Superintegrability of Sub-Riemannian Problems on Unimodular 3D Lie Groups

TL;DR

This paper proves Liouville integrability and superintegrability of Hamiltonian systems for sub-Riemannian geodesics on unimodular 3D Lie groups, advancing understanding of their geometric and dynamical properties.

Contribution

It establishes the superintegrability of sub-Riemannian Hamiltonian systems on unimodular 3D Lie groups, a novel result in geometric control theory.

Findings

Liouville integrability of the Hamiltonian system

Superintegrability of the Hamiltonian system

Enhanced understanding of sub-Riemannian geodesic dynamics

Abstract

Left-invariant sub-Riemannian problems on unimodular 3D Lie groups are considered. For the Hamiltonian system of Pontryagin maximum principle for sub-Riemannian geodesics, the Liouville integrability and superintegrability are proved.