aa r X i v : . [ m a t h - ph ] M a y ROLLING BALLS OVER SPHERES IN R n Boˇzidar Jovanovi´c
Mathematical Institute SANUSerbian Academy of Sciences and ArtsKneza Mihaila 36, 11000 Belgrade, Serbia
Abstract.
We study the rolling of the Chaplygin ball in R n over a fixed ( n − SO ( n )-Chaplygin reduction to S n − and prove the Hamiltonizationof the reduced system for a special inertia operator. Contents
1. Introduction 12. Chaplygin ball in R n
43. Rolling of the Chaplygin ball without slipping and twisting 84. Reduction of SO ( n )–symmetry 125. Hamiltonization of the reduced system 16References 21
1. Introduction
Let (
Q, L, D ) be a nonholonomic Lagrangian system, where Q is a n -dimensionalmanifold, L : T Q → R Lagrangian, and D nonintegrable ( n − k )-dimensional distribu-tion of constraints. Let q = ( q , . . . , q n ) be some local coordinates on Q in which theconstraints are written in the form(1) n X i =1 α ji ( q ) ˙ q i = 0 , j = 1 , . . . , k. The motion of the system is described by the Lagrange-d’Alembert equations(2) ddt ∂L∂ ˙ q i = ∂L∂q i + k X j =1 λ j α ji , i = 1 , . . . , n, where the Lagrange multipliers λ j are chosen such that the solutions q ( t ) satisfy con-straints (1). The sum P kj =1 λ j α ji represents the reaction force of the constraints. Mathematics Subject Classification.
The nonholonomic systems, generically, are not Hamiltonian systems. However,many constructions from the theory of Hamiltonian systems, such as Noether’s theoremand the reduction of symmetries, apply with certain modifications (e.g, see [
1, 4, 16,17, 23, 38, 39, 41, 45 ]). Besides, some systems have an invariant measure, whichputs them rather close to Hamiltonian systems and allow the integration using theEuler–Jacobi theorem (e.g., see [ ]).The existence of invariant measure for various nonholomic problems is studiedextensively (e.g., see [
29, 30, 33, 42, 43, 51 ]). The LR systems introduced by Veselovand Veselova [
48, 49 ] and L+R systems introduced by Kozlov and Fedorov [
26, 25 ]on unimodular Lie groups are one of the basic and remarkable examples.The closely related problem is the Hamiltonization of nonholonomic systems, inparticular, after the time reparametrisation by using the Chaplygin reducing multiplier(e.g., see [
2, 5, 14, 9, 17, 21, 27, 32, 44, 46 ]). In the case of integrability, thedynamics over regular invariant m –dimensional tori, in the original time, has the form(3) ˙ ϕ = ω / Φ( ϕ , . . . , ϕ m ) , . . . , ˙ ϕ m = ω m / Φ( ϕ , . . . , ϕ m ) , Φ > . Inspired by the study of the rolling of a of a balanced, dynamically asymmetric ballwithout slipping (after Chaplygin [ ] usually called the Chaplygin ball or the marbleChaplygin ball [ ]) and without slipping and twisting (referred as the rubber Chap-lygin ball in [ ]) over a fixed sphere in R , given by Borisov, Fedorov, and Mamaev[
7, 10, 11, 12 ] and Ehlers and Koiller [ ], we study the associated nonholonomic prob-lems in R n : the rolling of the Chaplygin ball in R n over a fixed ( n − ].Note that n –dimensional nonholonomic rigid body problems: the Veselova prob-lem [ ], the Suslov problem [ ], the rolling of the rubber Chaplygin ball [ ] andthe Chaplygin ball [ ] over hyperplane in R n (at the zero level set of the SO ( n − . In this paper we prove that the rolling of the rubber Chaplyginball over a sphere allows Chaplygin Hamiltonization, while, however, in general theproblem is not integrable.For a given nonintegrable distribution D on a Riemannian manifold Q , there is analternative, important, variational or sub-Riemannian problem, that is already Hamil-tonian. The variational problem for rolling of a ( n − ]. In Section 2 we consider a motion ofthe Chaplygin ball of radius ρ without slipping (the velocity of the contact point equalszero) over a fixed sphere in R n of radius σ in three variants of the problem. The firstone represents the motion of the ball over outside surface of the fixed sphere, the secondone is the rolling over inside surface of the fixed sphere, and the third one is the casewhere Chaplygin ball represents spherical shell with fixed sphere placed in its interior.The systems are described in Proposition 1. In all cases the configuration space is SO ( n ) × S n − and the nonholonomic distribution is diffeomorphic to T SO ( n ) × S n − . The Suslov problem studied in [ ] is an exception. There, the invariant manifolds not need tobe tori. OLLING BALLS OVER SPHERES IN R n It appears that these nonholonomic problems are examples of ǫ -modified L+Rsystems (see [ ]) with the parameter(4) ǫ = σσ ± ρ , and we directly obtain an invariant measure (see Theorem 2, item (i)), which takes thesimpler form for the inertia operator (Theorem 2, item (ii))(5) I ( E i ∧ E j ) = Da i a j D − a i a j E i ∧ E j . Here 0 < a i a j < D , i, j = 1 , . . . , n , E , . . . , E n is the standard base of R n :(6) E = (1 , , . . . , , T , . . . , E n = (0 , , . . . , , T , and D = mρ , where m and ρ are the mass and the radius of the rolling ball, respec-tively.The operator (5) is introduced in [ ] in the study of a related problem of rollingof the Chaplygin ball over a horizontal hyperplane in R n . Rolling over the horizontalplane can be seen as the limit case, where ǫ becomes 1, as the radius of the fixed sphere σ tends to infinity. Although we have the Hamiltonization of the system for ǫ = 1 (atthe zero level set of the SO ( n − ǫ = 1 is still an open problem.In Section 3, we study the rolling with additional constraints determined by thenon-twist condition of the ball at the contact point (the infinitesimal rotation of theball in the tangent plane to the contact point are forbidden), referred as the rubberChaplygin ball problem. The equations of motion are described in Proposition 3. Now,the distribution of constraints is ( n − SO ( n ) / / SO ( n ) × S n − π (cid:15) (cid:15) S n − with respect to the diagonal SO ( n )-action, i.e., the system is a SO ( n )-Chaplygin sys-tem.We also consider an appropriate extended system allowing the integrals that replacethe non-twist condition (the rubber Chaplygin ball problem is its subsystem, Subsection3.3). The obtained system is an example of ǫ -modified LR system (see [ ]), implyingthe form of an invariant measure described in Theorems 4 and 5. In particular, for theinertia operator(8) I ( E i ∧ E j ) = ( a i a j − D ) E i ∧ E j , the invariant measure, as in the case of non-rubber rolling and the operator (5), signif-icantly simplifies (see Theorem 5, item (ii)).Further, in Section 4, we derive the curvature of the nonholonomic distribution (seeLemma 7), describe the SO ( n )-Chaplygin reduction to S n − (Theorem 8), as well asthe reduced invariant measure (Theorem 10). Finally, we obtain the Hamiltonizationof the reduced system defined by the inertia operator (8) (Theorem 12, Section 5). BOˇZIDAR JOVANOVI´C
