Area-Constrained Planar Elastica

Details Area-Constrained Planar Elastica Area-Constrained Planar Elastica

2001-03-12

arxiv.org/abs/cond-mat/0103262v2

Physics

Condensed Matter

Soft Condensed Matter

23 pages, several figures. Version 2: New title. Changes in the introduction, addition of a new section with conclusions. Figu

Scientific paper

10.1103/PhysRevE.65.031801

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.

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