Knotted Elastic Curves in Ir

One of the oldest topics in the calculus of variations is the study of the elastic rod which, according to Daniel Bernoulli's idealization, minimizes total squared curvature among curves of the same length and first order boundary data. The classical term elastica refers to a curve in the plane or IR which represents such a rod in equilibrium. While the elastica and its generalizations have long been (and continue to be) of interest in the context of elasticity theory, the elastica as a purely geometrical entity seems to have been largely ignored. Recently, however, Bryant and Griffiths [1, 3] have found the elastica and its natural generalization to space forms (where arc length is generally not constrained) to be an interesting example in the context of the general theory of exterior differential systems. Independently, the present authors have studied 'free' elastic curves in space forms and have drawn connections to well-known problems in differential geometry [4, 5]. From the geometric point of view, the closed elasticae and their global behaviour are naturally of particular interest. In the present paper we maintain this emphasis but return to the classical setting of Euclidean curves with fixed arc length. Specifically, we give a complete classification of closed elastic curves in U" and determine the knottedness of these elasticae. We note that the integrability of the equations for a classical elastica was known already to Euler in the planar case and (essentially) to Radon in the case of IR (see Blaschke's Vorlesungen iiber Differentialgeometrie I); to determine the closed elasticae, however, the chief problem is to understand the dependence of the resulting elliptic integrals on certain parameters. Since the closed planar elasticae were well described already by Euler, and since uniqueness of solutions in the initial-value problem implies that any elastica in U" must in fact lie in U, it will suffice to present the following.