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On the Routh sphere problem

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IOP PUBLISHING J. Phys. A: Math. Theor. 46 (2013) 085202 (11pp)

JOURNA

L OF

PHYSICS A: MAT

HEMATICAL AND

THEORETICAL

doi:10.1088/1751-8113/46/8/085202

On the Routh sphere problem
I A Bizyaev 1 and A V Tsiganov 2
1 2

Institute of Computer Science, Udmurt State University, Izhevsk, Russia St Petersburg State University, St Petersburg, Russia

E-mail: bizaev-90@mail.ru and andrey.tsiganov@gmail.com

Received 29 October 2012, in final form 13 January 2013 Published 6 February 2013 Online at stacks.iop.org/JPhysA/46/085202 Abstract We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on sixdimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra e (3 ). It allows us to relate the nonholonomic Routh system with the Hamiltonian system on a cotangent bundle to the sphere with a canonical Poisson structure. PACS numbers: 02.30.Ik, 45.10.Na Mathematics Subject Classification: 34D20, 70E40, 37J35

1. Introduction Let us consider a smooth manifold M with coordinates x1 ,..., xm and a dynamical system defined by the following equations of motion: xi = Xi ,
m

i = 1,..., m. , xi

(1.1)

We can identify this system of ODEs with the vector field X=
i=1

Xi

(1.2)

which is a linear operator on a space of the smooth functions on M that encodes the infinitesimal evolution of any quantity F F = X (F ) = Xi xi along the solutions of the system of equations (1.1). In Hamiltonian mechanics, a Hamilton function H on M generates a vector field X describing a dynamical system X = XH P dH.
1751-8113/13/085202+11$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA

(1.3)
1


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

Here, dH is a differential of H and P is some bivector on the phase space M. By adding some other assumptions we can prove that P is a Poisson bivector. In fact, it is enough to add energy conservation H = XH (H ) = (P dH, dH ) = 0 and the compatibility of dynamical evolutions associated with two functions H1,2 XH1 (XH2 (F )) = XH2 (XH1 (F )) + XXH1
(H2 )

(F ),

see [1, 11] and references therein. For a considerable collection of dynamical systems, vector fields are created using a Hamilton function H and a nowhere vanishing smooth function g. They are of the form X = gP dH. (1.4) This vector field X (1.4) is the so-called conformally Hamiltonian vector field. If the Hamiltonian vector field X has the form (1.4) with respect to the second Poisson bivector, we have the so-called quasi-bi-Hamiltonian system. Among such systems are found the Kepler problem, the Euler problem of two fixed centres, the Jacobi system on the ellipsoid etc. In nonholonomic mechanics, conformally Hamiltonian vector fields appear for the socalled Chaplygin-type systems after a nonholonomic reduction by the symmetry group. These systems include the nonholonomic Veselova and Suslov problems, the Chaplygin ball and their generalizations [3­5, 2, 21, 22]. Below we discuss the nonholonomic Routh sphere problem with the vector field X , which is a sum of commuting vector fields P dHk determined by integrals of motion H1 ,..., Hn : X = g1 P dH1 + ··· + gn P dHn . (1.5) Recall that the Hamiltonian vector fields for separable systems always admit such decompositions with respect to the second Poisson structure. For example, such decompositions for the Lagrange and Kowalevski top, for the Toda lattice and Henon­Heiles systems and other integrable systems can be found in [15­18]. In nonholonomic mechanics for separable systems, we have similar decompositions [20, 22]. Moreover, we suppose that such generalized conformally Hamiltonian vector fields naturally appear for the Chaplygin-type systems after a partially nonholonomic reduction by some part of the symmetry group. We try to prove this fact by starting with the nonholonomic Routh sphere problem. Similar partially reduced nonholonomic systems, such as motion of a body of revolution on a plane, motion of a homogeneous ball on a surface of revolution and on a cylindrical surface, will be discussed in forthcoming publications. If we have decomposition (1.5), then we can see that the common levels of integrals of motion form a Lagrangian foliation associated with the Poisson bivector P with all the ensuing consequences. We want to highlight that we discuss only the properties of foliations and do not discuss the linearization of the corresponding flows. It is known that a proper momentum mapping for the non-Hamiltonian vector field X associated with the Routh sphere has a `focus­focus' singularity [8]. According to [9], the nontriviality of the corresponding monodromy is the coarsest obstruction to existence of the global action-angle variables. We will observe the marks of this singularity in the corresponding Poisson structure. 2. The Routh sphere Following [4, 7, 8, 14], let us consider a rolling of a dynamically symmetric and nonbalanced spherical rigid body, the so-called Routh sphere, over a horizontal plane without slipping
2


