) is the function giving the density of the invariant measure (77). (k ) Since the v1 are also involved in the definition of the Poisson bivectors, we write them out explicitly: v v
(1) 1 (2) 1

= f1 = R , = mR(R3 + a) , g

v2 = f2 = R3 + a, v2 =
(2)

(1)

1 = g ,
2

I1 + m(R3 + a) , g

2 = R3 + a.

The corresponding Poisson bivectors P (1,2) in (85) were found in [9]. Using these bivectors, we can construct different representations for the vector field X corresponding to (32): X= or X=
-1 -1

P

(1)

dH + 1 XS =
(1)

-1

P

(2)

dH + 2 XS I1 2 P mg 2
(2)

(99)

P

(1)

dH +

I1 1 P g2

dJ2 =

-1

P

(2)

dH -

dJ1 .

Remark 13. If instead of the Jellet integral J1 and the Routh integral J2 we consider the linear combinations I3 (R + a) (+) (+) J+ = J1 - J2 = v1 (1 M1 + 2 M2 ) + v2 M3 , 2+I m(R + a) 1 I3 (R - a) (-) (-) J- = J1 + J 2 = v1 ( 1 M 1 + 2 M 2 ) + v2 M 3 m(R - a)2 + I1 of them and use these combinations as Casimir functions of the bivectors P constructed in accordance with (86), then these bivectors will not have singularities at
(±)


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

the points where 3 = 1 or 3 = -1, respectively. We underscore that for the Routh sphere the geometric centre is different from the centre of mass, so the equilibria (73) for this dynamical system have a different character: one is stable while the other is not. Choosing suitable linear integrals of motion J± , we can construct (±) bivectors P in (86) which have singularities only at one of the equilibria. Discussions of various applications of the general theory of singularities on Poisson manifolds to the study of dynamical systems can be found in [67] and the papers cited there. The use of singularities of bi-Hamiltonian structures in the analysis of the stability of periodic solutions was the sub ject of [15]. Topological obstructions to the Hamiltonization of non-holonomic systems (non-Hamiltonian monodromy) were investigated in [14]. Let us now consider the Poisson map (91) which takes the bracket to a canonical form. For example, we look at the bracket for which the Jellet integral (97) is a Casimir function. In this case 1 = - I1 + m(R3 + a)2 , g I1 (R3 + a)

and for = const and c = 0 after the Poisson map the Hamiltonian H= 1 2 (2 L1 - 1 L2 )2 2 2 L2 I1 R2 (3 - 1) - (R3 + a)2 I3 + 3 2 2R2 (1 - 3 ) I1 + m(r, r) - 2g b(r3 + a)L3 I1 + m(R3 + a)2 b + 2 I1 I1
2

+ U (3 )

and the Routh integral J2 = I1 L3 will determine on the 2-sphere a dynamical system endowed with the canonical Poisson bracket. Using the standard spherical coordinates 1 = sin sin , 2 = cos sin , 3 = cos , L1 = sin cos sin cos cos L2 = sin L3 = -p , p - cos p , p + sin p , (100)

where and are the Euler angles and p and p are the conjugate momenta, with {, p } = {, p } = 1, {, } = {, p } = {, p } = 0,

we can write the integrals of motion in the standard form H= A()p2 + B ()p2 + bC ()p + b2 D() + U (), 2 J2 = -I1 p , (101)


Non-holonomic dynamics and Poisson geometry

519

where b is the value of the Jellet integral J1 , = const, and I3 (a2 + 2aR cos R2 sin2 2 B () = I1 + m(a2 + 2aR cos + R 2g (R cos + a) , C () = I1 R2 sin2 I1 + m(a2 + 2aR cos + R D() = 2 I1 R2 sin2 A() =
2

I1 +

+ R2 cos2 ) ,
2

)

,

2

)

.

