2010/08/02 by Joris Vankerschaver, Vankerschaver, Joris, Hiroaki Yoshimura +3 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Control and Dynamics of Mobile Robots #Control and Stability of Dynamical Systems #Differential Geometry (math.DG) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1008.0252
50 pages, v2: correction to prop. 6.1, typographical changes
openalex publication_date 2010/08/02 · arxiv created 2010/10/21 · arxiv updated 2010/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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.