{
  "id": "1008.0252",
  "version": "v2",
  "published": "2010-08-02T09:09:36.000Z",
  "updated": "2010-10-21T18:40:09.000Z",
  "title": "Multi-Dirac Structures and Hamilton-Pontryagin Principles for Lagrange-Dirac Field Theories",
  "authors": [
    "Joris Vankerschaver",
    "Hiroaki Yoshimura",
    "Jerrold E. Marsden"
  ],
  "comment": "50 pages, v2: correction to prop. 6.1, typographical changes",
  "categories": [
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining 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 field equations obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Furthermore, we show that any multi-Dirac structure naturally gives rise to a multi-Poisson bracket. We treat the case of field theories with nonholonomic constraints, showing that the integrability of the constraints is equivalent to the integrability of the underlying multi-Dirac structure. We finish with a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields and the electromagnetic field.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-21T18:40:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "field theory",
      "multi-dirac structure",
      "lagrange-dirac field theories",
      "hamilton-pontryagin principle",
      "standard dirac structures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.0252V"
    }
  }
}