The equation for the nutation angle is Hamiltonian, = {H, } + {J2 , } {H, }, and in a way quite similar to that for the Lagrange top, we can find a quadrature for the nutation angle: = B ()p =
2 B () 2E1 - A()E2 - bE2 C () - b2 D() - 2U () ,

where E1 = H and E2 = -J2 /(I1 ) are some fixed values of the integrals of motion. Substituting this solution in the equation for the angle of the proper rotation (which is not Hamiltonian), we obtain the second quadrature (I1 sin2 + I3 cos2 ) + aI3 cos cos d- b. = {H, } + {J2 , } = g ()I1 R sin2 I1 R sin2 Of course, these equations can also be deduced without finding Poisson structures and reducing them to the canonical form. However, by investigating the Poisson geometry of this system we can clear up various non-local properties of the behaviour of tra jectories in the phase space: for example, in this case the foliation by invariant manifolds is purely Hamiltonian (a Liouville foliation), and so on. The gyrostatic generalization of the Jellet integral remains an algebraic function:
s J1 = J1 + sR3 ,

whereas a non-algebraic term is added to the Routh integral: arctan s J2 = J2 + s m (RI1 3 - (R3 - a)I3 ) a g g I1 - I3 . - 2 3/2 m R2 (I1 - I3 )mI1 R (I1 - I3 )

We recall that this integral is a Casimir function for the Poisson bivector P in (96), whose entries are algebraic functions as before.

(2) s


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6. Dynamically symmetric ball on a surface of revolution Now we consider a ball with mass m, radius R, and inertia tensor I = µE, where E is the identity matrix, which rolls on a given motionless surface, with the following condition imposed on the shape of the surface: at each moment of time there is only one point of contact between the ball and the surface. In this case it is more convenient to use a fixed reference frame in which the surface is given by an equation ~(r) = 0. f (102) Since the motion takes place without slipping, the velocity of the point of contact is equal to zero: v + â a = 0, (103) where and v are the angular velocity and the velocity of the centre of the ball, and a is the vector connecting the centre of the ball and the point of contact (see Fig. 3).

Figure 3

If N is the reaction force induced by the constraint (103), and F and MF are the resultant of the external forces applied to the body and its moment, then the laws of conservation of momentum and angular momentum have the form mv = N + F and I = a â N + MF .

Let denote the normal vector to the surface. Then a = -R , and instead of the angular velocity we shall use the angular momentum vector with respect to the point of contact: M = µ + d â ( â ), d = mR2 .

Using (103), we next eliminate from these equations the velocity v of the centre of mass of the ball and the reaction force of constraint N , which gives us the following system of dynamical equations: M = d â ( â ) + MF , r + R = â R . (104)


Non-holonomic dynamics and Poisson geometry

521

This system determines a vector field X on the 6-dimensional phase space with the coordinates x = (1 , 2 , 3 , M1 , M2 , M3 ). In these equations the radius vector r of the point of contact can be found from (102): = f(r) . |~(r)| f

The system (104) must be supplemented with the equations defining the orientation of the ball: e1 = â e1 , e2 = â e2 , e3 = â e3 , where the ei are the unit basis vectors of the coordinate system attached to the body, taken in the fixed reference frame. In the case of potential external forces F generated by a potential U = U (r + R ), the moment of the external forces has the form U MF = R â , r = r + R , r and the vector field X corresponding to (104) has the first integrals H= 1 (M , ) + U (r ), 2 C = ( , ) = 1, (105)

where the radius vector r describes the surface along which the centre of the ball moves. (Recall that here all the vectors are taken in the fixed reference frame.) In the general case of an arbitrary surface the symmetry group G = SO(3) consists of all possible rotations of the ball. Obviously, the components of the vectors and (or of and r ) with respect to the fixed reference frame are invariant under the action of this group and form a closed system of invariants. Hence, the equations (104) split off from the full system and form a reduced system. Following [69] and [75], we consider a homogeneous ball rolling on a surface of revolution under the action of some external potential forces directed along the symmetry axis, for instance, gravitational forces. Then in addition to the symmetry of the system with respect to rotations of the ball we have rotations of the surface of revolution about its symmetry axis [95]. Thus, in this case the full symmetry group is G = SO(3) â SO(2), and the reduced vector field X corresponding to (104) is invariant under rotations about the axis OZ . Instead of the surface of rolling contact it will be more convenient in what follows to consider the surface along which the ball's centre, given by the radius vector r = r + R , moves. In the axially symmetric case it can be parametrized as follows: r1 = (f - R)1 , r2 = (f - R)2 , r3 =
2- f - (1 - 3 )g3 1 f

d3 - R3

(see a thorough discussion in [27]). The function f = f (3 ) is assumed to be sufficiently convex that the surface has only one point of contact with the ball. The vector field X corresponding to (104) is invariant under the symmetry subgroup SO(2) generated by the symmetry field XS =
1

- 2 + M1 - M2 . 2 1 M2 M1


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

Furthermore, X has the invariant measure f
3 2 f - (1 - 3 ) -1 3f

d dM .

