Yuri Fedorov Multidimensional Integrable Generalizations of the nonholonomic Chaplygin sphere problem. November 29, 2006

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1 Yuri Fedorov Multidimensional Integrable Generalizations of the nonholonomic Chaplygin sphere problem November 29,

2 Preservation of Invariant Measure ẋ = v(x), x = (x 1,..., x n ) ( ) The volume form I = µ(x) dx 1 dx n is said to be an invariant measure of (*) if L v I = 0 µ(x) + µ div(v) = 0. Some Integrability Theorems: Theorem 1. (Euler Jacobi). If the system (*) possesses an an invariant measure I and n 2 independent first integrals, then it is integrable by quadratures. Theorem 2. (Kozlov). Suppose that the system (*) has an invariant measure I(x), k 1 independent integrals f 1 (x),..., f k 1 (x), and there are vector fields u k+1 = v, u k+2,..., u n, independent at each point, commuting with each other, such that Then this system is integrable by quadratures. L uj I = 0, L uj f i = 0, i = 1,..., k, j = k + 1,..., n. 2

3 Example: Spherical Supporter (Yu. F., 1988) A dynamically non-symmetric sphere surrounded by N symmetric spheres C k J ω + ω Jω = N γ k R k, D k ω k = λ k γ k R k, k = 1,..., N. k=1 N nonholonomic constraints expressing absence of slipping at the contact points: λ k ω k = ω (ω, γ k )γ k λ k (ω k, γ k )γ k, λ k (ω k, γ k ) = c k = const. 3

4 Taking into account the constraints, the first equation gives rise to N Λ ω + ω Λω = Γ ω, Λ = J + (D k /λ 2 k)i, Γ = Setting Γ = aα α + bβ β + cγ γ, one obtains the generalized Euler top on T SO(3): First integrals: (K, K), k=1 N D k γ k γ k /λ 2 k, k=1 K = K ω, α = α ω, β = β ω, γ = γ ω, K = (Λ + Γ)ω = Λω + a(ω, α)α + b(ω, β)β + c(ω, γ)γ, (K, α) = (Λ a ω, α), (K, β) = (Λ b ω, β), (K, γ) = (Λ c ω, γ), Λ a = Λ + ai, Λ b = Λ + bi, Λ c = Λ + ci, of which any three integrals are independent. Besides, the system has the kinetic energy integral 1 2 (ω, (Λ + Γ)ω) = 1 2 (ω, Λω) + a 2 (ω, α)2 + b 2 (ω, β)2 + c 2 (ω, γ)2. It also possesses an invariant measure with density µ = det(λ + Γ) By the Euler Jacobi theorem, the system is integrable by quadratures, and its generic invariant manifolds are two-dimensional tori. For N = 1 we have Γ = D 1 γ γ/λ 2 1, γ = γ1, K = K ω, γ = γ ω, K = Λω D(ω, γ)γ, and the spherical support becomes equivalent to the celebrated Chaplygin sphere problem (Chaplygin, 1903). 4

5 The Suslov problem (A. Suslov, 1903) Nonholonomic constraint: the projection of the angular velocity vector ω R 3 to a certain fixed in the body unit vector γ equals zero: ( ω, γ ) = 0. The Lagrangian L = 1 2 ( ω, I ω ), the momentum M = (M 1, M 2, M 3 ) T = I ω. d dt (I ω ) = I ω ω + λ γ, which is equivalent to d dt (I ω ) = (I ω, γ ) ω I 1 γ. The Suslov system possesses the energy integral and a line of equilibria positions ( ω, I ω ) ( M, I 1 M) = h, h = const E = {( ω, γ ) = 0} {(I ω, γ ) = 0}. In the basis where γ = (0, 0, 1) T the energy integral can be replaced by I 22 M 2 1 2I 12M 1 M 2 + I 11 M

6 3 e E 1 2 6

7 Discrete nonholonomic mechanical system on Q Q (J.Cortés, S. Martínez, 1999) (1) a discrete Lagrangian L : Q Q R; (2) distribution D on TQ (D = { q TQ A j (q), q = 0, j = 1,..., s}); (3) a discrete constraint manifold D d Q Q (of the same dimension as D and (q, q) D d for all q Q) F j (q k, q k+1 ) = 0, j = 1,..., s Discrete Lagrange d Alembert equations with multipliers D 1 L(q k, q k+1 ) + D 2 L(q k 1, q k ) = s λ k ja j (q k ), F j (q k, q k+1 ) = 0 j=1 This defines a multi-valued map Q Q Q Q Discrete Euler Poincaré Suslov Equations (Yu. F., D. Zenkov, 2004) - Assume Q = G{g} and L(g g k, g g k+1 ) = L(g k, g k+1 ). - Introduce left displacement W k = g 1 k g k+1 G. There exists reduced discrete Lagrangian l d : (G G)/G = G R such that L d (g k, g k+1 ) = l d (W k ). - The discrete body momentum p k : G G g p k, ξ = d l d (exp( sξ)w k ), p k = RW ds s=0 k l d(w k ). - Left-invariant distribution D T G, D g = TL g d, d = {ξ g a j, ξ = 0, j = 1,..., s}, a j = const. Then a j, g 1 ġ = 0, j = 1,..., s. 7

