Metric Lines in the Special Euclidean group on the plane

TL;DR

This paper investigates the properties of sub-Riemannian geodesics and metric lines in the Special Euclidean group on the plane, providing conditions for periodic geodesics and characterizing metric lines using Hamilton-Jacobi theory.

Contribution

It offers new insights into the characterization of metric lines and periodic geodesics in SE(2) using Hamiltonian and Hamilton-Jacobi methods.

Findings

Necessary conditions for periodic sub-Riemannian geodesics in SE(2)

Characterization of metric lines in SE(2)

An alternative proof using Hamilton-Jacobi theory

Abstract

The Special Euclidean group on the plane $SE(2)$ has the left-invariant sub-Riemannian structure. Every sub-Riemannian manifold possesses a Hamiltonian function governing the sub-Riemannian geodesic flow. Two natural questions are: What are the necessary conditions for periodic sub-Riemannian geodesics? What geodesics are the metric lines in SE(2)? We answer both questions, and our method for the second is an alternative proof using the Hamilton-Jacobi theory.