Notes on equivalent formulations of Hamiltonian dynamics on multicotangent bundles

TL;DR

This paper demonstrates the equivalence of five different conditions characterizing Hamiltonian solutions in multicotangent bundles, including the Hamilton-Volterra equations and the Hamilton-de Donder-Weyl equation, extending to multisymplectic manifolds.

Contribution

It establishes the equivalence of multiple formulations of Hamiltonian dynamics on multicotangent bundles, generalizing to multisymplectic manifolds for various source dimensions.

Findings

Equivalence of five conditions for Hamiltonian solutions.

Extension of Hamilton-de Donder-Weyl equation to multisymplectic manifolds.

Framework unifies different approaches to Hamiltonian field theories.

Abstract

We show the equivalence of five different conditions on a classical field $\psi$ with values in a restricted multicotangent bundle to be a solution of the field equations, notably in terms of the Hamilton-Volterra equations, the principle of least action and several conditions based on the contraction of the multi-vector tangent to $\psi$ with canonical differential forms. Most prominently, we have equivalence to the "dynamical Hamilton-de Donder-Weyl equation", that can be vastly generalized to define Hamiltonian dynamics on multisymplectic manifolds, defined for sources of different dimensions.