Poisson bracket in classical field theory as a derived bracket

TL;DR

This paper introduces a Leibniz bracket on differential forms over jet bundles and demonstrates its application by formulating Maxwell's equations with sources in a covariant Hamiltonian framework.

Contribution

It constructs a Leibniz bracket on jet bundle forms and applies it to express Maxwell's equations in a covariant Hamiltonian form.

Findings

Leibniz bracket defined on differential forms over jet bundles

Maxwell equations expressed in covariant finite-dimensional Hamiltonian form

Provides a new geometric framework for classical field theories

Abstract

We construct a Leibniz bracket on the space $\Omega^\bullet (J^k (\pi))$ of all differential forms over the finite-dimensional jet bundle $J^k (\pi)$. As an example, we write Maxwell equations with sources in the covariant finite-dimensional hamiltonian form.