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

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.