March  2009, 1(1): 35-53. doi: 10.3934/jgm.2009.1.35

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

1. 

Section de Mathematiques, Station 8, EPFL, CH-1015 Lausanne, Switzerland, Switzerland

Received  November 2008 Revised  March 2009 Published  April 2009

Via compression ([18, 8]) 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.
Citation: Simon Hochgerner, Luis García-Naranjo. $G$-Chaplygin systems with internal symmetries, truncation, and an (almost) symplectic view of Chaplygin's ball. Journal of Geometric Mechanics, 2009, 1 (1) : 35-53. doi: 10.3934/jgm.2009.1.35
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