Rolling balls over spheres in R^n

We study the rolling of the Chaplygin ball in ${\mathbb{R}}^n$ over a fixed $(n-1)$–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 groups with an invariant measure. In the case of the rolling without slipping and twisting, we describe the $SO(n)$-Chaplygin reduction to $S^{n-1}$ and prove the Hamiltonization of the reduced system for a special inertia operator.