Hamiltonians for curves

Abstract

We examine the equilibrium conditions of a curve in space when a local energy\npenalty is associated with its extrinsic geometrical state characterized by its\ncurvature and torsion. To do this we tailor the theory of deformations to the\nFrenet-Serret frame of the curve. The Euler-Lagrange equations describing\nequilibrium are obtained; Noether's theorem is exploited to identify the\nconstants of integration of these equations as the Casimirs of the euclidean\ngroup in three dimensions. While this system appears not to be integrable in\ngeneral, it {\\it is} in various limits of interest. Let the energy density be\ngiven as some function of the curvature and torsion, $f(\\kappa,\\tau)$. If $f$\nis a linear function of either of its arguments but otherwise arbitrary, we\nclaim that the first integral associated with rotational invariance permits the\ntorsion $\\tau$ to be expressed as the solution of an algebraic equation in\nterms of the bending curvature, $\\kappa$. The first integral associated with\ntranslational invariance can then be cast as a quadrature for $\\kappa$ or for\n$\\tau$.\n