Rubber Rolling over a Sphere

"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 $Q = SO (3) \times S^2$ and base $S^2$, that can be reduced to an almost Hamiltonian system in $T * S^2$ with a non-closed 2-form $\omega_{NH}$. 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 $\omega_{NH}$ 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.