Extended Hamiltonian systems in multisymplectic field theories

TL;DR

This paper explores Hamiltonian systems in multisymplectic field theories, introducing an extended formulation, analyzing their geometric properties, and relating them to restricted systems, with applications to almost-regular theories.

Contribution

It introduces Hamiltonian systems in the extended multimomentum bundle and studies their geometric properties and relations to restricted systems in multisymplectic field theories.

Findings

Extended Hamiltonian systems are formulated in the extended multimomentum bundle.

The geometric properties of these systems are characterized.

Relations between extended and restricted Hamiltonian systems are established.

Abstract

We consider Hamiltonian systems in first-order multisymplectic field theories. We review the properties of Hamiltonian systems in the so-called restricted multimomentum bundle, including the variational principle which leads to the Hamiltonian field equations. In an analogous way to how these systems are defined in the so-called extended (symplectic) formulation of non-autonomous mechanics, we introduce Hamiltonian systems in the extended multimomentum bundle. The geometric properties of these systems are studied, the Hamiltonian equations are analyzed using integrable multivector fields, the corresponding variational principle is also stated, and the relation between the extended and the restricted Hamiltonian systems is established. All these properties are also adapted to certain kinds of submanifolds of the multimomentum bundles in order to cover the case of almost-regular field theories.