2. Chaplygin ball in R n We consider the Chaplygin ball type problem of rolling with-out slipping of an n -dimensional balanced ball, the mass center C coincides with thegeometrical center, of radius ρ in several nonholonomic models: (i) rolling over outer surface of the ( n − σ ,Figure 1a;(ii) rolling over inner surface of the ( n − σ ( σ > ρ ), Figure 1b;(iii) rolling over outer surface of the ( n − σ ,but the fixed sphere is within the rolling ball ( σ < ρ , in this case, the rollingball is actually a spherical shell), Figure 1c.We suppose that the origin O of R n coincides with the center of the fixed sphere.The configuration space is the direct product of Lie groups SO ( n ) and R n , where g ∈ SO ( n ) is the rotation matrix of the sphere (mapping a frame attached to the bodyto the space frame) and r = −−→ OC ∈ R n is the position vector of its center C (in thespace frame). The vector r belongs to the ( n − S defined by the holonomic constraint( r , r ) = ( σ ± ρ ) i.e, S is a sphere S n − . As usual, for a trajectory ( g ( t ) , r ( t )) we define angular velocities of the ball in themoving and the fixed frame, and the velocity of the center C of the ball in the fixedframe by ω = g − ˙ g, Ω = ˙ gg − = Ad g ( ω ) , V = ˙ r = ddt −−→ OC, respectively.Let A be the point of the rolling ball at the point of contact. The condition forthe ball to roll without slipping leads that the velocity of the contact point is equal tozero in the fixed reference frame: ddt −→ OA = ddt (cid:0) −−→ OC + −→ CA (cid:1) = V − ρ Ω Γ = 0 (the case (i)); ddt −→ OA = ddt (cid:0) −−→ OC + −→ CA (cid:1) = V + ρ Ω Γ = 0 (the cases (ii) and (iii)) , (9)where Γ ∈ R n is the unit normal to the fixed sphere at the contact point directedoutward, or, equivalently, the direction of the contact point in the fixed referenceframe:(10) Γ = 1 |−→ OA | −→ OA = 1 σ ± ρ r . Therefore, the nonholonomic distribution is D ± = { ( ω, V , g, r ) | V = ± ρ Ad g ( ω ) Γ = ± ρσ ± ρ Ad g ( ω ) r } . It would be also interesting to study a modified problem, where we assume that the ball rollsover a rotating n –dimensional sphere (for n = 3, see [
8, 23 ]). Rolling of a n –dimensional Chaplyginball over a rotating horizontal plane is considered in [ ]. From now on, whenever we have a sign ± , we take ”+” for the case (i) and ” − ” in the cases (ii)and (iii). Through the paper, we consider vectors in R n as columns and Ω Γ denotes the usual matrixmultiplication. The Euclidean scalar product of x, y ∈ R n is simply ( x, y ) = x T y , while the wedgeproduct is x ∧ y = x ⊗ y − y ⊗ x = xy T − yx T . OLLING BALLS OVER SPHERES IN R n It is clear that D ± is diffeomorphic to the product T SO ( n ) × S n − . b O b A b C Γ b O b C Γ b A b C b O b Γ A Figure 1a Figure 1b Figure 1c
In what follows we identify so ( n ) ∼ = so ( n ) ∗ by an invariant scalar product(11) h X, Y i = −
12 tr( XY ) . Let m be the mass of the ball and I : so ( n ) → so ( n ) ∗ ∼ = so ( n ) be the inertia tensorthat defines a left–invariant metric on SO ( n ). The Lagrangian of the system is thengiven by(12) L ( ω, V , g, r ) = 12 h I ω, ω i + 12 m ( V , V ) , where ( · , · ) is the Euclidean scalar product in R n .By the use of the constraints (9) we find the form of reaction forces in the right-trivialization of SO ( n ) in which the equations (2) become˙ M = − ( ± ρ Λ ∧ Γ ) , (13) m ˙ V = Λ , (14) ˙ g = Ω · g, (15) ˙ r = V . (16)where M = Ad g ( I ω ) ∈ so ( n ) ∗ ∼ = so ( n ) is the ball angular momentum in the fixed frameand Λ ∈ R n is the Lagrange multiplier. Differentiating the constraints (9) and using(14), we get(17) Λ = ± mρ ( ˙Ω Γ + Ω ˙ Γ ) . Further, (10) and (9) imply that the vector Γ in the fixed frame satisfies theequation:(18) ˙ Γ = 1 σ ± ρ V = ± ρσ ± ρ Ω Γ . Finally, from (17) and (18) we get that (13) takes the form(19) ˙ M = − D (cid:0) ˙Ω Γ ⊗ Γ + Γ ⊗ Γ ˙Ω (cid:1) − D (cid:0) ± ρσ ± ρ (cid:1)(cid:0) Ω Ω Γ ⊗ Γ − Γ ⊗ Γ Ω Ω (cid:1) , where D = mρ . BOˇZIDAR JOVANOVI´C
Both the Lagrangian L and the distribution D ± are invariant with respect to the left SO ( n )-action(20) a · ( ω, V , g, r ) = ( ω, a V , ag, a r ) , a ∈ SO ( n ) . Therefore, the system can be reduced to so ( n ) × S n − ∼ = ( T SO ( n ) × S n − ) /SO ( n ) ∼ = D ± /SO ( n ) . Note that the SO ( n )–action defines the principal bundle (7), where the submersion π is given by(21) γ = π ( g, r ) = 1 σ ± ρ g − r = g − Γ , that is, a base point of ( g, r ) is γ = g − Γ , the unit normal at the contact point to thefixed sphere (directed outward) in the frame attached to the ball.We can use ( g, γ ) instead of ( g, r ), for coordinates of a configuration space. Thenthe SO ( n )–action (20) takes the form:(22) a · ( ω, ˙ γ, g, γ ) = ( ω, ˙ γ, ag, γ ) . a ∈ SO ( n ) . From (18), we get the kinematic equation for γ ˙ γ = ddt (cid:0) g − (cid:1) Γ + g − ˙ Γ = − g − ˙ gg − Γ ± ρσ ± ρ g − Ω Γ = − ωγ ± ρσ ± ρ ωγ. By introducing parameter ǫ (see (4)), we can write it as a modified Poisson equation(23) ˙ γ = − ǫωγ. Let(24) k = κ ( ω ) = I ω + D ( ω γ ⊗ γ + γ ⊗ γ ω ) ∈ so ( n ) ∼ = so ( n ) ∗ be the angular momentum of the ball relative to the contact point (see [ ]). Proposition . (i) The complete set of equations on T ∗ SO ( n ) × S n − in variables ( k , g, γ ) is given by (25) ˙ k = [ k , ω ] , ˙ g = g · ω, ˙ γ = − ǫωγ. (ii) The reduction of the left SO ( n ) –symmetry (22) gives a system on so ( n ) ∗ × S n − defined by the equations (26) ˙ k = [ k , ω ] , ˙ γ = − ǫωγ. Proof.
