Lepage forms in Finsler geometry: Second-order generalization

Abstract

Projectability of Lepage forms, defined on higher-order jet spaces, onto the corresponding Grassmann fibrations, is a basic requirement for the extension of the theory of Lepage forms to integral variational functionals for submanifolds. In this paper, projectability of second-order Lepage forms is considered for variational functionals of the Finsler type. Underlying geometric concepts such as regular 2-velocities, contact elements and second-order Grassmann fibrations of rank 1 are discussed. It is shown that a Lepage form is projectable if and only if its Hamiltonian vanishes identically. In this case, explicit formulas for the Lagrange functions and the projected Lepage forms are given in terms of the adapted coordinates.