Spherical and Planar Ball Bearings — Nonholonomic Systems with Invariant Measures

We first construct nonholonomic systems of n homogeneous balls 𝐁_1,…,𝐁_n with centers O_1,…,O_n and with the same radius r that are rolling without slipping around a fixed sphere 𝐒_0 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 𝐒_0 rolls without slipping over the moving balls 𝐁_1,…,𝐁_n . 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 𝐁_1,…,𝐁_n with centers O_1,…,O_n and the same radius r that are rolling without slipping over a fixed plane Σ_0 , 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.