Rolling sphere problems on spaces of constant curvature

The rolling sphere problem on Euclidean space consists of determining the path of minimal length traced by the point of contact of the oriented unit sphere Sⁿ as it rolls on Eⁿ without slipping between two points of Eⁿ× SOₙ₊₁(R) . This problem is extended to situations in which an oriented sphere Sρⁿ of radius ρ rolls on a stationary sphere Sσⁿ and to the hyperbolic analogue in which the spheres Sρⁿ and Sσⁿ are replaced by the hyperboloids Hρⁿ and Hσⁿ respectively. The notion of “rolling” is defined in an isometric sense: the length of the path traced by the point of contact is measured by the Riemannian metric of the stationary manifold, and the orientation of the rolling object is measured by a matrix in its isometry group. These rolling problems are formulated as left invariant optimal control problems on Lie groups whose Hamiltonian extremal equations reveal two remarkable facts: on the level of Lie algebras the extremal equations of all these rolling problems are governed by a single set of equations, and the projections onto the stationary manifold of the extremal equations havingI4=0, whereI4is an integral of motion, coincide with the elastic curves on this manifold. The paper then outlines some explicit solutions based on the use of symmetries and the corresponding integrals of motion.