A New Geometric Proposal for the Hamiltonian Description of Classical Field Theories

TL;DR

This paper introduces a novel geometric Hamiltonian framework for classical field theories using composite fibered bundles, multisymplectic forms, and Hamiltonian connections, providing a covariant and extended description of the Hamiltonian formalism.

Contribution

It proposes a new geometric Hamiltonian formalism based on composite fibered bundles and multisymplectic forms, extending the standard polymomentum approach for field theories.

Findings

Defines covariant Hamilton equations using multisymplectic forms.

Introduces extended Legendre bundle and Hamiltonian connection concepts.

Relates the new framework to standard polymomentum formulations.

Abstract

We consider the geometric formulation of the Hamiltonian formalism for field theory in terms of {\em Hamiltonian connections} and {\em multisymplectic forms}. In this framework the covariant Hamilton equations for Mechanics and field theory are defined in terms of multisymplectic $(n+2)$--forms, where $n$ is the dimension of the basis manifold, together with connections on the configuration bundle. We provide a new geometric Hamiltonian description of field theory, based on the introduction of a suitable {\em composite fibered bundle} which plays the role of an {\em extended configuration bundle}. Instead of fibrations over an $n$--dimensional base manifold $\bX$, we consider {\em fibrations over a line bundle $\Tht$ fibered over $\bX$}. The concepts of {\em extended Legendre bundle}, {\em Hamiltonian connection}, {\em Hamiltonian form} and {\em covariant Hamilton equations} are introduced and put in relation with the corresponding standard concepts in the polymomentum approach to field theory.