Rubber rolling over a sphere

Abstract

``Rubber'' coated rolling bodies satisfy a no-twist in addition to the no slip satisfied by ``marble'' coated bodies. 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 exceptionalgroup G_2. The 2-3-5 nonholonomic geometries are classified in a companion paper via Cartan's equivalence method. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system with SO(3) symmetry group, total space Q = SO(3) X S^2 that can be reduced to an almost Hamiltonian system in T^*S^2 with a non-closed 2-form ω_{NH}. In this paper we present some basic results on this reduction and as an example we discuss the sphere-sphere problem. In this example the 2-form is conformally symplectic so the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure. Using sphero-conical coordinates we verify the results by Borisov and Mamaev that the system is integrable for a ball over a plane and a rubber ball with twice the radius of a fixed internal ball.