Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups

TL;DR

This paper analyzes the sub-Riemannian geometry on certain three-dimensional Lie groups, revealing symmetries in the Hamiltonian system and providing explicit bounds on geodesic optimality times based on pendulum dynamics.

Contribution

It characterizes Maxwell sets and cut times for sub-Riemannian problems on solvable, non-nilpotent Lie groups using phase-space analysis of the pendulum-like Hamiltonian.

Findings

Maxwell set characterized by symmetries in the Hamiltonian system

First Maxwell time equals the pendulum period for most geodesics

Explicit upper bound for cut time derived from pendulum dynamics

Abstract

In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian system takes the form of a perturbed pendulum. A qualitative phase-space analysis allows us to prove that this vertical component exhibits nontrivial symmetries. In particular, we are able to fully characterize the Maxwell set corresponding to these symmetries, and show that its first Maxwell time coincides with the period of the pendulum for almost all geodesics. This result yields an explicit upper bound for the cut time in terms of the period of the pendulum.