Exponential mapping in Euler's elastic problem

TL;DR

This paper analyzes the global structure of the exponential mapping in Euler's elastic problem, providing a framework for computing optimal elastic curves with fixed endpoints and tangents.

Contribution

It describes the global structure of the exponential mapping and proves diffeomorphic properties, enabling algebraic solutions for optimal elasticae.

Findings

Open domains are mapped diffeomorphically by the exponential map.

Optimal elasticae can be computed via algebraic equations with unique solutions.

Explicit solutions are provided for certain boundary conditions.

Abstract

The classical Euler's problem on optimal configurations of elastic rod in the plane with fixed endpoints and tangents at the endpoints is considered. The global structure of the exponential mapping that parameterises extremal trajectories is described. It is proved that open domains cut out by Maxwell strata in the preimage and image of the exponential mapping are mapped diffeomorphically. As a consequence, computation of globally optimal elasticae with given boundary conditions is reduced to solving systems of algebraic equations having unique solutions in the open domains. For certain special boundary conditions, optimal elasticae are presented.