Rubber rolling over a sphere

Abstract

Rubber coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by 'marble' coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at 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 G(2)). The 2-3-5 nonholonomic geometries are classified in a companion paper [2] via Cartan's equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4-8] with SO(3) symmetry group, total space Q = SO(3) x S-2 and base S-2, that can be reduced to an almost Hamiltonian system in T* S-2 with a non-closed 2-form w(NH). In this paper we present some basic results on the sphere-sphere 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 w(NH) is conformally symplectic [9]: 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 result by Borisov and Mamaev [10] that the system is integrable for p = - 1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = - 3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I-j are known.'