Chaplygin systems associated to Cartan decompositions of semi-simple Lie algebras

TL;DR

This paper explores the connection between Chaplygin systems and Cartan decompositions of semi-simple Lie groups, analyzing their structure, measure preservation, and conditions for Hamiltonization, with applications to rolling ball problems.

Contribution

It introduces a novel link between Chaplygin systems and Cartan decompositions, providing criteria for Hamiltonizability and methods for symmetry reduction.

Findings

The system possesses a preserved measure.

Internal symmetries can be factored out via truncation.

Criteria for Hamiltonizability are established.

Abstract

We relate a Chaplygin type system to a Cartan decomposition of a real semi-simple Lie group. The resulting system is described in terms of the structure theory associated to the Cartan decomposition. It is shown to possess a preserved measure and when internal symmetries are present these are factored out via a process called truncation. Furthermore, a criterion for Hamiltonizability of the system on the so-called ultimate reduced level is given. As important special cases we find the Chaplygin ball rolling on a table and the rubber ball rolling over another ball.