Legendre transformation for regularizable Lagrangians in field theory

TL;DR

This paper explores a generalized Legendre transformation for singular Lagrangians in field theory, establishing Hamilton equations equivalent to Euler-Lagrange equations, and discusses applications to Dirac and electromagnetic fields.

Contribution

It introduces a Legendre transformation for regularizable Lagrangians using Lepagean equivalents, extending Hamiltonian formalism to certain singular Lagrangians in field theory.

Findings

All affine or quadratic Lagrangians in first derivatives are regularizable.

Derived Hamilton equations are equivalent to Euler-Lagrange equations.

Applied framework to Dirac and electromagnetic fields.

Abstract

Hamilton equations based not only upon the Poincare--Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton--De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Euler--Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.