Geometry of Hamiltonean n-vectors in Multisymplectic Field Theory

TL;DR

This paper explores the algebraic structure of Hamiltonian functions in multisymplectic geometry, generalizing classical Hamiltonian vector fields to better understand solutions of classical field theories.

Contribution

It clarifies the algebraic correspondence between Hamiltonian functions and antisymmetric tensor products of vector fields in multisymplectic geometry.

Findings

Hamiltonian functions correspond to antisymmetric tensor products of vector fields.

Generalization of Hamiltonian vector fields to multisymplectic field theory.

Enhanced algebraic understanding of solutions to classical field equations.

Abstract

Multisymplectic geometry - which originates from the well known de Donder-Weyl theory - is a natural framework for the study of classical field theories. Recently, two algebraic structures have been put forward to encode a given theory algebraically. Those structures are formulated on finite dimensional spaces, which seems to be surprising at first. In this article, we investigate the correspondence of Hamiltonian functions and certain antisymmetric tensor products of vector fields. The latter turn out to be the proper generalisation of the Hamiltonian vector fields of classical mechanics. Thus we clarify the algebraic description of solutions of the field equations.