(106)

As in the case of a body of revolution rolling on the plane, here we have first integrals of X which are linear in M : Jk = v where the functions v
(k) 1

(1 M1 + 2 M2 ) + v

(k) 2

M3 ,

k = 1, 2,

(107)

1,2

(3 ) in the definitions of the integrals satisfy the equations

2 f (v1 (µ + 3 d) - 3 v2 d) , f (µ + d) 2 2 f (1 - 3 )(v1 (µ + 3 d) - 3 v2 d) v2 = v 1 - . 3 f (µ + d) v1 =

(108)

This system is derived from the equations Jk = X (Jk ) = 0, k = 1, 2,

by setting the coefficients of M1,2 and M3 , respectively, equal to zero. Equations (108) always have real analytic solutions, but the latter are not necessarily algebraic. Example 2. Assume that the centre of the ball moves along an ellipsoid of revolution. Then b1 f (3 ) f = - 2 2, b1 (1 - 3 ) + b3 3 where the b1,2 are the principal semi-axes of the ellipsoid, and equations (108) have the independent solutions 13 b1 (1,2) 3/2 v1 = b-1 3 f L(1,2) - , , , 1 22 f 1 13 b1 2 - 1 (1,2) 1/2 + L(1,2) - , , - (109) v2 = 3 f 3 b1 (b1 - f 2 )( + 1) 22 f 1/2 3 f 2 b1 13 (1,2) + L + , ,- , 2 )( + 1) 22 f b1 (b1 - f where L(1,2) are associated Lagrange functions of the first and second kinds which can be expressed in terms of a hypergeometric function, and = µ . µ+d
1/2

We note that, according to [27], an algebraic integral also exists here: H2 = f µ 3 ( , M ) - M b2 µ( , M )2 1 + ( b1 - b2 ) µ+d µ+d
3 2

,

which easily leads to a solution of the equations of motion. Thus , in this case the system of equations (104) has the four independent first integrals (105) , (107) and the invariant measure (106) , so it is integrable by the Euler­Jacobi theorem [19] , [53].


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523

6.1. The Poisson bracket for case of a body of revolution on and form a system of equations ficients of the Mi equal to zero. algebro-differential equations, we P where P
(k) (k )

a ball on a surface of revolution. As in the the plane, we substitute the ansatz (13) in (12) for the functions ck and dij by setting the coefij Solving the resulting overdetermined system of obtain two solutions, rank P
(k)

( = Pk) + P ,

= 4,

k = 1, 2,

(110)

and P are compatible rank-2 Poisson bivectors, [P
(k)

, P ] = 0,

which satisfy (12), and the and 3 , respectively, that is, The first components of (k) tors P in (86) for a body P where
(k)

coefficients and are arbitrary functions of 2 /1 = (2 /1 ) and = (3 ) as before. the solutions (110) have the same form as the bivecof revolution on the plane: , M
(k )

= k

0 -

rank P

(k )

= 2,

k = 1, 2,

(111)

1 2 v2 2 + 2 (k ) 1 2 v1 (k) 2 = 1 v2 - 2 + 2 (k) 1 2 v1 0 1 M 1 + 2 M 0 2 2 1 + 2 M = 0 up to replacement of the functions v (108) and replacement of k by k = exp Each of P P
(k ) (1) (1,2) 1,2



(k ) 2 2 v2 - 2 2 2 1 + 2 v ( k ) 1 (k) , 1 2 v2 -2 1 2 1 + 2 v (k) 1 0 0 (k ) 2 v2 -M 2 (k ) v1 , M1 0
(1,2) 1,2

satisfying (76) by functions v
2 (3 - 1) - 3 v (k ) 1 (k) 2

satisfying

df (v

(k) 1

)

d

v

f (µ + d)

3

.

(112)

and P

(2)

has four Casimir functions dJk = 0, P
(k )

dC = 0,

P

(k)

d3 = 0,

P

(k)

d(1 M2 - 2 M1 ) = 0

and determines a symmetry field by acting on M3 : P
(k)

dM3 = XS .

