Rubber rolling over a sphere

“Rubber” coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by “marble” coated bodies [28]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at the corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2-3-5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G2). The 2-3-5 nonholonomic geometries are classified in a companion paper [29] via Cartan’s equivalence method [19]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [21, 57, 35, 27, 39] with SO(3) symmetry group, total space Q = SO(3)× S2 and base S2, that can be reduced to an almost Hamiltonian system in T ∗S2 with a non-closed 2-form ωNH . In this paper we present some basic results on the spheresphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ωNH is conformally symplectic [59]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1 2 ( b a −1). Using sphero-conical coordinates we verify the results by Borisov and Mamaev [12, 13] that the system is integrable for p =−1/2 (ball over a plane) and p = −3/2 (rolling ball with twice the radius of a fixed internal ball). ∗On a CAPES-Fulbright visit to Caltech, Winter 2005-2006.