The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories

TL;DR

This paper develops a variational framework called the Hamilton-Pontryagin principle for classical field theories, introduces multi-Dirac structures as a geometric tool, and demonstrates their application through various physical examples.

Contribution

It introduces the Hamilton-Pontryagin principle for field theories and the concept of multi-Dirac structures, linking variational principles with geometric structures in a novel way.

Findings

Implicit Euler-Lagrange equations can be described using multi-Dirac structures.

Multi-Dirac structures are determined by multisymplectic forms.

The framework applies to diverse physical systems like electromagnetism and elastostatics.

Abstract

We introduce a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit Euler-Lagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Lastly, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics.