Integrable geodesic flows of non-holonomic metrics

TL;DR

This paper investigates the integrability of geodesic flows in non-holonomic metrics, specifically on the Heisenberg group, linking Hamiltonian systems with geometric control theory.

Contribution

It provides a detailed analysis of integrable geodesic flows for left-invariant metrics and distributions on Lie groups, connecting these flows to Euler equations and mechanics.

Findings

Geodesic flows on the Heisenberg group are integrable.

Reduction to Euler equations on Lie groups is demonstrated.

Connections to analytical mechanics are established.

Abstract

Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The flows corresponding to left-invariant metrics and left-invariant distributions on Lie groups are reduced to Euler equations on Lie groups. Relation of these constructions to problems of analytical mechanics is discussed.