Area-constrained planar elastica

We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties; not only can the Euler-Lagrange equation be integrated to provide a quadrature for the curvature but, in addition, the embedding itself can be expressed as a local function of the curvature. The configuration space is shown to be essentially one dimensional, with surprisingly rich structure. Distinct branches of integer-indexed equilibria exhibit self-intersections and bifurcations—a gallery of plots is provided to highlight these findings. Perturbations connecting equilibria are shown to satisfy a first-order ODE which is readily solved. We also obtain analytical expressions for the energy as a function of the area in some limiting regimes.