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

under the influence of a constant vertical gravitational force. Dynamically nonbalanced means that the geometric centre differs on the centre of mass, whereas dynamically symmetric means that two momenta of inertia coincide with each other, for instance I1 = I2 . The line joining the centre of mass and the geometric centre is an axis of inertial symmetry. The corresponding symmetry group G = E (2 ) â SO(2 ) consists of two subgroups. The outer E (2 ) symmetry of Routh's sphere is generated by translations of the horizontal plane and its rotations about a vertical axis. The inner SO(2 ) symmetry is generated by rotations of the sphere about the axis of inertial symmetry. The moving sphere is subject to two kinds of constraint: a holonomic constraint of moving over of a horizontal plane and no slip nonholonomic constraint associated with the zero velocity in the point of contact v + â r = 0. (2.1) Here, and v are the angular velocity and velocity of the centre of mass of the ball, respectively, r is the vector joining the centre of mass with the contact point and â means the vector product in R3 . All the vectors are expressed in the so-called body frame, which is firmly attached to the ball, its origin is located at the centre 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 ball with respect to the contact point is equal to M = IQ , IQ = I + mr2 E - mr r. (2.2) Here E is a unit matrix, m is a mass and I = diag(I1 , I2 , I3 ) is an inertia tensor of the rolling ball. If = (1 ,2 ,3 ) is the unit normal vector to the plane at the contact point, then r = (R1 , R2 , R3 + a ), where R is a radius of the ball and a is a distance from the geometric centre to the centre of mass. The phase space, initial equations of motion, reduction of symmetries and deriving of the reduced equation of motion are discussed in [4­6, 7, 8]. We omit this step and begin directly with the reduced equations of motion on the six-dimensional phase space M with local coordinates x = ( , M ) : U , = â . (2.3) M = M â + mr â ( â r ) + â Here, U = -mg(r, ) and g is a gravitational acceleration. These equations on the sixdimensional phase space were obtained using nonholonomic reduction by the E (2 ) symmetry subgroup [8]. The completely reduced equations and the corresponding rank-2 Poisson structures are discussed in [4, 12, 13]. A straightforward calculation shows that these equations (2.3) possess four integrals of motion: H1 = 1 (M, ) + U , 2 H2 = (M, M ) - mr2 (, M ) + 2I1U , H3 = (M, r ), H4 = ( , ), (2.4) and the following invariant measure: = g-1 ( ) d dM, g( ) = I1 I3 + I1 mR2 H4 -
2 3

+ I3 m(R3 + a )2 .

(2.5)

As for the symmetry Lagrange top, there are two linear in momenta integrals of motion. The first integral H3 = (M, r )
3


