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

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.

Uses RevTeX; this article supersedes arXiv:1008.0252

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Cited by in corpus (10)

• A Variational Formulation of Nonequilibrium Thermodynamics for Discrete Open Systems with Mass and Heat Transfer
• Tensor products of Dirac structures and interconnection in Lagrangian mechanical systems
• On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras
• Multisymplectic Hamiltonian Variational Integrators
• Routh reduction and Cartan mechanics
• Geometric Methods for Adjoint Systems
• Poisson-Poincaré reduction for Field Theories
• Graded Poisson and Graded Dirac structures
• Coisotropic reduction in Multisymplectic Geometry
• Dirac structures in nonequilibrium thermodynamics for simple open systems