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

Mauro Francaviglia·Marcella Palese·Ekkehart Winterroth
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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.

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Mauro Francaviglia, Marcella Palese, Ekkehart Winterroth. 2004-05-28. A New Geometric Proposal for the Hamiltonian Description of Classical Field Theories. https://arxiv.org/abs/math-ph/0311018

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