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

is a well-known Jellet integral [10], see also section 243, p 192 in [14]. The second integral ^ H2 = g( )3 (2.6) was found by Routh in 1884 [14] and recovered later by Chaplygin in [7]. These linear integrals are related to the quadratic in momenta integral of motion I3 - I1 ^ 2 m 2 H2 = 2I1 H1 + H2 - H3 . I1 I1 According to [4] this quadratic integral survives at I1 = I2 in contrast with the linear integral ^ H2 . The vector field X defined by equations (2.3) has homoclinic trajectories when the Routh sphere is either spinning very slowly about a vertical axis which passes through the centre of mass and the geometric centre, or the axis is in the same position but the value of the Jellet integral is slightly less than the threshold value [8]. 2.1. Poisson brackets For the Routh sphere, six equations of motion (2.3) possess four integrals of motion and an invariant measure and, therefore, they are integrable by quadratures according to the Euler­ Jacobi theorem. It allows us to suppose that common level surfaces of integrals form a direct sum of symplectic and Lagrangian foliations of a dual dynamical system which is Hamiltonian with respect to the Poisson bivector P, so that [P, P] = 0 , P dC1,2 P dHi, j = 0 , (P dH , dHk ) {H , Hk } = 0 . (2.7) Here [., .] is the Schouten bracket and (i, j, l , m ) is the arbitrary permutation of (1, 2, 3, 4 ).In fact, here we suppose that the Euler­Jacobi integrability of the non-Hamiltonian system (2.3) is equivalent to the Liouville integrability of the dual Hamiltonian dynamical system with the same integrals of motion, see [17]. The first equation in (2.7) guarantees that P is a Poisson bivector. In the second equation, we define two Casimir elements C1 = Hi and C2 = H j of bivector P. It is a necessary condition because by fixing its values one obtains the four-dimensional symplectic phase space of our dynamical system if we assume that rank P = 4. The third equation provides that the two remaining integrals lH and Hk are in involution with respect to the Poisson bracket associated with P. In this note, we discuss the solutions of equations (2.7) in the space of the linear in momenta Mi and bivectors P at the different choice of the Casimir functions: C2 = (M, r ), H = H1 , Hk = H2 ; Case 1. C1 = ( , ), Case 2. Case 3. C1 = ( , ), C1 = (M, r ), C2 = g( )3 C2 = g( )3 H = H1 , H = H1 , Hk = H3 ; Hk = H4 . (2.8) In the generic case, linear in momenta Casimir functions look like C1,2 = a1,2 ( , ) + b1,2 (M, r ) + c1,2 g( )3 , a1,2 , b1,2 , c1,2 C. (2.9) However, the corresponding complete solutions of (2.7) have the same properties as particular solutions obtained in the above listed three special cases. In [16], we have solved the same system of equations (2.7) for the symmetric Lagrange top and proved that solutions may be useful for the construction of the variables of separation and the recursion Lenard­Magri relations for this Hamiltonian system.
4


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I A Bizyaev and A V Tsiganov

If we have some solution P of (2.7), then we can obtain the decomposition of the initial vector field X by commuting Hamiltonian vector fields P dH and P dHk . The existence of such decomposition by the basis of the Hamiltonian vector fields requires one to impose one more condition rank P=4. 3. First Poisson bracket Substituting linear in momenta Mi ansatz for the entries of the Poisson bivector
3

Pij =
k=1

aijk ( )Mk + bij ( )

(3.1)

into (2.7) in case 1 in (2.8) and solving the resulting system of algebro-differential equations, one obtains the following proposition. Proposition 1. In this case, the generic solution of (2.7) is parameterized by two functions (1 /2 ) and (3 ) : P = g Here, matrices
,

0 -




M



+
2

0 -




M



.
12 3 12 +22 1 2 3 12 +22

(3.2) 0 0 , 0

are equal to
22 (R3 +a ) R(12 +22 ) a 2 R - 1 ((2 +3 +) ) R 1 22

=

1 2 (R3 +a ) R(12 +22 ) 2 (R3 +a ) - R1( 2 + 2 ) 1 2

-

0

0

1 , 0



-- 122 32 1+ 2 2 = - 2 3 2 12 +2 2

-

1

and skew symmetric matrices M, have the form M2 )( 0 (1 M1 +(2 2 + 2R3 +a) -M2 R1 2) M = 0 M1 , 0 R 1+ 0 M3 - 3 (1M+2 M2 ) - m R(g23 2 2 1 2 M = 0 where g g( ) and

+a )

m2 R2 g2 2 - m21 R g

,

0

= mR(m(r, )C2 + I3 (1 M1 + 2 M2 ) + I1 M3 3 ). The proof is a straightforward solution of (2.7) using linear in momenta ansatzs. The corresponding Poisson brackets read as {M1 ,1 } = - g12 (R3 + a ) 2 3 g1 2 (R3 + a ) 1 2 3 , {M1 ,2 } = +2 + 2 2 2, (12 + 22 )R 1 + 22 (12 + 22 )R 1 + 2 {M2 ,3 } = 1 , {M2 ,2 } = g1 2 (R3 + a ) 1 2 3 , -2 (12 + 22 )R 1 + 22 {i , j } = 0, (3.3)
5

