Poisson structures for two nonholonomic systems with partially reduced symmetries

We consider nonholonomic systems which symmetry groups consist of two subgroups one of which represents rotations about the axis of symmetry. After nonholonomic reduction by another subgroup the corresponding vector fields on partially reduced phase space are linear combinations of the Hamiltonian and symmetry vector fields. The reduction of the Poisson bivectors associated with the Hamiltonian vector fields to canonical form is discussed.


Introduction
In 1887-1911 S.A. Chaplygin proposed a theory of reducing multiplier for investigation of nonholonomically constrained mechanical systems with an invariant measure [9,10]. The first part of this theory is a nonholonomic reduction of the phase space M by a symmetry group G, which is based on the most famous and established method for finding exact solutions of differential equations, which is called the classical symmetries method or group analysis, which originated in 1881 by S. Lie. The second part states that nonholonomic system on the reduced phase space M/G became Hamiltonian after a suitable reparameterization of time, a process now referred to as Chaplygin Hamiltonization [6,13,15,17,25,28].
It is known that sometimes symmetry group G consists of various subgroups G = G 1 × G 2 associated either with the translations and the different kinds of rotations [11,16,41], or with external and internal symmetries, see [3,13,12] and reference within. The existence of some subgroups allows us to make a partial nonholonomic reduction M/G 1 by one of the subgroups [12], then to reduce the corresponding Poisson bracket to canonical one and only then to finish reduction by remaining subgroup G 2 . Thus, we want to construct an appropriate Poisson map which identifies the given dynamical system on the partially reduced phase space with a dynamical system on some well-known manifold equipped with the canonical Poisson brackets. This reduction to the canonical Poisson brackets may be useful for the comparison of the different nonholonomic system to each other and to the application of the well-studied methods of Hamiltonian mechanics to the nonholonomic systems. For instance, in [35,36] we used such reduction in order to prove the equivalence of the Chaplygin ball problem with the nonholonomic Veselova problem.
Below we consider the motion of the body of revolution on a plane and motion of the homogeneous ball on the surface of revolution. For both these systems initial phase space is M = T E (3), where E(3) is the Euclidean group of all rigid motions. The symmetry groups G = G 1 × G 2 consist of different subgroups G 1 = E(2) or G 1 = SO (3), and the common subgroup G 2 = SO (2) represents rotations about the corresponding axis of revolution. The main aim of this paper is to identify the common level surfaces of integrals of motion with lagrangian foliation with respect to some Poisson bivector. Later we will try to identify the common level surfaces of integrals of motion with bi-lagrangian foliation with respect to some Poisson pencil and then with some subvariety of the Jacoby variety associated with some algebraic curve related with this pencil.

Vector fields on partially reduced phase space
Let us consider some smooth manifold M with coordinates x 1 , . . . x m and a dynamical system defined by the following equations of motioṅ This system of ODE's defines a vector field which is a linear operator on a space of smooth functions on M that encodes the infinitesimal evolution of any quantityḞ = X(F ) = X i ∂F ∂x i along with the solutions of the system of equations (1). In Hamiltonian mechanics any Hamilton function H on M generates vector field X describing the dynamical system X = P dH .
Here dH is a differential of H and P is a Poisson bivector on the phase space M. For a lot of nonholonomic dynamical systems vector fields on the reduced phase space are created using the Hamilton function H and the reducing multiplier g X = gP dH .
This vector field X (4) is the so-called conformally Hamiltonian vector field, see examples of such fields in [6,7,34,35].
In this note we discuss the rolling motion of a heavy rigid body of revolution on a rough horizontal plane and the motion of a homogeneous ball on the surface of revolution [9,27]. For both of these nonholonomic systems vector fields X on the partially reduced six-dimensional phase space are linear combinations of the Hamiltonian field P dH and the symmetry vector field X S associated with the rotations about the axis of revolution: According to the Noether's theorem if the symmetry field X S is related with the integral of motion J, then we can rewrite (5) in the following form Below we present the corresponding Poisson bivectors P , rank P = 4, which are missed in the existing works on these nonholonomic systems. Moreover, we show that this bivector P is a deformation of the canonical Poisson bivector on the Lie algebra e * (3). According to [9] projection of X (6) on the four dimensional submanifold M/G is the Hamiltonian vector field. This Chaplygin idea of reduction has generated considerable interest and has been applied to various nonholonomic systems, see [3,7,17,19,25,40,41] and the references within. However, projection of P on the invariant submanifold M/G is usually a highly degenerate Poisson bivector. Thus we could have some problems with the definition of the symplectic leaves and the Casimir functions, which have to be solved before creating suitable Hamiltonian vector fields on these symplectic leaves.

