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

Francaviglia, Mauro; Palese, Marcella; Winterroth, Ekkehart

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

Authors

Francaviglia, Mauro

Palese, Marcella

Winterroth, Ekkehart

Description

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.
9 pages, VIII Int. Conf. Diff. Geom. Appl. (Opava 2001); O. Kowalski et al. eds., misprints corrected