Euler elasticae in the plane and the Whitney--Graustein theorem

TL;DR

This paper analyzes the shapes and stability of plane Euler elasticae using energy principles, providing a classification of their critical points, and offers a new proof of the Whitney--Graustein theorem with visual demonstrations.

Contribution

It introduces a stability analysis of Euler elasticae, identifies their normal forms as energy minima, and offers a mechanical proof of the Whitney--Graustein theorem with computational visualization.

Findings

Stable elastica shapes correspond to energy minima.

The set of stable elasticae matches the normal forms.

The work includes a software for visualizing curve evolution.

Abstract

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and show which of the critical points are stable equilibrium points of the energy given by U, and which are unstable. It turns out that the set of stable equilibrium points coincides with the set of minima of U, so that the corresponding shapes of the curves obtained may be regarded as normal forms of Euler elasticae. In this way, we find the solution of the Euler problem (set in 1744) for plane closed elasticae. As a by-product, we obtain a "mechanical" proof of the Whitney--Graustein theorem on the classification of regular curves in the plane up to regular homotopy (in the particular case of C^2 curves). Besides mathematical theorems, our work includes a computer graphics software which shows, as an animation, how any plane curve evolves to its normal form under a discretized version of gradient descent along the (discretized) Euler functional.