Covariant field theory on frame bundles of fibered manifolds (Q2738220)

The authors show that covariant field theory for sections of a fiber bundle \(\pi\colon E\to M\) lifts in a natural way to the bundle of vertically adapted linear frames \(L_{\pi}E\). The advantage gained by this reformulation is that it allows to utilize the \(n\)-symplectic geometry that is supported on \(L_{\pi}E\). On \(L_{\pi}E\) the canonical soldering 1-forms play the role of the contact structure of \(J^1\pi\). A lifted Lagrangian is used to construct modified soldering 1-forms, which are referred to as Cartan-Hamilton-Poincaré 1-forms. These forms on \(L_{\pi}E\) can be used to define the standard Cartan-Hamilton-Poincaré \(m\)-form on \(J^1\pi\). Then, the generalized Hamilton-Jacobi and Hamilton equations on \(L_{\pi}E\) are derived. Finally, as special cases of the generalized Hamilton-Jacobi equations, the Carathéodory-Rund and the de Donder-Weyl Hamilton-Jacobi equations are obtained.