$G$-Chaplygin systemswith internal symmetries, truncation, and an (almost) symplecticview of Chaplygin's ball

Abstract

Via compression ([18, 8])we write the $n$-dimensional Chaplygin sphere system as analmost Hamiltonian system on T*$\SO(n)$ with internal symmetry group$\SO(n-1)$. We show how this symmetry group can be factored out, andpass to the fully reduced system on (a fiber bundle over)T*$S^{n-1}$.This approach yieldsan explicit description of the reduced system in terms of thegeometric data involved. Due to this description we can studyHamiltonizability of the system. It turns out that the homogeneousChaplygin ball, which is not Hamiltonian at the T*$\SO(n)$-level,is Hamiltonian at the T*$S^{n-1}$-level. Moreover, the$3$-dimensionalball becomes Hamiltonian at the T*$S^{2}$-level aftertime reparametrization, whereby we re-prove a result of[4, 5] in symplecto-geometric terms.We also study compression followed by reduction of generalizedChaplygin systems.