Sympletic reduction of the sub-Riemannian geodesic flow for metabelian nilpotent groups

TL;DR

This paper investigates the symplectic reduction of the normal Hamiltonian flow in sub-Riemannian nilpotent Lie groups with abelian derived subgroups, establishing criteria for integrability and length-minimizing trajectories.

Contribution

It introduces a symplectic reduction approach to analyze the Hamiltonian flow, providing new integrability criteria and conditions for length-minimizing trajectories in specific nilpotent groups.

Findings

Normal trajectories correspond to polynomial Hamiltonians.

Flow is integrable in Engel-type groups.

Conditions for length-minimizing trajectories in certain Carnot groups.

Abstract

We consider nilpotent Lie groups for which the derived subgroup is abelian. We equip them with subRiemannian metrics and we study the normal Hamiltonian flow on the cotangent bundle. We show a correspondence between normal trajectories and polynomial Hamiltonians in some euclidean space. We use the aforementioned correspondence to give a criterion for the integrability of the normal Hamiltonian flow. As an immediate consequence, we show that in Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot groups that are semidirect products of two abelian groups, we give a set of conditions that normal trajectories must fulfill to be globally length-minimizing. Our results are based on a symplectic reduction procedure.