1.2
Euler-Jacobi theorem and rank-two Poisson structures Let us consider the following system of differential equations where X i are the functions on variables x 1 , . . . , x k . If we know the Jacobi multiplier µ defined by and know k − 2 independent first integrals H 1 , . . . , H k−2 , we can integrate this system by quadratures. Namely, according to the Euler-Jacobi theorem, in this case we can introduce new variables y 1 , . . . , y k by rule In y-variables the initial system of equations has the following form or Using the definition (7) of the Jacobi multiplier µ we can prove that µ(Y 2 dy 1 − Y 1 dy 2 ) is the total differential. So, there is one more independent first integral It is well-known that any k-dimensional dynamical system with the independent k − 1 first integrals and the Jacobi multiplier µ may be rewritten in the Hamiltonian form according to the Vallée-Poussin theorem on functional determinants [39]. Here P (y) is a ranktwo Poisson bivector and H is a function on H 1 , . . . , H k−1 , see details in [5] . For instance, we can put P (y) so that Other brackets are equal to zero. The corresponding rank-two bivector P (y) is the Poisson bivector iff If we choose variables y 1 , . . . , y k so that the ratio Y 2 /Y 1 is independent on y 3 , then the Hamiltonization of dynamical equations (9) is equivalent to the existence of invariant measure because Here [., .] is a Schouten bracket. This construction of rank-two Poisson bivectors for the nonholonomic Stübler model is discussed in [38]. Such rank-two Poisson structures are well-studied [7,14,16,21,26] and, therefore, we will only consider rank-four Poisson structures below.

Rolling motion without sliding
In [1,2,9,7,20,23,24,27] there are variants of the derivation of the equations of rolling motion of a heavy rigid body, which touches a surface at one point. For brevity we reproduce only the necessary facts about the motion of bodies of revolution on the plane and facts about the motion of the homogeneous ball on the surface of revolution. In both cases the equations of motion on six-dimensional phase space have a rotational symmetry about the axis of revolution.

Body of revolution on a plane
The moving body is subject to two kinds of constraints: a holonomic constraint of moving over of a horizontal plane and no slip nonholonomic constraint associated with the zero velocity at the point of contact v + ω × r = 0.
Here ω and v are the angular velocity and velocity of the center of mass of the body, r is the vector joining the center of mass with the contact point and × means the vector product in R 3 . All the vectors are expressed in the so-called body frame, which is firmly attached to the body, its origin is located at the center of mass of the body, and its axes coincide with the principal inertia axes of the body.
In the body frame the angular momentum M of the body with respect to the contact point is equal to M = Jω , Here E is a unit matrix, m is a mass and I = diag(I 1 , I 2 , I 3 ) is an inertia tensor of the rolling body.
After elimination of the Lagrange multiplier and reduction by E(2) one gets the following equations of motionṀ where γ is the unit vector orthogonal to the plane at the contact point and M F is the moment of external forces with respect to the contact point depending on γ only. Vector r is expressed as a function of normal vector γ using the following relation that defines the Gauss transformation, if f(r) = 0 is an equation of the body surface. Intermediate five dimensional reduced space M/G 1 may be identified with the submanifold ||γ|| = 1 in the six-dimensional space (γ, M ) ∈ R 3 × R 3 . Similar to the Lagrange top we will study the vector field X defined by equations (13) on the six-dimensional manifold with local coordinates x = (γ 1 , γ 2 , γ 3 , M 1 , M 2 , M 3 ). Restriction ||γ|| = 1 we will consider as an integral of motion for these equations in order to have some additional freedom for the search of the Poisson bivectors, see Section 2.3.
If the moment of external forces is described by potential U depending only on γ we have In this case vector field X (13) possesses two integrals of motion In some cases vector field X has an invariant measure and additional integrals of motion. In the next paragraph we consider one of these particular cases.
If we want to study a rigid body of revolution we have to impose some restrictions on the tensor of inertia I = (I 1 , I 2 , I 3 ) and on the equation of the body surface f(r) = 0. It means that the surface of the body and its central ellipsoid of inertia are coaxial surfaces of revolution. Because (ṙ, γ) = 0 functions f 1,2 (γ 3 ) in (15) satisfy the equation that defines a meridional section. Below we do not use this equation at all. We also assume that U is an arbitrary function of γ 3 , i.e. that it depends only on the slope of the revolution axis of the body to the vertical [9]. In particular, it means that in the case of gravity field the center of mass must be situated on the axis of revolution.
Under the above conditions the symmetry group G consist of two independent parts [11]. The first part of the symmetry group G 1 = E(2) is generated by translations of the horizontal plane and rotations about the vertical axis while the second part G 2 = SO(2) is generated by rotations about the axis of revolution. On the six-dimensional phase space vector field X (13) has a symmetry field associated with the second part of the symmetry group and an invariant measure According to [1,2,9,7] there are also two additional integrals of motion, which are linear in momenta functions In generic case coefficients v are the real analytic non algebraic functions on γ 3 , which satisfy to the following system of equations: Here and everywhere below we omit dependence of functions on γ 3 , i.e.
For example, let us consider a rolling disk of radius R with the center of mass displaced on a along the axis of dynamical symmetry. In this case At a = 0 functions v are the Legendre functions of the first and second kind, respectively. However, at a = 0 we have only implicit definitions of the additional integrals of motion (18).