{M1 ,3 } = -2 , {M2 ,1 } = - g22 (R3 + a ) 2 3 - 2 1 2, 2 + 2 )R (1 1 + 2 2

{M3 ,1 } = g2 , {M3 ,2 } = - g1 ,

{M3 ,3 } = 0,


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

and

{M1 , M2 } =

g(1 M1 + 2 M2 )(R3 + a ) 3 (1 M1 + 2 M2 ) (R3 + a ) , - + M3 - 2 + 2 )R g2 (1 12 + 22 2 R R {M1 , M3 } = - gM2 + 2 2 , {M2 , M3 } = gM1 - 2 1 . g g In the generic case, rank P = 4; however, if = 0 or = 0 there are additional Casimir function 3 and 1 /2 , respectively. A particular form of these brackets was obtained in [13]. Using this Poisson bivector, we can obtain the basis of the commuting Hamiltonian vector fields X1 = P dH1 and X2 = P dH2 , and try to expand the initial non-Hamiltonian vector field X (2.3) by these vector fields.

Proposition 2. Using Poisson brackets (3.3), we can rewrite the reduced equations of motion for the Routh sphere (2.3) in the following form: xk = g1 {xk , H1 }+ g2 {xk , H2 }, k = 1,..6, (3.4) if and only if a (1 /2 ) = const, . (3.5) ( 3 ) = g 1 + R3 In this case, coefficients are equal to (R3 + a )I1 - R3 I3 a g1 = - , g2 = . g(I1 - I3 )(R3 + a ) 2 g(I1 - I3 )(R3 + a ) The proof is a straightforward verification of equations (3.4). There are other special values of the functions and according to the following: g(I1 R3 - I3 (R3 + a ) , (3.6) R (I1 - I3 )3 + am(r, ) then the Poisson bivector P (3.2) is compatible with the canonical Poisson bivector P0 on the Lie algebra e (3 ) 0 P0 = , M - where 0 3 -2 0 M3 -M2 0 1 , 0 M1 . = -3 M = -M3 2 -1 0 M2 -M1 0 (1 /2 ) = const and (3 ) = The proof is a calculation of the Schouten bracket [P0 , P] = 0. Recall that compatibility means that a linear combination of these bivectors P = P0 + P, C, is a Poisson bivector at any value of . It also means that P (3.2) is a trivial deformation of P0 , see details in [19]. It is interesting that this condition of compatibility allows us to expand the initial vector field (2.3) by a basis of Hamiltonian vector fields ^ xk = g1 {xk , H1 }+ g2 {xk , H2 }, ^ ^ k = 1,..6, (3.7) ^ 2 (2.6). Here, coefficients associated with the linear in momenta Routh integral H 1 a 3 (1 M1 + 2 M2 ) M3 - g2 = ^ g1 = - , ^ 12 + 22 I1 I1 R3 - I3 (R3 + a ) depend on coordinates and momenta in contrast with the previous decomposition.
6

Proposition 3. If


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

3.1. Properties of the first Poisson brackets Similar to the Chaplygin sphere problem [21] and nonholonomic Veselova problem [22], we can reduce this Poisson bracket to the canonical Poisson brackets on the Lie algebra e (3 ) and identify the Routh sphere model with the Hamiltonian system on the two-dimensional sphere. One of the possible reductions is given by the following proposition. Proposition 4. After a change of momenta L1 = L2 = L3 = 1 +
2 1 2 2 - 1 3 R(1 M1 + 2 M2 ) - bI1 1 (I1 + m(R3 + a )2 2 (2 M1 - 1 M2 ) + + c g(R3 + a ) - 2 3 R(1 M1 + 2 M2 ) - bI1 1 (I1 + m(R3 + a )2 1 (2 M1 - 1 M2 ) - + c g(R3 + a ) 1

, ,

1 2 + 1

2 2

2

M3 bm(R3 + a ) , b = (M, r ), c = (L, ), (3.8) + g gI1 the Poisson brackets {., .} (3.3) coincide with the canonical Poisson brackets on the Lie algebra e (3 ) {Li , L j }0 = ijk Lk , {Li , j }0 = ijk k , {i , j }0 = 0 , (3.9) where ijk is a completely antisymmetric tensor. It is easy to see that the Poisson map (3.8) is locally defined in the region 12 + 22 1 - 32 = 0. In this region of the phase space the vector field for the Routh sphere X (2.3) does not have homoclinic orbits [8]. At = const and c = 0 the images of the initial Hamiltonian are the nonhomogeneous second-order polynomial in momenta H= 1 2 (2 L1 - 1 L2 ) 2 2 L3 I1 R2 32 - 1 - (R3 + a )2 I3 + 2) 2R2 (1 - 3 I1 + m(r, r ) - 2 gb(r3 + a )L3 I1 + m(R3 + a )2 b2 + 2 I1 I1 + U (3 )
2

(3.10)

which defines a Hamiltonian system on cotangent bundle T S2 to the sphere S2 . As for the Lagrange top, the existence of the linear in momenta integral of motion (2.6) ^ H2 = I1 L3 , allows us to explicitly integrate the corresponding Hamiltonian equations of motion by quadratures. Let us introduce spherical coordinates on the sphere sin cos 1 = sin sin , p - cos p , L1 = sin cos cos , (3.11) p + sin p , L2 = 2 = cos sin , sin L3 = - p 3 = cos , where , are the Euler angles, p and p are the canonically conjugated momenta {, p } = {, p } = 1, {, } = {, p } = {, p } = 0. For these variables, the initial integrals of motion are equal to A( ) p2 + B( ) p2 + bC ( ) p + b2 D( ) H= + U ( ) , J2 = - I1 p . 2

(3.12)
7


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

Here, b is a value of the Jellet integral J1 , = const and 2 I3 (a2 + 2aR cos + R2 cos2 ) , B( ) = , A( ) = 2 I1 + I1 + m(a2 + 2aR cos + R2 ) R2 sin2 2 g(R cos + a ) I1 + m(a2 + 2aR cos + R2 ) C ( ) = , D( ) = . 2 I1 R2 sin2 I1 R2 sin2 Using the expansion of the initial vector field (3.7), one obtains = {H, }. Thus, similar to the Lagrange top, we have a standard equation on the nutation angle = B( ) p =
2 B( ) 2E1 - A( )E2 - bE2C ( ) - b2 D( ) - 2U ( ) ,

where E1 = H and E2 = -J2 / I1 are constants of motion. Solving this equation by quadrature, one obtains the equation for the second Euler angle (I1 sin2 + I3 cos2 ) + aI3 cos cos = d- b. 2 g( )I1 R sin I1 R sin2 3.2. Conformally Hamiltonian equations of motion If the Jellet integral is equal to zero C2 = (M, r ) = 0, i.e. if b = 0, then integrals of motion H1,2 (3.10) become homogeneous quadratic polynomials in momenta. In this case, we can easily find variables of separation q1,2 in the corresponding Hamilton­Jacobi equation, if we diagonalize simultaneously two quadratic forms H1,2 (3.10). Then, using these variables of separation, we can rewrite the initial vector field X (1.5) in the conformally Hamiltonian form (1.4). At C2 = b = 0 integrals of motion H1,2 satisfy the following separated relations:
i

(qi , pi , H1 , H2 ) = 0,

k = 1, 2.

Here, q1,2 and p1,2 are canonically conjugated variables of separation. In this case, according to [15] , these integrals H1,2 are in involution {H1 , H2 } f = 0 with respect to the Poisson brackets {q1 , p1 } f = f1 (q1 , p1 ), {q2 , p2 } f = f2 (q2 , p2 ),
1,2

{q1 , q2 } f = { p1 , p2 } f = 0,

labelled by two arbitrary functions f of motion H1,2 satisfy the equations Here, functions Fij depend on f
1,2

. The corresponding Poisson bivector Pf and integrals i = 1, 2. (3.13)

Pf dHi = Fi1 P dH1 + Fi2 P dH2 ,

and form the so-called control matrix [15, 17].

Proposition 5. If X (1.5) is a linear combination of the commuting Hamiltonian vector fields X = g1 P dH1 + g2 P dH2 , and coefficients g1,2 are special combinations of Fij gi = g(a1 Fi1 + a2 Fi2 ), i = 1, 2, then there is a Poisson bivector Pf , which allows us to rewrite X in the conformally Hamiltonian form X = g1 P dH1 + g2 P dH2 = gPf dH, H = a1 H1 + a2 H2 . In this case, H is a sum of initial physical integrals of motion H1,2 .
8


