Geometry of multisymplectic Hamiltonian first-order field theories

TL;DR

This paper compares different multisymplectic Hamiltonian formalisms in first-order field theories, establishing their equivalence and clarifying the role of geometric structures and connections in the covariant Hamiltonian approach.

Contribution

It provides a detailed comparison and unification of various Hamiltonian formalisms in multisymplectic field theories, including the role of connections and the derivation of field equations.

Findings

Different Hamiltonian formalisms are shown to be equivalent.

The geometric structures for Hamiltonian formalism are characterized and compared.

The role of connections in Hamiltonian field theories is clarified.

Abstract

In the jet bundle description of Field Theories (multisymplectic models, in particular), there are several choices for the multimomentum bundle where the covariant Hamiltonian formalism takes place. As a consequence, several proposals for this formalism can be stated, and, on each one of them, the differentiable structures needed for setting the formalism are obtained in different ways. In this work we make an accurate study of some of these Hamiltonian formalisms, showing their equivalence. In particular, the geometrical structures (canonical or not) needed for the Hamiltonian formalism, are introduced and compared, and the derivation of Hamiltonian field equations from the corresponding variational principle is shown in detail. Furthermore, the Hamiltonian formalism of systems described by Lagrangians is performed, both for the hyper-regular and almost-regular cases. Finally, the role of connections in the construction of Hamiltonian Field theories is clarified.