By applying the identities˙ ω = Ad g − ( ˙Ω) , I ˙ ω − [ I ω, ω ] = Ad g − ( ddt (cid:0) Ad g ( I ω ) (cid:1) = Ad g − ( ˙ M ) , to (19), in the left trivialization of SO ( n ) we obtain the equation: I ˙ ω − [ I ω, ω ] = − D (cid:0) ˙ ω γ ⊗ γ + γ ⊗ γ ˙ ω (cid:1) − D (cid:0) ± ρσ ± ρ (cid:1)(cid:0) ω ω γ ⊗ γ − γ ⊗ γ ω ω (cid:1) (27) = − D (cid:0) ˙ ω γ ⊗ γ + γ ⊗ γ ˙ ω (cid:1) + D (1 − ǫ )[ ω γ ⊗ γ + γ ⊗ γ ω, ω ] . Next, from (23) we have ddt ( ωγ ⊗ γ + γ ⊗ γω ) = ˙ ωγ ⊗ γ + γ ⊗ γ ˙ ω − ǫωωγ ⊗ γ + ǫωγ ⊗ γω − ǫωγ ⊗ γω + ǫγ ⊗ γωω (28) = ˙ ωγ ⊗ γ + γ ⊗ γ ˙ ω + ǫ [ ωγ ⊗ γ + γ ⊗ γω, ω ] . OLLING BALLS OVER SPHERES IN R n As a result, from (27) and (28) we obtain:˙ k = I ˙ ω + D ( ˙ ωγ ⊗ γ + γ ⊗ γ ˙ ω ) + Dǫ [ ωγ ⊗ γ + γ ⊗ γω, ω ]=[ I ω, ω ] + D (1 − ǫ )[ ωγ ⊗ γ + γ ⊗ γω, ω ] + Dǫ [ ωγ ⊗ γ + γ ⊗ γω, ω ]=[ k , ω ] . (cid:3) Remark . If the radius σ of the fixed sphere (the case (i)) tends to infinity, theparameter ǫ tends to 1, and the above equations reduce to the equations of the rollingof the Chaplygin ball over a horizontal hyperplane in R n (see [
25, 36 ]). Also, notethat the rolling of a Chaplygin ball over a sphere (26) is an example of a modified L+Rsystem on the product of so ( n ) and the Stiefel variety V n,r for r = 1, see Section 4.1of [ ]. Remark . Note that the mapping ξ ( ξ Γ ) ∧ Γ = ξ Γ ⊗ Γ + Γ ⊗ Γ ξ is the orthogonal projection pr v : so ( n ) → v with respect to the scalar product (11),while ξ ( ξγ ) ∧ γ = ξγ ⊗ γ + γ ⊗ γξ is the orthogonal projection pr v γ to v γ , wherethe subspaces v and v γ are defined by(29) v = R n ∧ Γ and v γ = Ad g − ( v ) = R n ∧ γ. Then we have(30) ddt pr v γ = ǫ [pr v γ , ad ω ] , where [ · , · ] is the standard Lie bracket in the space of linear operators of so ( n ). Thus,equivalently, we can derive (28) from the identity ddt (pr v γ ω ) = pr v γ ˙ ω + ddt (pr v γ ) ω. Remark . The operator κ = I + D pr v γ : so ( n ) → so ( n ) ∼ = so ( n ) ∗ can be alsodefined by the use of the constrained Lagrangian (31) L = L | V = ± ρ Ad g ( ω ) Γ = 12 h I ω, ω i + D g ( ω ) Γ , Ad g ( ω ) Γ ) =: 12 h κ ( ω ) , ω i , which represents the kinetic energy, preserved along the flow of the system. Based on general observations given for ǫ -modifiedL+R systems (see Theorems 4 and 5, [ ]) we have that for the rolling over a sphere,the density of an invariant measure keeps the same form as in the case of the rollingover a horizontal hyperplane (see Fedorov and Kozlov [
25, 26 ]).Let(32) µ ( γ ) = p det( κ ) = q det( I + D pr v γ ) , and let A = diag( a , . . . , a n ), where a , . . . , a n are parameters of the inertia operator(5). Also, by d k and d γ we denote the standard volume forms on so ( n ) ∗ and S n − ,respectively, and by Ω the canonical symplectic structure on T ∗ SO ( n ), d = dim SO ( n ). Theorem . (i) The problem of the rolling of a ball over a sphere (25) on T ∗ SO ( n ) × S n − in variables ( k , g, γ ) has an invariant measure (33) µ − Ω d ∧ d γ = 1 / p det( κ ) Ω d ∧ d γ = 1 / q det( I + D pr v γ ) Ω d ∧ d γ, BOˇZIDAR JOVANOVI´C while the reduced flow (26) in variables ( k , γ ) has an invariant measure (34) µ − d k ∧ d γ = 1 / q det( I + D pr v γ ) d k ∧ d γ. (ii) For the inertia operator (5) , the density (32) is proportional to ( γ, A − γ ) ( n − . Remark . Since d k = det( κ )d ω , the invariant measure of the reduced systemconsidered in variables ( ω, γ ) is µ ( γ )d ω ∧ d γ . In the case n = 3, under the isomorphism between R and so (3)(35) ~X = ( X , X , X ) X = − X X X − X − X X , from (26), we obtain the classical equations of rolling without slipping of the Chaplyginball over a sphere(36) ddt~ k = ~ k × ~ω, ddt~γ = ǫ~γ × ~ω, where ~ k = I ~ω + D~ω − D ( ~ω, ~γ ) ~γ and I = diag( I , I , I ) is the inertia operator of theball. In the space R ( ~ω, ~γ ) the density (32) of an invariant measure is equal to(37) µ ( ~γ ) = q det( I + D E ) (cid:0) − D ( ~γ, ( I + D E ) − ~γ ) (cid:1) , the expression given by Chaplygin for ǫ = 1 [ ] (see Remark 1), and by Yaroshchukfor ǫ = 1 [ ]. Here E = diag(1 , , F = ( ~γ, ~γ ) = 1 , F = 12 ( ~ k , ~ω ) , F = ( ~ k , ~ k ) . For ǫ = 1, there is the fourth integral F = ( ~ k , ~γ ) and the problem is integrableby the Euler-Jacobi theorem: the phase space is almost everywhere foliated by two-dimensional invariant tori with quasi-periodic, non-uniform motion (3) (see Chaplygin[ ]). Moreover, Borisov and Mamaev proved that the system (36) is Hamiltonizablewith respect to certain nonlinear Poisson bracket on R ([ ], see also [
14, 47 ]).Remarkably, for ǫ = − ρ = 2 σ ) Borisov and Fedorov (see [ ])found the integrable case with the fourth integral˜ F = ( I + I − I + D ) k γ + ( I + I − I + D ) k γ + ( I + I − I + D ) k γ . The system is integrated on an invariant hypersurface ˜ F = 0 [ ]. Furthermore, itstopological analysis and a representation as a sum of two conformally Hamiltonianvector fields are given in [ ] and [ ], respectively. We feel that it would be veryinteresting to have similar results in a dimension greater then 3.
