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arXiv · 1207.2814

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

Joris Vankerschaver·Hiroaki Yoshimura·Melvin Leok
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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.

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Joris Vankerschaver, Hiroaki Yoshimura, Melvin Leok. 2013-06-18. The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories. https://doi.org/10.1063/1.4731481

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