Homogeneous ball on a surface of revolution
Let us consider the homogeneous ball with mass m, radius R and tensor of inertia I = µE, where E is a unit matrix. We are going to study the case when the ball rolls without sliding on the surface having only one point in common during all motion. The surface is defined by equation and the rolling ball is subject to two kinds of constraints: a holonomic constraint of moving over the surface and no slip nonholonomic constraint associated with the zero velocity at the where we denote the velocity of the ball's center by v, the angular velocity of the ball by ω and vector joining the ball's center with the contact point by a.
If N is the reaction force at the contact point, F and M F are the external force and its moment with respect to the contact point, then the conservation principles of linear and angular momentum read as After substituting a = −Rγ, where γ is a normal to the surface, and eliminating the reaction force N one gets the following equations of motioṅ which define vector field X on the six-dimensional phase space with local coordinates x = (γ 1 , γ 2 , γ 3 , M 1 , M 2 , M 3 ). Here d = mR 2 and vectors ω and r are functions on x defined by the following relations If external force F is a potential force associated with potential U = U (r + Rγ), then and equations (21) possess first integrals Below we consider the homogeneous ball rolling on the inner side of the surface of revolution obtained by rotation of a smooth curve about a vertical axis. Following [8,24,27], we use parameterization of the surface on which the ball's center of mass is moving, so that where f = f (γ 3 ) is a sufficiently smooth function. It means that the curvature of the meridian of surface is smaller than 1/R. Moreover, for brevity, we consider free motion at U = 0, because free motion and potential motion are usually associated with the same Poisson brackets.
Apparently Routh was the first to explore this problem. In [27] he described the family of stationary periodic motions, obtained a necessary condition for stability of these motions and also noticed that the integration of the equations of motion may be reduced to an integration of a system of two linear differential equations with variable coefficients and considered several cases when the equations of motion can be solved by quadratures.
In this case the symmetry group consists of two independent parts G = SO(3) × SO(2) [14,41]. The first part is generated by rotations of the ball about its center while the second part is generated by rotations about the vertical axis of revolution. In contrast with the body of revolution on a plane all the vectors are expressed in the space frame. In some sense change of frames allows us to "exchange" inner and external symmetry subgroups according to [12].
As above we will extend the five dimensional reduced space M/G 1 to the six-dimensional space (γ, M ) ∈ R 3 × R 3 and will consider restriction ||γ|| = 1 as an integral of motion. On the six-dimensional phase space vector field X (13) has a symmetry field (16) and invariant measure that means that vector field X may be integrated by quadratures.
As above, there are linear in momentum integrals of motion Here functions v 1,2 (γ 3 ) are the solutions for the following system of equations Relations of these integrals with symmetries is discussed in [8,14,27,41]. For example, let us suppose the ball's center is moving on the ellipsoid of revolution with the principal semi-axes b 1/2 1,3 . In this case and equations (24) have the following two independent solutions v (1,2) 1 where L (1,2) are the associated Legendre functions of the first and second kind and According to [8] there is also the second order in momenta algebraic integral of motion and equations of motion may be solved in the elementary functions.

