Multivector Field Formulation of Hamiltonian Field Theories: Equations and Symmetries

TL;DR

This paper reformulates Hamiltonian field theories using multivector fields, jet fields, and connections, exploring solutions, symmetries, and the relation to Lagrangian formalism in a geometric framework.

Contribution

It introduces a unified geometric formulation of Hamiltonian field theories and analyzes solutions, symmetries, and the equivalence with Lagrangian formalism.

Findings

Equivalent geometric formulations of Hamiltonian equations

Analysis of existence and uniqueness of solutions

Extension of Noether's theorem to higher-order symmetries

Abstract

We state the intrinsic form of the Hamiltonian equations of first-order Classical Field theories in three equivalent geometrical ways: using multivector fields, jet fields and connections. Thus, these equations are given in a form similar to that in which the Hamiltonian equations of mechanics are usually given. Then, using multivector fields, we study several aspects of these equations, such as the existence and non-uniqueness of solutions, and the integrability problem. In particular, these problems are analyzed for the case of Hamiltonian systems defined in a submanifold of the multimomentum bundle. Furthermore, the existence of first integrals of these Hamiltonian equations is considered, and the relation between {\sl Cartan-Noether symmetries} and {\sl general symmetries} of the system is discussed. Noether's theorem is also stated in this context, both the ``classical'' version and its generalization to include higher-order Cartan-Noether symmetries. Finally, the equivalence between the Lagrangian and Hamiltonian formalisms is also discussed.