3. Rolling of the Chaplygin ball without slipping and twisting3.1. Rubber rolling.
Three–dimensional rubber Chaplygin ball problems areintroduced in [ ] and [ ], while the multidimensional rubber rolling over a horizontalhyperplane is considered in [ ]. For a given normal vector γ = g − Γ , let E , . . . , E n − , Γ , and e = g − E , . . . , e n − = g − E n − , γ = g − Γ OLLING BALLS OVER SPHERES IN R n be orhonormal bases of R n in the fixed frame and in the body frame, respectively.Rubber Chaplygin ball is defined as a system (9), (12) subjected to the additionalconstraints(39) φ ij = h Ω , E i ∧ E j i = h ω, e i ∧ e j i = 0 , ≤ i < j ≤ n − ω has rank 2 and thecorresponding admissible plane of rotation contains the normal vector γ to the rollingsphere at the contact point.Alternatively, note that E i ∧ E j , e i ∧ e j = Ad − g ( E i ∧ E j ) , ≤ i < j ≤ n − h and h γ = Ad g − h , orthogonal complements to v and v γ (see (29)) with respect to the scalar product (11). Thus, the constraints (39) can berewritten as(40) pr h Ω = 0 , i.e., pr h γ ω = 0 ⇐⇒ Ω ∈ v , i.e., ω ∈ v γ . As a result, we obtain ( n − F ± = { ( ω, V , g, r ) | V = ± ρσ ± ρ Ad g ( ω ) r , pr h γ ω = 0 } ⊂ D ± . Let E be the identity operator on so ( n ). We have the relation(42) k = I ω + Dω = I ω, for ω ∈ v γ = R n ∧ γ, where k is given by (24) and I = I + E . Let m = I ω ∈ so ( n ) ∼ = so ( n ) ∗ be the angularmomentum with respect to the modified inertia operator I . After the identification D ± ∼ = T SO ( n ) × S n − , we obtain a natural phase space of the problem: G = { ( m, g, γ ) ∈ T ∗ SO ( n ) × S n − | pr h γ I − m = pr h γ ω = 0 } . Using Proposition 1 and (42), we can write the equations of a motion in the vari-ables ( m, g, γ )(43) ˙ m = [ m, ω ] + λ , ˙ g = g · ω, ˙ γ = − ǫωγ. The Lagrange multiplier λ ∈ h γ is determined from the condition that the angularvelocity ω satisfies (40). From (30) and the identity pr h γ + pr v γ = E , we have ddt pr h γ = ǫ [pr h γ , ad ω ] . Thus, 0 = ddt (cid:0) pr h γ ω (cid:1) = ǫ (pr h γ ad ω − ad ω pr h γ ) ω + pr h γ ˙ ω = pr h γ ddt (cid:0) I − [ m, ω ] + I − λ (cid:1) , and the multiplier λ ∈ h γ is the solution of the equation(44) I − ([ m, ω ] + λ ) − γ ⊗ γ I − ([ m, ω ] + λ ) − I − ([ m, ω ] + λ ) γ ⊗ γ = 0 . Thus, we obtain.
Proposition . The equations of a motion of the rubber Chaplygin ball on G are given by (43) , where m = I ω = I ω + Dω , and λ ∈ h γ is the solution of (44) .The reduction of the left SO ( n ) –symmetry (22) induces a system on the space G = G /SO ( n ) = { ( m, γ ) ∈ so ( n ) ∗ × S n − | pr h γ ω = 0 } given by (23) and (45) ˙ m = [ m, ω ] + λ . The proof of the next theorem follows from considerations given in Subsection 3.3below.
Theorem . The problem of the rubber rolling of a ball over a sphere (43) andthe reduced system (23) , (45) possess invariant measures µ ǫ ( γ ) Ω d ∧ d γ | G , µ ǫ ( γ ) d m ∧ d γ | G , respectively, where the density µ ǫ ( γ ) is given by (46) µ ǫ ( γ ) = (det I − | h γ ) ǫ . Remark . Since d m = det( I )d ω = const · d ω , contrary to remark 4, here thereduced system considered in variables ( ω, γ ) has the invariant measure with the samedensity as in the variables ( m, γ ): µ ǫ d ω ∧ d γ . For n = 3, under the isomorphism (35) between R and so (3) and the identification of ~γ with ~ e ∧ ~ e in (39), we have(47) G = { ( ~m, ~γ ) ∈ R × S | φ = ( ~γ, ~ω ) = 0 } and the reduced system (45), (23) reads(48) ˙ ~m = ~m × ~ω + λ~γ, ˙ ~γ = ǫ~γ × ~ω, where(49) ~m = ( I + D E ) ~ω = I ~ω, λ = − ( ~m, I − ( ~m × ~ω )) / ( ~γ, I − ~γ ) . The density (46) reduces to the well known expression(50) µ ǫ ( ~γ ) = ( I − ~γ, ~γ ) ǫ (see [ ] for ǫ = 1 and [ ] for ǫ = 1). Apart of the integrability of the rolling over ahorizontal plane ( ǫ = 1) [ ], as in the case of non-rubber rolling, Borisov and Mamaevproved the integrability for ǫ = − ]. Note that for ǫ = 1, the above equationscoincide with the equations of nonholonomic rigid body motion studied by Veselov andVeselova [
48, 49 ].The problem is Haminltonizable for all ǫ [
21, 22 ]. On the other hand, the rub-ber rolling of the ball where the mass center does not coincide with the geometricalcenter over a horizontal plane provides an example of the system having the followinginteresting property (see [
6, 13 ]). The appropriate phase space is foliated on invarianttori, such that the foliation is isomorphic to the foliation of integrable Euler case of therigid body motion about a fixed point, but the system itself has not analytic invariantmeasure and is not Hamiltonizable.