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

For the Routh sphere at C2 = 0, variables of separation q1,2 are functions only on coordinates i . Thus, the desired bivector Pf may be directly obtained from P (3.2) at g(r, Ir ) = -R, =- , ( , Ir ) so we have 1 X = g1 P dH1 + g2 P dH2 = - Pf dH1 . 2 At C2 = 0, variables of separation q1,2 have to be functions on coordinates i and momenta Mi and, therefore, entries of Pf have to be more complicated functions on Mi . Unfortunately, we do not know how to obtain variables of separation for the nonhomogeneous polynomial integrals of motion (3.10) on the sphere. 4. Second and third Poisson brackets Let us substitute linear in momenta Mi ansatzs (3.1) into equations (2.7) in case 2 in (2.8). Proposition 6. In this case, integrals of motion are in involution {H , Hk } = the bivector 0 P= - M is labelled by three arbitrary functions i ( ) entering into the matrix mR2 (1 1 + 2 2 )(R3 + a ) -2 (1 + 3 3 ) -2 2 + 3 1 3 I1 + m(R3 + a )2 mR1 (1 1 + 2 2 )(R3 + a = 1 1 1 2 - I1 + m(R3 + a )2 2 1 2 3 -1 3 0 and into the skew symmetric matrix M1,2 = -1 M1 - 2 M2 + 3 (1 M3 - 3 M1 ) - M1,3 = 0 if and only if (4.1) ) ,

3 1 R(R3 + a )(m2 ( , r )H3 + m(I3 (1 M1 + 2 M2 ) + I1 3 M3 )) , g2

mR(R3 + a )(1 1 + 2 2 ) M2 I1 + m(R3 + a )2 3 1 2 R2 (m2 ( , r )H3 + m(I3 (1 M1 + 2 M2 ) + I1 3 M3 )) + , g2

mR(R3 + a )(1 1 + 2 2 ) M1 I1 + m(R3 + a )2 3 12 R2 (m2 ( , r )H3 + m(I3 (1 M1 + 2 M2 ) + I1 3 M3 )) - . g2 If we impose additional restriction rank P = 4, then the third equation [P , P ] = 0 in (2.7) has the following single solution: 2 I1 R I1 + ma(R3 + a ) I1 + m(R3 + a )2 , +2 1 = - 1 1 (I, r ) 2 I1 R I1 + ma(R3 + a ) (4.2) 2 = - (I, r ) ( 2 + 22 )RI1 I1 + ma(R3 + a ) I1 + m(R3 + a )2 . 3 = -1 1 3 1 3 (I, r ) M2,3 = -
9


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

The proof is a straightforward solution of the differential equations using substitution (3.1). Using this Poisson bivector, we can obtain a basis of Hamiltonian vector fields and an expansion of the initial vector field
2 X = g1 P dH1 + g2 P dH3

(4.3) I1 (I1 + ma(R3 + a )) - mR( , Ir ) . 2 2g2 RI1 (R3 + a )

by these vector fields. The corresponding coefficients are equal to g1 = - 1 , 21 3 I1 g2 =

Similar to the first Poisson bracket (3.3), there is a transformation of momenta Mi Li , which reduces this Poisson brackets {., .} to canonical Poisson brackets on the Lie algebra e (3 ) . Now let us consider the third possible choice of the linear in momenta Casimir functions, i.e. case 3 in (2.8). In this case, the generic solution of (2.7) coincides with the previous solution P (4.1) P =P|
1 1 +2 2 =0

,

at 1 1 + 2 2 = 0. It is easy to see that rank P = 3, and there is a third Casimir function H4 = ( , ) : P dC3 = 0, C3 = H4 .

Consequently, in this case we have only one nontrivial Hamiltonian vector field P dH1 , which does not form a basis. In the similar manner, we can consider generic linear in momenta Casimir functions (2.9). Proposition 7. We cannot rewrite equations of motion (2.3) on the six-dimensional phase space in the conformally Hamiltonian form X = gP dF (H1 , H2 , H3 , H4 ) (4.4)

using linear in momenta Poisson bivector P satisfying equations (2.7). Here, F (H1 , H2 , H3 , H4 ) is an arbitrary function on integrals of motion for the Routh sphere. The proof is a straightforward verification of the fact that the common system of equations (4.4) and (2.7) is inconsistent if the Poisson bivector P has linear in momenta entries. 5. Conclusion It is well known that equations of motion for the nonholonomic Routh sphere are integrable by quadratures according to the Euler­Jacobi theorem. We identify the corresponding level sets of integrals of motion with the Lagrangian foliations associated with two different Poisson bivectors P and P . The corresponding expansions of the initial vector field X (3.4) and (4.3)
2 X = g1 P dH1 + g2 P dH2 = g1 P dH1 + g2 P dH3

may be considered as a counterpart of the standard Lenard­Magri recurrence relations X = P dH1 = f1 P dH2 + f2 P dH3 for two-dimensional bi-Hamiltonian systems ( f1 = 1, f2 = 0 ), quasi bi-Hamiltonian systems ( f2 = 0 ) or bi-integrable systems ( f1,2 ), which appear in Hamiltonian mechanics [15­17].
10


