Covariant Hamiltonian formalism for the calculus of variations with several variables: Lepage--Dedecker versus de Donder--Weyl

TL;DR

This paper develops a covariant Hamiltonian formalism for field theories that aligns with relativity, comparing Lepage-Dedecker and de Donder-Weyl approaches, emphasizing the Legendre correspondence's role in the geometric framework.

Contribution

It provides a detailed geometric comparison between Lepage-Dedecker and de Donder-Weyl formalisms, highlighting the Legendre correspondence's significance in covariant Hamiltonian theory.

Findings

Lepage-Dedecker formalism generalizes the Legendre transform to a correspondence.

The approach clarifies the geometric structure of covariant Hamiltonian field theories.

The formalism accommodates degenerate Lagrangians through the Legendre correspondence.

Abstract

The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) de Donder-Weyl (dDW) one. One of the main points is the fact that the Legendre transform in the dDW approach is replaced by a Legendre correspondence in the LP theory (This correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).