Note that we can consider equations (48), (49) on the product R × S as well. Thesystem also has an invariant measure with density (50) and the reduced system on (47)is its subsystem ( φ = ( ~ω, ~γ ) is the first integral). Similarly, the system (45), (23) canbe extended and the invariant measure given in Theorem 4 is the restriction to G ofan invariant measure of the extended system. In order to define the extended systemsuch that we can use the results of [ ], we need to add some additional variables.Firstly, consider the system (45), (23) on G . We can choose vectors e i ( t ), i =1 , . . . , n − m ( t ) , γ ( t )), such that e ( t ) , . . . , e n − ( t ) , e n ( t ) = γ ( t ) isa orthonormal base of R n and that(51) ˙ e i = − ǫω e i , i = 1 , . . . , n. OLLING BALLS OVER SPHERES IN R n Indeed, we can take a base e ( t ) , . . . , e n ( t ) at some initial time t (it is defined modulothe orthogonal transformations of the hyperplane γ ( t ) ⊥ ). From the modified Poissonequations (51) it follows that the scalar products ( e i ( t ) , e j ( t )) are conserved.Further, the equations (51) imply(52) ( e i ∧ e j ) · = ǫ [ e i ∧ e j , ω ] , ≤ i < j ≤ n. We can determine the reaction force λ starting from the expression(53) λ = X ≤ i 1. We have the momentum equation (see [ ], i.e, [ ] for ǫ = 1)(58) ˙ m = ǫ [ m , ω ] + (1 − ǫ ) pr v γ [ I ω, ω ] . Thus, we obtain an alternative description of the extended system on so ( n ) ∗ × S n − given by (23) and (58). It leads to the dual expression for an invariant measure (seeTheorems 2 and 4, [ ]). Let A = diag( a , . . . , a n ), where a , . . . , a n are parameters of the special inertiaoperator (8). Theorem . (i) The extended system (23) , (58) of the rubber rolling of a ball overa fixed sphere in variables ( m , γ ) has an invariant measure ˜ µ ǫ d m ∧ d γ , (59) ˜ µ ǫ ( γ ) = (det I | v γ ) ǫ − . (ii) For I defined by (8) , i.e., I ( E i ∧ E j ) = a i a j E i ∧ E j , the density (59) is propor-tional to ( γ, Aγ ) ( ǫ − n − . It is also clear that the momentum equation (58), together with ˙ g = gω and (23),defines extended system on T ∗ SO ( n ) × S n − with an invariant measure ˜ µ ǫ Ω d ∧ d γ . 4. Reduction of SO ( n ) –symmetry4.1. Chaplygin reduction to T S n − . As we already mentioned, the problemof the rubber rolling of a ball over a fixed sphere is a SO ( n )-Chaplygin system withrespect to the action (20). We have the principal bundle (7), (21), together with theprincipal connection T ( g, r ) SO ( n ) × S n − = F ± ( g, r ) ⊕ ker dπ ( g, r ) , (60) ker dπ ( g, r ) = so ( n ) · ( g, r ) . The system reduces to the tangent bundle T S n − ∼ = F ± /SO ( n ). The procedureof reduction for rubber rolling over a sphere for n = 3 is given by Ehlers and Koiller[ ]. Note that in this case the system is always Hamintonizable due to the fact thatit has an invariant measure and that the reduced configuration space is 2–dimensional.We proceed with a reduction of n –dimensional variant of the problem.Recall that the vector in F ± ( g, r ) are called horizontal , while the vectors in ker dπ ( g, r ) vertical . The horizontal lift ˙ γ h of the base vector ˙ γ ∈ T γ S n − to the horizontal space F ± at the point ( g, r ) ∈ π − ( γ ) is the unique vector in F ± ( g, r ) satisfying dπ ( γ h ) = ˙ γ . Lemma . The reduced Lagrangian on T S n − = F ± /SO ( n ) reads L red ( ˙ γ, γ ) = 12 ǫ h I ( γ ∧ ˙ γ ) , γ ∧ ˙ γ i = − ǫ tr( I ( γ ∧ ˙ γ ) γ ∧ ˙ γ ) . Proof. The horizontal lift ˙ γ h | ( g, r ) = ( ω, V ) is given by: ω = 1 ǫ γ ∧ ˙ γ = σ ± ρσ γ ∧ ˙ γ, V = ˙ r = ( σ ± ρ ) ddt ( gγ ) = ( σ ± ρ )( ˙ gγ + g ˙ γ ) = ( σ ± ρ )( g ǫ ( γ ∧ ˙ γ ) γ + g ˙ γ )= ( σ ± ρ ) (cid:0) − ǫ (cid:1) g ˙ γ = − ( σ ± ρ ) (cid:0) ± ρσ (cid:1) g ˙ γ. As a result, the reduced Lagrangian is L red ( ˙ γ, γ ) = L ( ˙ γ h | ( g, r ) , g, r )) ( g, r ) ∈ π − ( γ ) = 12 ǫ h I ( γ ∧ ˙ γ ) , γ ∧ ˙ γ i + D ǫ ( ˙ γ, ˙ γ ) , which proves the statement. (cid:3) The reduced Lagrange–d’Alembert equation describing the motion of the systemon a sphere S n − takes the form(61) (cid:16) ∂L red ∂γ − ddt ∂L red ∂ ˙ γ , ξ (cid:17) = h J ( g, r ) ( ˙ γ h ) , K ( g, r ) ( ˙ γ h , ξ h ) i , ξ ∈ T γ S n − , OLLING BALLS OVER SPHERES IN R n where ( g, r ) ∈ π − ( γ ), K ( · , · ) is so ( n )–valued curvature of the connection, and J is themomentum mapping of SO ( n )–action (20) (see [ 41, 4 ]).It is well known that the momentum mapping J : T ( SO ( n ) × S n − ) → so ( n ) ∼ = so ( n ) ∗ of the action (20) is given by J ( g, r ) ( ω, V ) = Ad g ( I ω ) + m V ∧ r . Therefore, J ( g, r ) ( ˙ γ h ) = 1 ǫ Ad g I ( γ ∧ ˙ γ ) − mǫ ( σ ± ρ ) (cid:0) ± ρσ (cid:1) g ˙ γ ∧ r = Ad g (cid:16) ǫ I ( γ ∧ ˙ γ ) ± m ( σ ± ρ ) ρσ ( γ ∧ ˙ γ ) (cid:17) = 1 ǫ Ad g (cid:16) I ( γ ∧ ˙ γ ) ± D σ ± ρρ ( γ ∧ ˙ γ ) (cid:17) = 1 ǫ Ad g (cid:16) I ( γ ∧ ˙ γ ) + D − ǫ ( γ ∧ ˙ γ ) (cid:17) . Let ξ , ξ ∈ F ± ( g, r ) . By definition, the curvature K ( g, r ) ( ξ , ξ ) is the element η ∈ so ( n ), such that η · ( g, r ) is the vertical component of the commutator of vector fields[ X , X ] at ( g, r ), where X and X are smooth horizontal extensions of ξ and ξ . Lemma . Let ξ , ξ ∈ T γ S n − and ( g, r ) ∈ π − ( γ ) . Then K ( g, r ) ( ξ h , ξ h ) = (1 − ρ σ ) Ad g ( ξ ∧ ξ ) = 2 ǫ − ǫ Ad g ( ξ ∧ ξ ) . In particular, for ǫ = 1 / , i.e, ρ = σ , the curvature vanish and the constraints areholonomic. Remark . Note that the factor 1 − ρ σ equals to 1 − K /K where K and K are curvatures of the fixed and rolling sphere, respectively. The same factor appears inthe case of rubber rolling of arbitrary two surfaces in R (see [ ]).Since h γ ∧ ˙ γ, ˙ γ ∧ ξ i = 0, we can replace J by ǫ Ad g ( I ( γ ∧ ˙ γ )) at the right hand sideof (61), and we get the J-K term in the form h J ( g, r ) ( ˙ γ h ) , K ( g, r ) ( ˙ γ h , ξ h ) i = 2 ǫ − ǫ h I ( γ ∧ ˙ γ ) , ξ ∧ ˙ γ i = − ǫ − ǫ tr( I ( γ ∧ ˙ γ ) · ( ξ ⊗ ˙ γ − ˙ γ ⊗ ξ )) = 2 ǫ − ǫ ( I ( γ ∧ ˙ γ ) ˙ γ, ξ ) . We have ∂L red ∂γ = 1 ǫ I ( γ ∧ ˙ γ ) ˙ γ, ∂L red ∂ ˙ γ = − ǫ I ( γ ∧ ˙ γ ) γ. Therefore, we obtain the following statement. Theorem . The Lagrange–d’Alembert equation describing the motion of the re-duced system are given by (62) (cid:16) ǫ ddt (cid:0) I ( γ ∧ ˙ γ ) γ (cid:1) + (1 − ǫ ) I ( γ ∧ ˙ γ ) ˙ γ, ξ (cid:17) = 0 , ξ ∈ T γ S n − . The above reduction slightly differs from the Chaplygin SO ( n − ]. Proof of Lemma 7. In the coordinates ( g, γ ), the SO ( n )-action takes the form(22). Let η ∈ so ( n ). The associated vector field on SO ( n ) × S n − with respect to theaction (22) is given by η · ( g, γ ) ∼ = (Ad g − η, ∈ T ( g,γ ) SO ( n ) × S n − , where, as above, we use the left trivialization of T SO ( n ). Further, the horizontal andvertical components of the vector ( ω, ξ ) ∈ T ( g,γ ) ( SO ( n ) × S n − ), respectively, simplyread ( ω, ξ ) H = ( 1 ǫ γ ∧ ξ, ξ ) , ( ω, ξ ) V = ( ω − ǫ γ ∧ ξ, . Now, let ξ , ξ be vector fields, the extensions of ξ , ξ ∈ T γ S n − defined in aneighborhood U of γ , and X , X their horizontal lifts to SO ( n ) × U : X i ( g, γ ) = ( 1 ǫ γ ∧ ξ i , ξ i ) = Y i + Z i , Y i = ( 1 ǫ γ ∧ ξ i , , Z i = (0 , ξ i ) , i = 1 , . Then, by definition h K ( g,γ ) ( ξ h , ξ h ) , η i = h− [ X , X ] | V ( g,γ ) , Ad g − η i , i.e.,(63) K ( g,γ ) ( ξ h , ξ h ) = − Ad g [ X , X ] | V ( g,γ ) . We shall prove(64) [ X , X ] V = [ X , X ] = (cid:0) ǫ − ǫ (cid:1) ( ξ ∧ ξ , , which, according to (63), proves the lemma.Without loosing a generality we may suppose that γ = (0 , , . . . , , T . Let( q , . . . , q n − ) ∈ U be the local coordinates on the upper half-sphere S n − = { γ ∈ S n − | γ n > } defined by γ i = q i , i = 1 , . . . , n − ,γ n = q − q − · · · − q n − ,U = { ( q , . . . , q n − ) ∈ R n − | q + · · · + q n − < } . The given vectors ξ , ξ ∈ T γ S n − have the form ( ξ i , . . . , ξ n − i , T , i = 1 , 2. By taking ξ ji = const ,(65) ξ i = n − X j =1 ξ ji ∂∂q j , i = 1 , , define their natural commutative extensions to U . Note that ∂/∂q i corresponds to thevector field E i − q i q − q − · · · − q n − E n = E i − γ i γ n E n in redundant variables on S n − ⊂ R n , where we consider (6) as vector fields on R n .Whence, in redundant variables the vector fields (65) read ξ i = ( ξ i , . . . , ξ n − i , ξ ni ) T = ( ξ i , . . . , ξ n − i , − γ n ( ξ i γ + · · · + ξ n − i γ n − )) T , OLLING BALLS OVER SPHERES IN R n i = 1 , 2, implying the identities ξ i ( γ j ) = ξ i ( q j ) = ξ ji , j = 1 , . . . , n − ,ξ i ( γ n ) = ξ i ( q − q − · · · − q n − ) = − ξ i q + · · · + ξ n − i q n − q − q − · · · − q n − = ξ ni . (66)Let E kl = ( E k ∧ E l , Y i = P k 0) + X i 0) + X i 1) ˙ γ ∧ p + (1 − ǫ )( ˙ γ ∧ p + γ ∧ Υ) ⇐⇒ ǫγ ∧ ˙ p + ( ǫ − γ ∧ Υ) = 0 . As a bi-product, we get the following statement. Theorem . The reduced equations (72) has an invariant measure (det I | v γ ) ǫ − w n − , where w is the canonical symplectic form (73) w = dp ∧ dγ + · · · + dp n ∧ dγ n | T ∗ S n − Proof. The mappingΦ : ( γ, p ) ( γ, m ) , m = ǫγ ∧ p, together with ω = ǫ γ ∧ ˙ γ , maps the reduced system (72) to the subsystem of (23),(58), and the pull-back Φ ∗ (d m ∧ d γ ) is the standard volume form w n − on T ∗ S n − (up to the multiplication by a constant). Now the statement follows from Theorem 5,item (i). (cid:3) 5. Hamiltonization of the