The second, `horizontal' component P of the solutions (110) has the form P = 0 - M , rank P = 2, (113)


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

where is the same matrix as in (87), but 2 d2 f (M3 (3 - 1) + (1 M1 + 2 M2 )3 ) 0 -M3 + f (µ + d) 2 d1 f (M3 (3 - 1) + (1 M1 + 2 M2 )3 M = 0 - f (µ + d) 0 where =
2 2 (1 - 3 )(µ + 3 d)f M3 + 3 f ( µ + d) 2 (µ + 3 d)f 3 - f (µ + d) 1- 2 3

) ,

(1 M1 + 2 M2 ).

The bivector P has the following Casimir functions: P dC = 0, P dJ1 = 0, P dJ2 = 0, P d 2 = 0. 1
(k )

As in the case of a body of revolution on the plane, the bivectors P and P have formal singularities for 3 = ±1. Using the freedom we have in choosing the coefficients and and the freedom in choosing the linear first integrals which play the role of Casimir functions, we can construct non-singular solutions P (k) in (110). In the general case, on the regular part of the Poisson manifold each of the P (k) in (110) has two Casimir functions: P
(k )

dC = 0,

P

(k)

dJk = 0,

k = 1, 2.

As before, the vanishing of the Schouten bracket is equivalent to the existence of the linear integrals of motion J1,2 in (107): [P
(k )

,P

(k)

]=0



Jk = X (Jk ) = 0.

This means that both of these conditions lead to the same system of equations for (k ) the functions v1,2 in (108). The Poisson structure on the partially reduced phase space found enables us to study the structure of the vector field X . Prop osition 7. For a bal l on a surface of revolution , the vector field X corresponding to (104) on the 6-dimensional partial ly reduced phase space is a sum of a Hamiltonian field and the symmetry field XS in (74) : 3 R P (k) dH + k XS , 2 (3 f - (1 - 3 )f ) u1 (1 M1 + 2 M2 ) + u2 M3 k = , f (d + µ) X=- and the coefficients in the expansion u1 = -
2 2 3 R k 3 Rf (d3 (1 - 3 )v1 + (3 d - µ - d)v 2+ (k ) 2 2 1 - 3 µv1 (1 - 3 )(3 f - (1 - 3 )f ) (k ) (k ) 2

(114)

)

,

u2 = R +

k 3 Rf (v

(k ) 1 (µ (k) µv1 (3 f

2 + 3 d) - d3 v

(k) 2

)

- (1 - )f )

2 3




Non-holonomic dynamics and Poisson geometry

525

depend on the functions v motion Jk . Setting = const and =-
2 (1 - 3 )(3 v (k) 1

(k ) 1

and v

(k ) 2

in the definition (107) of the integrals of

-v

(3 f - (1 - X=-

(k ) 2 2 )d 2 3 )f )

+ µ((v (v
(k) 1

(k) 2 1)

2 + (1 - 3 )(v (k ) 3 v2

(k) 2 2)

) 3 f µv1
(k)

k

(1 - ) +
(k)

2 3

,

(115)

)

we get that 3 R P 2 (3 f - (1 - 3 )f ) R(3 v
2 (1 - 3 )(3 v (k ) 1 (k) 1

dH + Jk XS ,

(116)

where = -v
(k) 2 2)

-v

d + µ((v

(k) 2) (k ) 2 1)

2 + (1 - 3 )(v

(k ) 2 2)

. )

Prop osition 8. For a bal l on a surface of revolution the vector field X corresponding to (104) is conformal ly Hamiltonian , X=- 3 R P 2 (3 f - (1 - 3 )f )
(k)

dH,

with respect to a rank- 4 Poisson bivector if Jk = 0, k = 1, 2.

6.2. Hamiltonization on a reduced submanifold. Following [27], we now consider the subspace with coordinates 3 , K 1 = f ( , M ), K2 = µ3 = µM2 + d3 ( , M ) , µ+d K3 = 2 M1 - 2 M 1-
2 3 1

,

which is invariant under the action (72) of the symmetry group SO(2). As before, these coordinate functions are defined only for 3 = ±1. The two Poisson bivectors P (1,2) in (110) have the same pro jection on this reduced 4-dimensional space. For =- R 2 f - ((1 - 3 )/3 )f


and =

2 1 - 3 R 2 )/ )f , f - ((1 - 3 3

this pro jection has the form 000 0 0 0 P = 0 0 0

1




f K2 3 , d K1 (µ + d)f 0

rank P = 2.

