Eur. Phys. J. B 29, 163-166 (2002)
https://doi.org/10.1140/epjb/e2002-00278-6

Elastica hypoarealis

¹ Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN, Apdo. Postal 14-740, 07000 México DF, México
² Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, 04510 México DF, México

Corresponding authors: ^a capo@fis.cinvestav.mx - ^b chryss@nuclecu.unam.mx - ^c jemal@nuclecu.unam.mx

Received: 18 October 2001
Revised: 31 May 2002
Published online: 2 October 2002

Abstract

We examine the equilibria of a rigid loop in the plane, characterized by an energy functional quadratic in the curvature, subject to the constraints of fixed length and fixed enclosed area. Whereas the only non self-intersecting equilibrium corresponding to the fixed length constraint is the circle, the area constraint gives rise to distinct equilibria labeled by an integer. These configurations exhibit self-intersections and bifurcations as the area is reduced. In addition, not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but the embedding itself can be expressed as a local function of the curvature. Perturbations connecting equilibria are shown to satisfy a first order ODE which is readily solved. Analytical expressions for the energy as a function of the area are obtained in the limiting regimes.