A variational study on a natural Hamiltonian for curves

Abstract

A variational study of finding critical points of the total squared torsion functional for curves in Euclidean 3-spaces is presented. Critical points of this functional also known as one of the natural Hamiltonians of curves are characterized by two Euler-Lagrange equations in terms of curvature and torsion of a curve. To solve these balance equations, the curvature of the critical curve is expressed by its torsion so that equations are completely solved by quadratures. Then two Killing fields along the critical curve are found for integrating the structural equations of the critical curve and this curve is expressed by quadratures in a system of cylindrical coordinate. Finally, the problem is generalized to finding extremals of total squared torsion functional for nonnull curves in Minkowski 3-space.