Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces

TL;DR

This paper studies a class of nonholonomic systems on Lie groups, showing they can be viewed as generalized Chaplygin systems with invariant measures, and demonstrates integrability in specific cases related to rigid body motion.

Contribution

It establishes that LR systems on Lie groups can be regarded as generalized Chaplygin systems with invariant measures and explores their integrability and geometric properties.

Findings

LR systems on Lie groups have invariant measures after reduction.

Special cases of LR systems are integrable and relate to geodesic flows on spheres.

Explicit reconstruction of motion on SO(n) is achieved.

Abstract

We consider a class of dynamical systems on a Lie group $G$ with a left-invariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle $G \to Q=G/H$, $H$ being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space $Q$ always possess an invariant measure. We study the case $G=SO(n)$, when LR systems are multidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion, which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties $V(k,n)$ as the corresponding homogeneous spaces. For $k=1$ and a special choice of the left-invariant metric on SO(n), we prove that under a change of time, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere $S^{n-1}$. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable. In this case we also explicitly reconstruct the motion on the group SO(n).