Rolling balls over spheres in $ \newcommand{\m}{\mathfrak m} {\mathbb{R}^n}$

We study the rolling of the Chaplygin ball in over a fixed -dimensional sphere without slipping and without slipping and twisting. The problems can be naturally considered within a framework of appropriate modifications of the L + R and LR systems—well known systems on Lie groups with an invariant measure. In the case of the rolling without slipping and twisting, we describe the -Chaplygin reduction to S n−1 and prove the Hamiltonization of the reduced system for a special inertia operator.