(117)

The original vector field X in (114) is Hamiltonian on the reduced submanifold with respect to this rank-2 Poisson structure with the Casimir functions J1,2 : X = P dH.


526

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

As before, dividing the two Hamiltonian equations K1 = {H, K1 } = - by the third equation 3 = {H, g3 } = - K3 , µ+d f K2 K3 , 3 ( µ + d) K2 = {H , K 2 } = - d K1 K3 f (µ + d)2

we obtain a system of first-order linear non-autonomous differential equations dK1 f = K2 , d3 3 dK2 d K1 , = d3 f (µ + d) (118)

an analogue of which was obtained by Routh [75] in the variables connected with a semimovable coordinate system. 6.3. Reducing the Poisson bracket to a canonical form. We now consider reduction of the bivector P (k) in (110) to a canonical form using the Poisson maps k : ( , M ) ( , L), which coincide in form with the maps (91) but involve
(k ) 1 (k ) 2 2 ) k

k =

v

1-

(v

2 2 (1 - 3 )(µ + 3 d)f 3 f (µ + d)



d3 ,

(119)

the functions k defined by (112), and the functions v1,2 satisfying equations (108). The maps k corrected in this way reduce the Poisson bivectors (110) to the canonical bivector (10) on the Lie algebra e (3) for = const. By using this map we can identify the original dynamical system (104) with a dynamical system on the sphere, endowed with the canonical Poisson bracket. The corresponding Hamiltonian has the standard form H = AL2 + B (L1 2 - L2 1 )2 + bC L3 + b2 D, 3 (120)

(1,2)

where A, B , C , and D are functions only of 3 , while the second integral of motion is linear in L3 . Since the functions are arbitrary, we can always identify the coefficients B in the Hamiltonians (93) and (120), which correspond to different non-holonomic systems. However, in the general case the question of identifying the Hamiltonian (120) corresponding to a ball on some concrete surface of revolution with the Hamiltonian (93) corresponding to a body of revolution of special form on the plane (up to canonical coordinate changes on the sphere) is still open.


Non-holonomic dynamics and Poisson geometry

527

6.4. Motion along a parab oloid of revolution. Assume that the centre of a ball moves along a paraboloid of revolution z = c(x2 + y 2 ). In this case f ( 3 ) = - and equations (108) have the solutions v
(1,2) 1

1 2c

3

=

± 3

,

v

(1,2) 2

=

1± 3

±

1±

± -1 3

,

(121)

where is a real number depending on the mass of the ball, its radius, and the magnitude of the inertia tensor: = µ , µ+d d = mR2 .

Remark 14. According to [27], the product H2 = J1 J2 of the integrals of motion is an algebraic integral of motion of second order in the momenta. In this case the definitions of the Poisson bivector and the map reducing it to canonical form involve the functions
1,2

=

1± 3

and

1,2

=±

2 2 (1 )2 3 (1 - 3 ) 2 2 2 3 ± (1 - 3 )

2 3

.

Replacing the integrals of motion J1,2 , for example, with their linear combinations -1 +1 (±) (±) J1 ± J2 v1 (1 M1 + 2 M2 ) + v2 M3 , J± = 2 2nu we obtain bivectors P (see (111)), which have no singularities for 1,2 = 0 and 3 = ±1. Thus, using the freedom we have in choosing the linear integrals of motion, we can obtain Poisson structures which are non-singular at the points in the Poisson manifold which correspond to stable equilibria. Now we consider the Poisson map (91) reducing the bracket to a canonical form. Setting c = 0 in (91) and applying the Poisson map to the bivector P (1) with Casimir function J1 = b, we obtain the following expressions for integrals of motion on the sphere: H=
2 2 3 L2 1 - 2 3 + 2 (2 L1 - 1 L2 ) 2 2d(1 - 3 ) (1 - )2 2 (±)

+ and

2 (23 - 1)bL3 (1 - )2 b2 + 4 2

-2 3

J2 =

2 L3 . 2 - 1

In the standard spherical coordinate system (100) we can, as above, integrate the equations of motion in quadratures in a completely standard way.