The Poisson structure for integrable nonholonomic systems
In the Chaplygin theory of reducing multiplier we have a constructive algorithm for calculation of the Poisson structures on the completely reduced submanifold M/G only. In order to get the Poisson structure on the partially reduced six-dimensional phase space M/G 1 with the coordinates x = γ, M we have to apply other methods. In this paper we use the brute force approach proposed in [32]. Namely, six equations of motion (13) and (21) possess four integrals of motion and the invariant measure (17,22) and, therefore, they are integrable by quadratures according to the Euler-Jacobi theorem. If we identify the common level surfaces of integrals H 1 , . . . , H 4 with the Lagrangian foliation of symplectic leaves of some unknown rank-four Poisson bivector P , then the following equations have to be fair If [P, P ] = 0, bivector P is a Poisson bivector. The brute force method consists of a direct solution of the equations (26) with respect to entries of bivector P using an appropriate anzats. Remind, that a'priory these equations have infinitely many solutions [30] and using of anzats allows us to get some partial solutions in a constructive way. Below we use the following simple anzats where c k ij and d ij are unknown functions on γ. Points on the phase space manifold where the rank of P is full are called regular points, and those where the rank is less than full, singular. Usually the change in rank governs physical properties like the presence of extra equilibria, or the stability of existing equilibria. Center of the Poisson bracket algebra consists of the Casimir functions which are in the involution with all other functions. The level sets of these Casimir functions would locally carve out symplectic leaves of even dimension equal to the rank of the Poisson bivector. When rank changes occur, these leaves drop in dimension by an even integer: extra Casimir functions, called subcasimir functions, arise and the new symplectic leaves of reduced dimension (lower dimensional strata or so-called thin orbits) are defined by the intersection of level sets of both the Casimir and the subcasimir functions. Physical systems tend to equilibrate towards states of greater symmetry, which occur on dynamical leaves of greater codimension. Thus, singular leaves become relevant as arenas where actual stability issues of equilibria must be addressed, see [22] and references within.
In [31] we successfully solved equations (26) for the Lagrange top, which has the same symmetry vector field X S and well-known linear integrals of motion J 1 = M 3 and J 2 = (γ, M ). For one of these solutions function C = (γ, γ) is not the Casimir function and, therefore, we will not consider a five-dimensional submanifold defined by the physical condition C = 1 in order to preserve generic mathematical construction, see also [29]. In [4,32,34,35] reader can also find other solutions of the equations (26) for various nonholonomic systems.

Poisson brackets for body of revolution on a plane
Substituting explicit definitions of the Hamilton function H and the geometric integral C and implicit definitions of the remaining two integrals of motion J 1,2 (18) into the equations (26) one gets overdetermined system of algebro-differential equations on coefficients c k ij and d ij of no more than linear polynomials in momenta P ij (28).
In contrast with the Lagrange top in the nonholonomic case the resulting equations have only two solutions with rankP (k) = 4 if γ 3 = ±1. Here P α and P β are compatible rank-two Poisson bivectors satisfying (26), whereas α and β are arbitrary functions on γ 2 /γ 1 and γ 3 , respectively. It allows us to say that solutions (29) of (26) are decomposed into "horizontal" and "vertical" parts with respect to the action of the symmetry subgroup, similar to decomposition of the vector field X, see [25] and [3,13,19]. First parts P where These rank-two Poisson bivectors have the following Casimir functions and give rise to the symmetry field acting on M 3 We can consider P (k) α as the "vertical" parts of P (k) (29), which will be lost taking the quotient by symmetry subgroup SO(2).
Second part of (29) looks like where This rank-two bivector P β has the following four Casimir functions It depends only on the form of the body, i.e. only on f 1,2 and it may be considered as "horizontal" part of the complete solution (29), which will survive in the reduction by SO(2) symmetry.
In fact bivector P β coincides with the standard rank-two bivector P (y) (9) if we put Here g is defined by (17). Using freedom in a choice of functions β and v 1,2 we can get Poisson bivectors P (k) with well defined limit at γ 1,2 → 0. The corresponding bivectors have rank two at γ 3 = ±1 and functions x 3 and M 3 play the role of the subcasimir functions. This change in rank indicates the possible presence of equilibria or the stability of existing equilibria. Of course, for an existing of equilibria we have impose certain restrictions on the angular velocity of the body, see discussion of the Rouths sphere in [11].
Equality of the Schouten brackets to zero is equivalent to the existence of integrals of motion J 1,2 (19) [P (k) , P (k) ] = 0 ⇔J k = X(J k ) = 0 .
We can say that the existence of the Casimir function is equivalent to the existence of the integral of motion of the corresponding dynamical system..

