The critical points of the elastic energy among curves pinned at endpoints

TL;DR

This paper explicitly identifies all critical points of elastic energy among fixed-length, pinned-end curves, showing that minimizers are loop-free elasticae with no interior inflections, using the shooting method.

Contribution

It provides a complete classification of critical points and minimizers for elastic energy among pinned curves, including explicit solutions via the shooting method.

Findings

Critical points are wavelike elasticae.

Minimizers have no loops or interior inflection points.

All critical points can be explicitly identified.

Abstract

In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.