Symmetries and order reduction for elasticae in surfaces of constant curvature.

Abstract

Parameter invariance of the variational problem associated to the squared curvature Lagrangian, whose extremals are the elasticae, allows us to find an equivalent,nonparametric variational problem which is regular. Hamiltonian formalism is then applied to the new Lagrangian, and the order of the resulting Hamilton equations is reduced by using the invariance under isometries of the problem. For a constant curvature surface the number of isometries allows us to perform this reduction to obtain a planar nonlinear ordinary differential equation of the first order.