Proposition 1
For the rigid body of revolution vector field X (13) is a linear combination of the Hamiltonian vector field and the symmetry field X S (16) where coefficients depend on a pair of functions v (k) 1,2 entering into J k (18). The proof is a straightforward verification of the equations (32).
It is easy to prove that 1 . It means that vector field X is the Hamiltonian vector field only if there is integral of motion J = v On the other hand, if we take α = const and then we get where It allows us to prove the following Proposition 2 For the rigid body of revolution vector field X (13) is the conformal Hamiltonian vector field on the zero level of integrals of motion where β is given by (33).
In generic case in order to get the Hamiltonian vector field we can use the Chaplygin Hamiltonization, which is discussed in the next paragraph.

The Chaplygin Hamiltonization
Following to [9,7] let us consider the ultimate four-dimensional reduced phase space M/G with coordinates γ 3 and where These coordinates are defined only at γ 3 = ±1. Both bivectors P (1,2) (29) have a common projection on this subspace of the six-dimensional space with coordinates x = (γ, M ). At β = 1 it looks likeP Projection of the initial vector field X (13,32) on this subspace becomes the Hamiltonian vector fieldX =P dH with respect to this degenerate Poisson structure. Dividing two Hamiltonian equations of motionK by the third Hamiltonian equation of motioṅ one gets a system of linear non autonomous first order differential equations which was obtained by Chaplygin in [9]. The generic solution of this system reads as where Φ 1,2 are fundamental solutions of (37), whereas constants c 1,2 are the values of integrals of motion J 2,3 (18). Solving (38) with respect to c 1,2 we could get these integrals which are the Casimir functions of the Poisson bivectorP , see details in [7]. The Poisson bivectorP was obtained in [7] using Chaplygin's reducing multiplier theory. Similar rank-two Poisson structures are discussed in [14,16,21,26].

Reduction to the canonical Poisson brackets
Let us consider the following transformation of momenta and where ε ijℓ is a completely antisymmetric tensor.
The proof consists of substituting momenta L i (39) into the brackets (40). The Poisson map (39) is defined on the regular part of the Poisson manifold without the singular points γ 3 = ±1. However we suppose that this singularity isn't connected with a possible existence of equilibria because the same singularity at the Poisson map we found for the Chaplygin ball, Veselova systems [35,36] and generalized Chaplygin ball [37]. It will be interesting to study such common properties of various deformations of the canonical Lie-Poisson brackets associated with different nonholonomic models.
We can use this reduction of the Poisson bracket to a canonical one in order to prove equivalence of some nonholonomic systems. For instance, in [35,36] such reduction of the Poisson brackets associated with the Chaplygin ball and Veselova system allowed us to prove trajectory equivalence of these nonholonomic systems.
We can also use this reduction in order to solve equations of motion. Namely, it is wellknown that symplectic leaf of e * where A, B, C and D are functions on γ 3 . The corresponding Hamiltonian equation of motion for γ 3 is solved completely similar to the Lagrange top. Becauseγ 3 = {H, γ 3 } and coefficients of (32) depend only on γ 3 we can solve the remaining equations of motion by quadratures too.
However, our main aim will be identification of the common level surfaces of these integrals with bi-lagrangian foliation using a concept of the natural Poisson bivectors on the Riemannian manifolds [33]. After that we could to apply inverse to (39) transformation to the corresponding Poisson pencil and to get bi-Hamiltonian description of the initial non-holonomic model.

