Rolling Problems on Spaces of Constant Curvature

Abstract

The rolling sphere problem on \( \mathbb{E}^n \) 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 without slipping between two boundary points of \( \mathbb{E}^n \times SO_n \). This problem is extended to the following cases of rolling: ℍn on \( \mathbb{E}^n ,\mathbb{S}_\rho ^n \) on \( \mathbb{S}_\sigma ^n \), and ℍ ρ n on ℍ σ n , where σ≠ρ are the radii of the spheres or hyperboloids. The term “rolling” is generalized to an isometric sense: the length of a curve is measured using the Riemannian metric of the stationary manifold while the orientation of the rolling object is described by a matrix from its isometry group. These problems constitute left-invariant optimal control problems on Lie groups, whose Hamiltonian equations reveal certain integrals of motion and show, on the level of Lie algebras, that all of the above problems are governed by a single set of equations.