Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures

Abstract

We first construct nonholonomic systems of n homogeneous balls (Formula presented.) with centers (Formula presented.) and with the same radius r that are rolling without slipping around a fixed sphere S0 with center O and radius R. In addition, it is assumed that a dynamically nonsymmetric sphere S of radius R + 2r and the center that coincides with the center O of the fixed sphere S0 rolls without slipping over the moving balls B1, ..., Bn. We prove that these systems possess an invariant measure. As the second task, we consider the limit, when the radius R tends to infinity. We obtain a corresponding planar problem consisting of n homogeneous balls B1, ..., Bn with centers B1, ..., Bn and the same radius r that are rolling without slippingover a fixed plane Σ0, and a moving plane Σ that moves without slipping over the homogeneous balls. We prove that this system possesses an invariant measure and that it is integrable in quadratures according to the Euler – Jacobi theorem.