Gyrostatic generalizations
According to S.A. Chaplygin [9] we can add to the body the uniformly rotating balanced rotor. The corresponding system can be interpreted as a nonholonomic gyrostat. The gyrostatic effect can also be obtained by an adding multiply connected cavities completely filled with the ideal incompressible liquid possessing nonzero circulation in the body.
In this case the equations of motion are equal tȯ where S is the constant three-dimensional vector of gyrostatic moment. For the body of revolution we suppose that gyrostatic moment is directed along the axis of revolution. In this case linear integrals of motion (18) are shifted by linear in s term J (s) Here v (k) 1 are functions on γ 3 from the definition of integrals J k (18) for the body without rotor. In this case even for a disk (20) one gets non algebraic integrals of motion J k because integrals L (1,2) (b − , γ 3 ) dγ 3 on the Legendre functions L (1,2) are non algebraic functions.

Proposition 4 For the nonholonomic gyrostat solutions of the equations (26) have the form
is a 3 × 3 skew symmetric matrix associated with three-dimensional vector S. In our case (44) S 3 = s and S 1 = S 2 = 0 . The corresponding vector field is a sum of the Hamiltonian vector fields and symmetry vector field with the same coefficients η k as for the body without rotor.
It is interesting that for the gyrostatic generalisation of the Chaplygin ball we can use the absolutely same shift of the corresponding Poisson bivector.

Example
Let us consider the Routh sphere, which is one of the well known examples of the rigid body of revolution [4,7,10,11,16,26]. The center of mass of this sphere is shifted with respect to its geometric center and the line joining the center of mass and the geometric center is an axis of inertial symmetry. It means that in the plane perpendicular to this axis the moment of inertia tensor has two equal principal moments of inertia. This sphere rolls on a horizontal sphere under the influence of a constant vertical gravitational force.
where R is a radius of the ball and a is a distance from the geometric center to the center of mass. As for the symmetry Lagrange top there are two linear in momentum integrals of motion. The first integral is a well-known Jellet integral [18], see also §243, p. 192 in Routh's book [27] . The second integral was found by Routh in 1884 [27] and recovered later by Chaplygin in [9].
In this case we have v (1) The corresponding Poisson bivectors P (1,2) (29) were found in [4]. These bivectors P (1,2) (29) allows us to get the following representations of the initial vector field X (13) or The last equation may be considered as a nonholonomic counterpart of the standard Lenard-Magri recurrence relations for two dimensional bi-Hamiltonian systems (f 1 = 1, f 2 = 0), quasi bi-Hamiltonian systems (f 2 = 0) or bi-integrable systems (∀f 1,2 ), which appear in Hamiltonian mechanics [30,31,32]. Let us consider the Poisson map (39) associated with the Jellet integral of motion (46) where At α = conts and c = 0 the Hamilton function and the Routh integral of motion J 2 = αI 1 L 3 , define an integrable system on the cotangent bundle T * S. In standard spherical coordinates where φ, θ are the Euler angles, p φ and p θ are the canonically conjugated momenta Here b is a value of the Jellet integral J 1 , α = const and A(θ) = α 2 I 1 + I3(a 2 +2aR cos θ+R 2 cos 2 θ) I1+m(a 2 +2aR cos θ+R 2 ) , Using the expansion of the initial vector field (48) one getṡ Thus, similar to the Lagrange top, we have a standard equation for the nutation anglė where E 1 = H and E 2 = −J 2 /αI 1 are constants of motion. Solving this equation by quadrature one gets equation for the second Euler anglė φ = (I 1 sin 2 θ + I 3 cos 2 θ) + aI 3 cos θ g(θ)I 1 R sin 2 θ d − cos θ I 1 R sin 2 θ b .
Of course, we can obtain these equations without the notion of the Poisson structure and without decomposition (48) of the initial vector field X on the Hamiltonian and symmetric components. Gyrostatic generalization of the Jellet integral of motion looks like The Routh integral we have to shift by the non algebraic term It is an example of the non algebraic Casimir function for the Poisson bivector P

