Relation between Euler's Elasticae and Sub-Riemannian Geodesics on SE(2)

TL;DR

This paper explores the connection between Euler's elasticae and sub-Riemannian geodesics on SE(2), revealing they coincide only for straight line segments through Hamiltonian analysis.

Contribution

It establishes a precise relation between elasticae and geodesics on SE(2), clarifying their equivalence only in the case of straight lines.

Findings

Elasticae and sub-Riemannian geodesics coincide only for straight lines.

Hamiltonian analysis reveals the conditions for their equivalence.

The study clarifies the geometric relationship on SE(2).

Abstract

In this note we describe a relation between Euler's elasticae and sub-Riemannian geodesics on SE(2). Analyzing the Hamiltonian system of Pontryagin maximum principle we show that these two curves coincide only in the case when they are segments of a straight line.