Covariant field theory on frame bundles of fibered manifolds

We show that covariant field theory for sections of π : E→M lifts in a natural way to the bundle of vertically adapted linear frames LπE. Our analysis is based on the fact that LπE is a principal fiber bundle over the bundle of 1-jets J1π. On LπE the canonical soldering 1-forms play the role of the contact structure of J1π. A lifted Lagrangian ℒ: LπE→R is used to construct modified soldering 1-forms, which we refer to as the Cartan–Hamilton–Poincaré 1-forms. These 1-forms on LπE pass to the quotient to define the standard Cartan–Hamilton–Poincaré m-form on J1π. We derive generalized Hamilton–Jacobi and Hamilton equations on LπE, and show that the Hamilton–Jacobi and canonical equations of Carathéodory–Rund and de Donder–Weyl are obtained as special cases.