J. Phys. A: Math. Theor. 46 (2013) 085202

I A Bizyaev and A V Tsiganov

Acknowledgment We would like to thank AV Bolsinov, AV Borisov and IS Mamaev for useful discussion of applications of the Poisson geometry to the different nonholonomic systems. References
[1] Abraham R and Marsden J E 1978 Foundations of Mechanics 2nd edn (Reading, MA: Addison-Wesley) [2] Bolsinov A V, Borisov A V and Mamaev I S 2011 Hamiltonization of nonholonomic systems in the neighborhood of invariant manifolds Regular Chaotic Dyn. 16 443­64 [3] Borisov A V and Mamaev I S 2001 The Chaplygin problem of the rolling motion of a ball is Hamiltonian Math. Notes 70 720­3 [4] Borisov A V and Mamaev I S 2002 The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics Regular Chaotic Dyn. 7 177­200 [5] Borisov A V, Mamaev I S and Kilin A A 2002 The rolling motion of a ball on a surface. New integrals and hierarchy of dynamics Regular Chaotic Dyn. 7 201­19 [6] Borisov A V and Mamaev I S 2008 Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems Regular Chaotic Dyn. 13 443­90 [7] Chaplygin S A 1948 On motion of heavy rigid body of revolution on horizontal plane Collected Works vol 1 (Moscow: GITTL) pp 57­75 (in Russian) Chaplygin S A 2002 On a motion of a heavy body of revolution on a horizontal plane Regular Chaotic Dyn. 7 119­30 (Engl. transl.) [8] Cushman R 1998 Routh's sphere Rep. Math. Phys. 42 47­70 [9] Duistermaat J J 1986 On global action-angle variables Commun. Pure Appl. Math. 33 687­706 [10] Jellet J H 1872 A Treatise on the Theory of Friction (London: MacMillan) [11] Jost R 1964 Poisson brackets (an unpedagogical lecture) Rev. Mod. Phys. 36 572­9 [12] Moshchuk N K 1987 Reducing the equations of motion of certain nonholonomic Chaplygin systems to Lagrangian and Hamiltonian form J. Appl. Math. Mech. 51 172­7 [13] Ramos A 2004 Poisson structures for reduced non-holonomic systems J. Phys. A: Math. Gen. 37 4821­42 [14] Routh E J 1884 Advanced Rigid Bodies Dynamics (London: MacMillan) Routh E J 1960 Advanced Dynamics of a System of Rigid Bodies (New York: Dover) (reprint) [15] Tsiganov A V 2007 On the two different bi-Hamiltonian structures for the Toda lattice J. Phys. A: Math. Theor. 40 6395­406 [16] Tsiganov A V 2008 On bi-Hamiltonian geometry of the Lagrange top J. Phys. A: Math. Theor. 41 315212 [17] Tsiganov A V 2011 On bi-integrable natural hamiltonian systems on Riemannian manifolds J. Nonlinear Math. Phys. 18 245­68 [18] Tsiganov A V 2011 On natural Poisson bivectors on the sphere J. Phys. A: Math. Theor. 44 105203 [19] Tsiganov A V 2011 Integrable Euler top and nonholonomic Chaplygin ball J. Geom. Mech. 3 337­62 [20] Tsiganov A V 2012 One invariant measure and different Poisson brackets for two non-holonomic systems Regular Chaotic Dyn. 17 72­96 [21] Tsiganov A V 2012 On the Poisson structures for the nonholonomic Chaplygin and Veselova problems Regular Chaotic Dyn. 17 439­50 [22] Tsiganov A V 2012 One family of conformally Hamiltonian systems Theor. Math. Phys. 173 1481­97

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