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 DonderWeyl are obtained as special cases. The manifold LπE emerges as a natural arena for a unified theory that contains, in addition to the sector for sections of π, dynamical sectors for a geometry for M and a geometry for the fibers of E.