Dirac constraints in field theory and exterior differential systems

TL;DR

This paper develops a new formulation for classical field theories that avoids primary constraints by using non-standard variational problems and exterior differential systems, with applications to electromagnetism and Poisson sigma models.

Contribution

It introduces a novel multisymplectic framework for non-standard variational problems, providing a new characterization of constraint manifolds in field theories.

Findings

The new formulation avoids primary constraints in electromagnetism.

A multisymplectic structure for non-standard variational problems is established.

The approach relates constraints to exterior differential systems.

Abstract

The usual treatment of a (first order) classical field theory such as electromagnetism has a little drawback: It has a primary constraint submanifold that arise from the fact that the dynamics is governed by the antisymmetric part of the jet variables. So it is natural to ask if there exists a formulation of this kind of field theories which avoids this problem, retaining the versatility of the known approach. The following paper deals with a family of variational problems, namely, the so called non standard variational problems, which intends to capture the data necessary to set up such a formulation for field theories; moreover, we will formulate a multisymplectic structure for the family of non standard variational problems, and we will relate this with the (pre)symplectic structure arising on the space of sections of the bundle of fields. In this setting the Dirac theory of constraints will be studied, obtaining among other things a novel characterization of the constraint manifold which arises in this theory, as generators of an exterior differential system associated to the equations of motion and the chosen slicing. Several examples of application of this formalism are discussed: Two of them motivated from the physical point of view, that is, electromagnetism and Poisson sigma models, and two examples of mathematical application. In the case of electromagnetism, it is shown that this formulation avoids the problems arising in the usual approach.