reduced system5.1. Equations for the special inertia operator. Based on the Hamiltoniaza-tion and integrability of the reduced Veselova system [ ], we have the Hamiltonizationand integrability of the rubber rolling of a Chaplygin ball over a horizontal hyper-plane for a special inertia operator (8) (see [ ]). Namely, under the time substitution OLLING BALLS OVER SPHERES IN R n dτ = 1 / p ( Aγ, γ ) dt , the reduced system becomes an integrable Hamiltonian systemdescribing a geodesic flow on S n − of the metric(74) ds A = 1( γ, Aγ ) (cid:0) ( Adγ, dγ )( Aγ, γ ) − ( Aγ, dγ ) (cid:1) , where dγ = ( dγ , . . . , dγ n ) T [ ].Now we proceed with a rolling over a sphere and, as in the case of the horizon-tal rolling, we suppose that the inertia operator is given by (8). Then the reducedLagrangian L red ( ˙ γ, γ ) and the Legendre transformation (68) take the form L red = 12 ǫ (cid:0) ( A ˙ γ, ˙ γ )( Aγ, γ ) − ( Aγ, ˙ γ ) (cid:1) . (75) p = ∂L red ∂ ˙ γ = 1 ǫ ( γ, Aγ ) A ˙ γ − ǫ ( ˙ γ, Aγ ) Aγ. (76)Under conditions (69), relations (76) can be uniquely inverted to yield(77) ˙ γ = ǫ ( γ, Aγ ) (cid:0) A − p − ( p, A − γ ) γ (cid:1) implying that the angular velocity in terms of ( p, γ ) takes the form ω ( p, γ ) = 1 ǫ γ ∧ ˙ γ = ǫ ( γ, Aγ ) γ ∧ A − p, and we get: Υ( p, γ ) = 1 ǫ ( Aγ ∧ A ˙ γ ) ˙ γ = 1 ǫ (( A ˙ γ, ˙ γ ) Aγ − ( Aγ, ˙ γ ) A ˙ γ )= ǫ ( γ, Aγ ) ( p − ( p, A − γ ) Aγ, A − p − ( p, A − γ ) γ ) Aγ − ǫ ( γ, Aγ ) ( γ, p − ( p, A − γ ) Aγ ) (cid:2) p − ( p, A − γ ) Aγ (cid:3) = ǫ ( γ, Aγ ) (cid:0) ( A − p, p ) + ( Aγ, γ )( p, A − γ ) (cid:1) Aγ + ǫ ( γ, Aγ ) ( p, A − γ )( Aγ, γ ) (cid:2) p − ( p, A − γ ) Aγ (cid:3) , that is Υ( p, γ ) = ǫ ( γ, Aγ ) (cid:0) ( A − p, p ) Aγ + ( p, A − γ )( Aγ, γ ) p (cid:1) . In particular, (Υ( p, γ ) , γ ) = ǫ ( A − p, p ) / ( γ, Aγ ), and the right hand side of equa-tion (71) reads X p ( p, γ ) = (1 − ǫ ) ǫ Υ + ( ǫ − ǫ (Υ , γ ) γ − ( ˙ γ, p ) γ = (1 − ǫ ) ǫ ( γ, Aγ ) (cid:0) ( A − p, p ) Aγ + ( p, A − γ )( Aγ, γ ) p (cid:1) + ǫ ( ǫ − γ, Aγ ) ( A − p, p ) γ − ǫ ( γ, Aγ ) (cid:0) ( A − p, p ) − ( p, A − γ )( γ, p ) (cid:1) γ. Finally, we obtain the equation(78) ˙ p = ǫ (1 − ǫ )( γ, Aγ ) (cid:0) ( A − p, p ) Aγ + ( p, A − γ )( Aγ, γ ) p (cid:1) − ǫ ( γ, Aγ ) ( p, A − p ) γ. By combing Theorems 5 and 10, we get. Theorem . The reduced flow of the rubber Chaplygin ball rolling over a spherewith a inertia operator (8) on the cotangent bundle T ∗ S n − realized with constraints (69) is given by equations (77) and (78) . The system has an invariant measure (79) ( Aγ, γ ) n − ǫ +2 − n w n − . The Hamiltonian function of thereduced system takes the form(80) H = ǫ p, A − p )( γ, Aγ ) , which is unique only on the subvariety (69).At the points of T ∗ S n − , the system (77), (78) can be written in the almostHamiltonian form(81) ˙ x = X H = ( X p , X γ ) , i X H ( w + Σ) = dH, where Σ is a semi-basic perturbation term, determined by the J-K term at the righthand side of (61) (e.g, see [ 21, 17, 45, 46 ]). The form w + Σ is non-degenerate, but,in general, it is not closed.The Chaplygin multiplier is a nonvanishing function ν such that ˜ w = ν ( w + Σ)is closed. The Hamiltonian vector field ˜ X H of the function H on ( T ∗ S n − , ˜ w ) isproportional to the original vector field:˜ X H = 1 ν X H , i ˜ X H ˜ w = dH. Thus, applying the time substitution dτ = νdt , the system (81) becomes the Hamil-tonian system ddτ x = ˜ X H . On the other hand, a classical way to introduce the Chaplygin reducing multiplierfor our system is as follows (e.g., see [ 19, 27 ]). Consider the time substitution dτ = ν ( γ ) dt , and denote γ ′ = dγ/dτ = ˙ γ/ν . Then the Lagrangian function transformsto L ∗ ( γ ′ , γ ) = L red ( νγ ′ , γ ) and we have the new momenta ˜ p = ∂L ∗ /∂γ ′ = νp . Thefactor ν is Chaplygin reducing multiplier if under the above time reparameterizationthe equations (77), (78) become Hamiltonian in the coordinates (˜ p, γ ).The existence of the Chaplygin reducing multiplier ν implies that the originalsystem has an invariant measure ν n − w n − (e.g., see Theorem 3.5, [ ]). From theexpression of an invariant measure (79) we get the form of a possible Chaplygin mul-tiplier: ν ( γ ) = const · ( Aγ, γ ) ǫ − . Remarkably, we have. Theorem . Under the time substitution dτ = ǫ ( Aγ, γ ) ǫ − dt and an appropri-ate change of momenta, the reduced system (77) , (78) becomes a Hamiltonian systemdescribing a geodesic flow on S n − with the metric (82) ds A,ǫ = ( γ, Aγ ) ǫ − (cid:0) ( Adγ, dγ )( Aγ, γ ) − ( Aγ, dγ ) (cid:1) . Remark . Note that, while reductions and invariant measures of considered non-holonomic systems