8 Discrete Euler Poincaré Suslov Equations (continuation) Discrete left-invariant constraints F j (g g k, g g k+1 ) = F j (g k, g k+1 ) there exist functions f j : G R, j = 1,..., s, such that F j (g k, g k+1 ) = f j (W k ). D d G G is completely defined by the admissible displacement subvariety S = {f 1 (W) = 0,..., f s (W) = 0} G; This implies that the discrete momentum p k is restricted to the admissible momentum subvariety U = {p g p = L Wl d(w), W S} g. p k+1 Ad W k p k = s λ j k+1 aj, where W k, W k+1 S, p k, p k+1 U g. j=1 Our choice of S G: S = exp d. Usually exp d = G! Assume g = h d, h being a subalgebra, such that [h, h] h, [d, d] h, [h, d] d. In this case S = exp d is a smooth submanifold of G homeomorphic to either the symmetric space G/H or to a quotient of G/H by a finite group action. 8

9 Discrete Suslov system G = SO(3) (SO(n) ), {R k } SO(3) (SO(n)) Finite rotations Ω k = R 1 k R k+1 (discrete analogue of the body angular velocity ω = R 1 Ṙ) Discrete Lagrangian is the same as for the discrete Euler top (A. Veselov, J.Moser, 1991) L(R k, R k+1 ) = 1 2 Tr(R kjr T k+1 ), l d(ω k ) = 1 2 Tr (Ω kj), The discrete body angular momentum M k = Ω k J JΩ T k so(3) Continuous constraints are defined by the subspace ω 1n 0 0 ω d = ω n 1,n so(n), d = 0 0 ω 23 so(3) ω 13 ω 23 0 ω 1n... ω n 1,n 0 Discrete constraints are defined by the admissible finite rotations S = exp d = {Ω SO(n) Ω ij = Ω ji, Ω in = Ω ni, 1 i, j n 1.} S is diffeomerphic to RP n 1 = S n 1 /Z 2. For n = 3, 2(q0 2 + q2 1 ) 1 2q 1q 2 2q 0 q 2 Ω = 2q 1 q 2 2(q0 2 + q2 2 ) 1 2q 0q 1, q 2q 0 q 2 2q 0 q 1 2q q2 1 + q2 2 =

10 Discrete Euler Poincaré Suslov equations on so (3). M k+1 = Ω T k M k Ω k + λ k , Admissible momentum locus in so (3), U = {Ω k J JΩ T k Ω k S} M k = Ω k J JΩ T k (a) (b) Theorem 3. Regardless to the branch of the map M k M m+1, the discrete Suslov map preserves the reduced constrained energy E c (M 23, M 13 ) = (J 11 + J 33 )M J 12M 23 M 13 + (J 22 + J 33 )M

11 0.2 3 e E (c) (d) 11

12 A discretization of the Chaplygin sphere {R k, r k }, R k SO(n), r k R n Discrete Lagrangian L = 1 2 Tr(R kjr T k+1 ) + m 2 r k, r k, r k = r k+1 r k. The discrete momentum of the sphere in the body frame M k = Ω k J JΩ T k, Ω k = R T k R k+1 SO(n), Continuous constraints expressing abcence of slipping at the contact point ṙ + ρω γ = 0. Discrete Euler Poincaré equations with multipliers M k = Ω T k 1M k 1 Ω k 1 + ρ F k γ T k, m( r k r k 1 ) = f k. where F k = Rk T f k, γ k = Rk T γ. Our choice of discrete constraints that mimic the continuous constraints r k + ρ 2 ( Ω k Ω T k ) γ = 0, Ωk = R T k+1r k SO(n). Proposition 4. The map admits the following compact representation where K k = Ω T k 1 K k 1Ω k 1, Γ k = Ω T k 1 Γ k 1Ω k 1, K k = Ω k (J + D ) 2 Γ k ( J + D ) 2 Γ k Ω T k + D 2 (Γ kω k Ω T k Γ k) M k + D 2 (Ω kγ k Γ k Ω T k ) + D 2 (Γ kω k Ω T k Γ k), D = mρ 2, = Ω k (J + D ) ( 2 (Γ k+1 + Γ k ) J + D ) 2 (Γ k+1 + Γ k ) Ω T k. 12

13 The classical case n = 3 Let M = (M 1, M 2, M 3 ) T (M 32, M 13, M 21 ) T, K = (K1, K 2, K 3 ) T (K 32, K 13, K 21 ) T. Then the map reads and preserves 3 independent integrals K k = Ω T k 1 K k 1, γ k = Ω T k 1 γ k 1 γ, γ = 1 K, γ = h, K, K = n. The special case K γ. ( K k = hγ k, h =const) This defines map G h : S 2 S 2, G h (γ k ) = γ k+1 Proposition 5. Regardless to branch of the map G h, it has the quadratic integral γ, Λ 1 γ = l. Hence, γ admits the elliptic parameterization, e.g., γ 1 = C 1 cn(u k), γ 2 = C 2 sn(u k), γ 3 = C 3 dn(u k), Therefore, for a fixed l, the map G h is reduced to one-dimensional map u k+1 = u k + u k (u k, l) u k depens non-trivially on u k! 13