LR and L+R Systems

TL;DR

This paper studies coupled nonholonomic LR systems on Lie groups, demonstrating integrability of the rubber Chaplygin sphere and relating L+R systems to coupled LR systems through reduction.

Contribution

It introduces a framework for coupled LR systems on Lie groups, proves integrability of the rubber Chaplygin sphere, and connects L+R systems to coupled LR systems via reduction.

Findings

Rubber Chaplygin sphere becomes integrable after reduction and reparametrization.

L+R systems can be derived as reduced systems of coupled LR systems.

Examples include n-dimensional variants of spherical support and rubber Chaplygin sphere.

Abstract

We consider coupled nonholonomic LR systems on the product of Lie groups. As examples, we study $n$-dimensional variants of the spherical support system and the rubber Chaplygin sphere. For a special choice of the inertia operator, it is proved that the rubber Chaplygin sphere, after reduction and a time reparametrization becomes an integrable Hamiltonian system on the $(n-1)$--dimensional sphere. Also, we showed that an arbitrary L+R system introduced by Fedorov can be seen as a reduced system of an appropriate coupled LR system.