Invariantised Euler-Lagrange equations and conserved quantities for nonconservative Herglotz variational problems

Abstract

In this paper the structures of the generalised Euler-Lagrange equations and their associated conserved quantities are derived for one-dimensional Herglotz variational problems of order $n$. Their derivations use the framework of moving frames and invariant calculus of variations. The knowledge of these structures not only offers a geometric insight, it may provide a more efficient path for the determination of extremals. This is exemplified with a Herglotz problem invariant under the restricted Lorentz group $SO^+(1,2)$.