Rolling sphere problems on spaces of constant curvature

Abstract

AbstractThe 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$\mathbb{S}^{n}$as it rolls on$\mathbb{E}^{n}$without slipping between two points of$\mathbb{E}^{n}\times SO_{n+1}(\mathbb{R})$. This problem is extended to situations in which an oriented sphere$\mathbb{S}_{\rho}^{n}$of radius ρ rolls on a stationary sphere$\mathbb{S}_{\sigma}^{n}$and to the hyperbolic analogue in which the spheres$\mathbb{S}_{\rho}^{n}$and$\mathbb{S}_{\sigma}^{n}$are replaced by the hyperboloids$\mathbb{H}_{\rho}^{n}$and$\mathbb{H}_{\sigma}^{n}$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.