$G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball

TL;DR

This paper analyzes the $n$-dimensional Chaplygin sphere as an almost Hamiltonian system, reduces it via symmetry, and demonstrates Hamiltonization at a lower-dimensional level, providing new geometric insights and re-proving known results.

Contribution

It introduces an explicit geometric reduction framework for Chaplygin systems, showing Hamiltonization at the reduced level and offering a new symplectic perspective on Chaplygin's ball.

Findings

Homogeneous Chaplygin ball is Hamiltonian after reduction to $T^*S^{n-1}$.

3D Chaplygin ball becomes Hamiltonian after time reparametrization on $T^*S^2$.

Explicit geometric description of reduced systems facilitates Hamiltonizability analysis.

Abstract

Via compression ([11, 7]) we write the $n$-dimensional Chaplygin sphere system as an almost Hamiltonian system on $T^*SO(n)$ with internal symmetry group $SO(n-1)$. We show how this symmetry group can be factored out, and pass to the fully reduced system on (a fiber bundle over) $T^*S^{n-1}$. This approach yields an explicit description of the reduced system in terms of the geometric data involved. Due to this description we can study Hamiltonizability of the system. It turns out that the homogeneous Chaplygin ball, which is not Hamiltonian at the $T^*SO(n)$-level is Hamiltonian at the $T^*S^{n-1}$-level. Moreover, the 3-dimensional ball becomes Hamiltonian at the $T^*S^2$-level after time reparametrization, whereby we re-prove a result of [4, 5] in symplecto-geometric terms. We also study compression followed by reduction of generalized Chaplygin systems.