Geometry and integrability of Euler-Poincare-Suslov equations

TL;DR

This paper explores the geometry and integrability of nonholonomic geodesic flows on Lie groups, introducing new methods to identify integrable cases of Euler-Poincare-Suslov equations using algebraic structures.

Contribution

It presents novel algebraic constructions that lead to integrable cases of nonholonomic Euler-Poincare-Suslov equations on Lie groups.

Findings

Constructed new integrable cases using chains of subalgebras.

Identified nonholonomic geodesic flows as restrictions of integrable sub-Riemannian flows.

Developed methods to find first integrals for nonholonomic systems.

Abstract

We consider nonholonomic geodesic flows of left-invariant metrics and left-invariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler-Poincare-Suslov equations on the corresponding Lie algebras. The Poisson and symplectic structures give raise to various algebraic constructions of the integrable Hamiltonian systems. On the other hand, nonholonomic systems are not Hamiltonian and the integration methods for nonholonomic systems are much less developed. In this paper, using chains of subalgebras, we give constructions that lead to a large set of first integrals and to integrable cases of the Euler-Poincare-Suslov equations. Further, we give examples of nonholonomic geodesic flows that can be seen as a restrictions of integrable sub-Riemannian geodesic flows.