are given for arbitrary inertia tensors (Sections 2 and 3), the Hamil-tonization is performed only for the special one (8). This assumption implies that I = I + D E preserves the subset of bivectors in so ( n ). For n ≥ 4, it is a restrictiveproperty, while for n = 3 an arbitrary inertia operator can be written in the form (8)and we reobtain the result of Ehlers and Koiller [ ]. This is expected since only if the OLLING BALLS OVER SPHERES IN R n reduced configuration space is two-dimensional, the existence of an invariant measureis equivalent to the existence of a Chaplygin multiplier (e.g., see [ ]). Proof. We take ν ( γ ) = ǫ ( Aγ, γ ) ǫ − , so the Lagrangian (75) in the new timebecomes(83) L ∗ ( γ ′ , γ ) = 12 ( γ, Aγ ) ǫ − (cid:0) ( Aγ ′ , γ ′ )( Aγ, γ ) − ( Aγ, γ ′ ) (cid:1) . Following the method of Chaplygin reducing multiplier, we introduce the newmomenta by considering the mapping(84) ( p, γ ) (˜ p, γ ) , ˜ p = νp = ǫ ( Aγ, γ ) ǫ − p . Under (84), the Hamiltonain (80) transforms to(85) H (˜ p, γ ) = 12 ( γ, Aγ ) − ǫ (˜ p, A − ˜ p ) . Now, we realize the cotangent bundle T ∗ S n − within R n (˜ p, γ ):(86) ψ = ( γ, γ ) = 1 , ψ = (˜ p, γ ) = 0 , endowed with the symplectic structure˜ w = d ˜ p ∧ dq + · · · + d ˜ p n ∧ dq n | T ∗ S n − . It is convenient to obtain the Hamiltonian vector field ˜ X H = ( ˜ X ˜ p , ˜ X γ ) of H on( T ∗ S n − , ˜ w ) by using the Lagrange multipliers (e.g., see [ ]). Let H = H − λψ − µψ . Then the equations of the geodesic flow of the metric ds A,ǫ can be written as γ ′ = ˜ X γ = ∂ H ∂ ˜ p = ( γ, Aγ ) − ǫ A − ˜ p − µγ, ˜ p ′ = ˜ X ˜ p = − ∂ H ∂γ = 1 − ǫǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p ) Aγ + 2 λγ + µ ˜ p, where the multipliers λ and µ are determined by taking the derivative of the constraints(86). The straightforward calculations yield ψ ′ = 2( γ, Aγ ) − ǫ ( A − ˜ p, γ ) − µ ( γ, γ ) = 0 ,ψ ′ = ( γ, Aγ ) − ǫ ( A − ˜ p, ˜ p ) − µ ( γ, ˜ p )+ 1 − ǫǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p )( Aγ, γ ) + 2 λ ( γ, γ ) + µ (˜ p, γ ) = 0 , implying µ =( γ, Aγ ) − ǫ ( A − ˜ p, γ ) , λ = − ( γ, Aγ ) − ǫ ( A − ˜ p, ˜ p ) + ǫ − ǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p )= − ǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p ) . Therefore, the Hamiltonian flow of H on ( T ∗ S n − , ˜ w ) takes the form γ ′ = ( γ, Aγ ) − ǫ (cid:0) A − ˜ p − ( A − ˜ p, γ ) γ (cid:1) , (87) ˜ p ′ = 1 − ǫǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p ) Aγ (88) − ǫ ( γ, Aγ ) − ǫ (˜ p, A − ˜ p ) γ + ( γ, Aγ ) − ǫ ( A − ˜ p, γ )˜ p. In the time t , after inverting the mapping (84), the equation (87) takes the form˙ γ · ǫ ( Aγ, γ ) − ǫ = ǫ ( Aγ, γ ) ǫ − ( γ, Aγ ) − ǫ (cid:0) A − p − ( A − p, γ ) γ (cid:1) , which coincides with (77). Further, from ddτ ˜ p = ddτ (cid:16) ǫ ( Aγ, γ ) ǫ − p (cid:17) = ddt (cid:16) ǫ ( Aγ, γ ) ǫ − p (cid:17) ǫ ( Aγ, γ ) − ǫ = (cid:16) p ddt ( Aγ, γ ) ǫ − + ˙ p ( Aγ, γ ) ǫ − (cid:17) ( Aγ, γ ) − ǫ , and ddt ( Aγ, γ ) ǫ − = 2 (cid:0) ǫ − (cid:1) ( Aγ, γ ) ǫ − ( Aγ, ˙ γ ) = (2 ǫ − ǫ ( Aγ, γ ) ǫ − ( p, A − γ ) , we get(89) ddτ ˜ p = (2 ǫ − ǫ ( Aγ, γ ) − ( p, A − γ ) p + ˙ p. Finally, by combining (89) with the right hand side of (88) written in variables( p, γ ), ˜ X ˜ p ( p, γ ) = ǫ (1 − ǫ )( γ, Aγ ) − ( p, A − p ) Aγ − ǫ ( Aγ, γ ) − ( p, A − p ) γ + ǫ ( Aγ, γ ) − ( A − p, γ ) p, we obtain the equation (78):˙ p =(1 − ǫ ) ǫ ( Aγ, γ ) − ( p, A − γ ) p + ǫ (1 − ǫ )( γ, Aγ ) − ( p, A − p ) Aγ − ǫ ( Aγ, γ ) − ( p, A − p ) γ + ǫ ( Aγ, γ ) − ( A − p, γ ) p = ǫ (1 − ǫ )( γ, Aγ ) (cid:0) ( A − p, p ) Aγ + ( p, A − γ )( Aγ, γ ) p (cid:1) − ǫ ( γ, Aγ ) ( p, A − p ) γ. We proved that the vector field defining the motion is proportional to the Hamil-tonian vector field X H = ( X p , X γ ) = ǫ ( Aγ, γ ) ǫ − ( ˜ X ˜ p , ˜ X γ ) = ǫ ( Aγ, γ ) ǫ − ˜ X H , and, whence, ν = ǫ ( Aγ, γ ) ǫ − is the Chaplygin multiplier of the system. (cid:3) Remark . For ǫ = 1, (82) becomes the metric for the horizontal rolling (74). Thegeodesic flow of the metric (74) is completely integrable [ ]. As in the 3-dimensionalcase, it is possible to prove the complete integrability of the reduced systems for ǫ = − A , as well as for ǫ = − A having additional symmetries.We shall consider the integrability aspects of the problem and a geometrical setting byusing nonholonomic connections following [ 3, 20, 41 ] in a separate paper. Acknowledgments. The author is very grateful to Yuri Fedorov, Borislav Gaji´c,and the referees for many valuable suggestions that helps the author to improve theexposition of the results. 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