Poisson brackets for ball on a surface of revolution
Like in the previous Section equations we have to substitute integrals of motion and anzats (28) into the system of equations (26) and to solve the resulting equations with respect to the functions c k ij and d ij on γ. The desired solutions of (26) consist of two parts with rankP (k) = 4 if γ 3 = ±1. Here P α and P β are compatible rank-two Poisson bivectors satisfying (26), whereas α and β are arbitrary functions on γ 2 /γ 1 and γ 3 , respectively. First parts of (52) look like the previous bivectors P where As above these rank-two Poisson bivectors have the following Casimir functions and generate the symmetry field acting on M 3 The difference between the Poisson bivectors P (k) α for the body of revolution on the plane and for the ball on a surface of revolution is related to nonholonomic reduction by different subgroups E(2) and SO(3), respectively.
Here g is defined by (22). Using freedom in a choice of functions β and v 1,2 we can get Poisson bivectors P (k) with well defined limit at γ 1,2 → 0. The corresponding bivectors have rank two at γ 3 = ±1 and functions x 3 and M 3 play the role of the subcasimir functions. As above this change in rank indicates only on the possibility of existence of equilibria.
Bivectors P (k) α and P β have singularities at the points γ 3 = ±1. Using a freedom in the choice of the functions α and β we can get well-defined bivector P (k) (52) such that rank P (k) =2 at γ 3 = ±1. These points correspond to relative equilibria where the ball sits at the bottom of the surface of revolution spinning about its axis of symmetry, see [14]. At these points x 3 and M 3 play the role of the subcasimir functions.
In generic case the rank-four Poisson bivectors P (k) (52) have the following Casimir functions Equality of the Schouten brackets to zero is equivalent to the existence of integrals of motion J 1,2 (24) [P (k) , P (k) ] = 0 ⇔J k = X(J k ) = 0 .
Proposition 5 For the ball on a surface of revolution vector field X (21) is a linear combination of the Hamiltonian vector field and the symmetry field X S (16) where The proof is a straightforward verification of the equations (56). Like in the previous section at α = const and one gets where It allows us to prove the following Proposition 6 For the ball on a surface of revolution vector field X (21) is the conformal Hamiltonian vector field 3 )f ′ P (k) dH on the zero level of integrals of motion where β is given by (57).

The Chaplygin Hamiltonization
Following to [8] let us consider the ultimate four-dimensional reduced phase space M/G with coordinates γ 3 and These coordinates are defined only at γ 3 = ±1. As above both bivectors P (1,2) (52) have a common projection on this subspace of the six-dimensional space with coordinates (γ, M ). If It is the Poisson bivector having two Casimir functions J 1,2 (23) that allows us to rewrite the projection of the initial vector field X (56) on this subspace in the Hamiltonian form X =P dH .
Dividing two Hamiltonian equations of motioṅ by the third Hamiltonian equation of motioṅ one gets a system of linear non autonomous first order differential equations which was obtained by Routh [27] in other variables associated with the so-called semifixed frame, see discussion in [8]. This system of linear equations always possesses two integrals which are linear in K 1,2 and proportional to J 1,2 (23). The Poisson bivectorP was obtained in [8] using Chaplygin's reducing multiplier theory, see also [16].

Reduction to canonical Poisson brackets
Like in the previous Section we can study the reduction of the Poisson bivectors P (k) (52) to the canonical Poisson bivector on the Lie algebra e * (3).

Proposition 7
The mapping ψ reduces the Poisson structures (52) associated with the ball on a surface of revolution to the canonical Poisson structure (40) on the Lie algebra e * (3) at α = const.
As above using this Poisson map we can identify initial dynamical system (21) with a dynamical system on the cotangent bundle T * S to sphere possessing the Hamilton function and linear in momenta L 3 integral of motion J 1 or J 2 . Using the freedom in choosing the function β we can always identify the coefficients B in the Hamiltonians (41)

Example
Let us suppose that the ball's center of mass is moving on the paraboloid of revolution z = c(x 2 + y 2 ). In this case