528

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

7. The Suslov system Let us consider the motion of a rigid body with a fixed point on which we impose the non-holonomic constraint ( , a) = 0, where is the angular velocity vector and a is an arbitrary fixed vector in the coordinate system attached to the body. Several ways to bring about such a constraint (which was proposed by Suslov) were discussed in [18] and [77]. The equations of motion have the following form in the body frame rigidly attached to the body: d I = I â + a, dt d = â , dt

where I is the inertia tensor, is an indeterminate Lagrange multiplier, and is the Poisson vector corresponding to the unit vector on the fixed axis. We can always choose an attached coordinate system such that a = (0, 0, 1) and the corresponding centrifugal moment of inertia vanishes: I12 = 0. Then the constraint equation takes the form 3 = 0, and the equations of motion can be written as I11 1 = -2 (I13 1 + I23 2 ), 1 = - 3 2 , I22 2 = 1 (I13 1 + I23 2 ), 3 = 1 2 - 2 1 . 3 = 0, (123) (122)

2 = 3 1 ,

The corresponding vector field X has the integrals of motion H1 = and H4 = (f , ), f = f1 ( ), f2 ( ), f3 ( ) . (125) The last integral of motion involves a vector f which is a particular solution of the equation f + â f = 0, so that H4 = (f, ) + (f , ) = (f, ) + (f , â ) = (f + â f , ) = 0. Besides these four integrals of motion, the field X corresponding to (123) has the invariant measure µ=g
-1

1 2 2 (I11 1 + I22 2 ), 2

H2 = ( , ),

H3 = 3 ,

(124)

d d ,

g = I13 1 + I23 2 .

Its properties and conditions ensuring that a single-valued integral of motion H4 exists are discussed in detail in [18]. As before, we substitute the expressions for the integrals of motion H1 , . . . , H4 and the ansatz
3

Pij =
k=1

ck ( )k + dij ( ), ij


Non-holonomic dynamics and Poisson geometry

529

which is linear in and in which ck ij equations (12) and solve the resulting vious cases we single out the integral integral is the Casimir function of the

and dij are unknown functions of , into equations. However, in contrast to the preH3 = 3 and assume that this particular required Poisson bivector:

P dH3 = P d3 = 0. Thus, the constraint (122) fixes some symplectic leaf on which the dynamical system under consideration evolves. 7.1. Poisson structures for inertial motion. Equations bivector have several rank-2 solutions. We write out only one -g1 I22 2 g1 I11 0 W P= , W = -g2 I22 2 g2 I11 -W 0 -g I g I
3 22 2

3 11

(12) for the Poisson of them: 1 0 1 0 , 1 0

where the gk are arbitrary functions of connected by the equality g1 1 + g2 2 + g3 3 = 0. Equations (12) also have rank-4 solutions. We write out one such solution in the case when the additional first integral H4 in (125) is a linear polynomial in the velocities and the ansatz we are using for the Poisson bivector is also polynomial in . We recall [18] that for I
13

= 0,

I

23

=

(I

11

- I22 )I

22

the integral of motion H4 has the following form: H4 = (I , ) = I11 1 1 + I22 2 2 + I23 2 3 . Then the solution of (12), P = P + P , (126) is a sum of two compatible Poisson bivectors of rank 2, P and P . The first has the form W 0 P = , 0 0 where 0 (I23 3 + I22 2 )2 -I23 2 W = I23 (I23 2 - I22 3 ) I22 2 P = where
- ( (I23 3 + I22 2 )1 2 2 (I23 3 + I22 2 )(I23 (1 + 3 ) + I22 2 3 ) I23 2 - I22 3 2 2 I23 3 + I22 2 )(I23 2 3 + I22 (1 + 2 )) I23 2 - I22 3
2 -I11 1 2 (I23 ( + 3 ) + I22 2 3 )I11 1 I23 2 - I22 3 2 2 (I23 2 3 + I22 (1 + 2 ))I11 1 - I23 2 - I22 3 2 1

I23 2 0 -I11 1

-I22 2 I11 1 . 0

The second bivector is 0 - , W
0

=

0 0


530

A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

and

0 W = -H 0



4

H4 0 0

0 0 . 0

The coefficients and in the decomposition (126) are arbitrary functions of the Casimir functions of the bivectors P and P , respectively. For example, P has the Casimir functions P dH4 = 0, P di = 0, i = 1, 2, 3,

and hence is an arbitrary function of . For P we could find only three Casimir functions out of four: P dH4 = 0, P dH2 = 0, P d3 = 0.

