Invariant measures of modified LR and L+R systems

TL;DR

This paper introduces a class of modified Lie group systems, including LR and L+R systems, that possess invariant measures, with applications to nonholonomic rigid body problems like Veselova's system.

Contribution

It defines and analyzes invariant measures for modified LR and L+R systems on Lie groups, extending known models and providing new insights into nonholonomic dynamics.

Findings

Modified systems have invariant measures.

Application to Veselova's problem and nonholonomic balls.

Analysis of systems on Lie algebra and Stiefel variety.

Abstract

We introduce a class of dynamical systems having an invariant measure, the modifications of well known systems on Lie groups: LR and L+R systems. As an example, we study modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra $so(n)$ with the Stiefel variety $V_{n,r}$, as well as the associated $\epsilon$L+R system on $so(n)\times V_{n,r}$. In the 3--dimensional case, these systems model the nonholonomic problems of a motion of a ball and a rubber ball over a fixed sphere.