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Covariant field theory on frame bundles of fibered manifolds. (English) Zbl 0995.70025
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.

70S05Lagrangian formalism and Hamiltonian formalism
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
70G45Differential-geometric methods for dynamical systems
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