In the general case a full solution (126) is a Poisson bivector of rank 4 with Casimir functions P dH4 = 0, P d3 = 0. Substituting the explicit meromorphic solutions of Poisson's equations which were written out for this case in [40], into the definition (126) of the bivector P , one can prove that the singularity at I23 2 - I22 3 = 0 can never be attained in a physical motion. Prop osition 9. On the regular part of the Poisson manifold the vector field X corresponding to (123) can be represented as X = 1 P dH1 + 2 P dH2 , where 1 = - and 2 (I23 (1 2 - 2 1 ) + (I11 - I22 )1 3 ) I11 H4 (I22 (1 2 - 2 1 ) + I23 1 3 )

2 2 1 I23 (I23 (1 1 2 - 2 1 - 2 3 ) - (1 1 + 2 2 )I22 3 ) . 2 H4 (I23 3 + I22 2 )2 A discussion of possible applications of this decomposition of the original vector field into Hamiltonian ones lies outside the scope of our survey. Thus, in this special case we have shown that also in the Suslov problem the common level surfaces of the integrals of motion can be identified with Lagrangian submanifolds.

2 =

7.2. A sp ecial case of a motion in a p otential field. If a = (0, 0, 1) is an eigenvector of the inertia tensor: Ia = µa, µ = I33 , that is, if I
13

= I23 = 0,

then we can consider an integrable system which describes the motion in a potential field [18]. In this particular case the equations of motion have the form I11 1 = U U U U - 3 , I22 2 = 3 - 1 , 3 = 0, 3 g2 1 g3 1 = -3 2 , 2 = 3 1 , 3 = 1 2 - 2 1 .
2

(127)


Non-holonomic dynamics and Poisson geometry

531

According to [18] the corresponding vector field X is Hamiltonian, X = P dH, if U ( ) = µ3 + V (1 , 2 ). In this case the Poisson bivector is 0 0 0 -1
- I111

H=

1 ( , ), 2

-2
1

0
- -I111 3 U

I

-1 22

3 U U

0 2 P = 0 -1 -I22 3 U 0

0 -
- I221 1

0
- -I111 2 U -1 I22 1 U

I

-1 11

2 U

3 U

0 0 0

0 0 0

0 0

0

0 0 0 , 0 0 0

where k = / k . If µ = 0, then this is a rank-4 Poisson structure: rank P = 4. As before, this means that for this non-holonomic system the common levels of the integrals of motion can be identified with Lagrangian submanifolds.

8. Conclusion A study of a non-holonomic motion of a rigid body usually begins with consideration of a free motion of the body, then imposition of holonomic and non-holonomic constraints, and reduction with respect to the symmetries present in the problem. We recall that for a free motion of a rigid body the phase space is the 12-dimensional space T E (3), where E (3) is the group of motions of the Euclidean 3-space. This phase space is a Poisson manifold, endowed with the canonical Poisson bivector, so that we can write the equations of motion of the free rigid body in Hamiltonian form. At each stage in the transition from the Hamiltonian equations of motion of the free body to the equations describing the non-holonomic motion on the phase space reduced with respect to the symmetries, we can find traces (non-linear deformations) of the original Poisson structure on T E (3). A comprehensive investigation of all the manifestations of the original Poisson structure has yet to be carried out, but we have collected in this survey all the basic information currently available about non-linear Poisson structures arising in non-holonomic mechanics in 3-dimensional Euclidean space. The authors are indebted to A. V. Bolsinov, I. A. Bizyaev, and A. A. Kilin for valuable discussions and comments, and to V. V. Kozlov for useful advice contributing to essential improvements of this paper.


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A. V. Borisov, I. S. Mamaev, and A. V. Tsiganov

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Received 23/DEC/13 Translated by N. KRUZHILIN

Alexey V. Borisov Udmurt State University E-mail : borisov@rcd.ru Ivan S. Udmurt Izhevsk E-mail : Mamaev State University; State Technical University mamaev@rcd.ru

Andrey V. Tsiganov St. Petersburg State University E-mail : andrey.tsiganov@gmail.com