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 HamiltonPontryagin principle, and we show that the resulting field equations are the EulerLagrange equations in implicit form. Second, 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 EulerLagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Finally, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics. C 2012 American Institute of Physics.

Clebsch Variational Principles in Field Theories and Singular Solutions of Covariant Epdiff Equations

“…Since it is locally given by E(x µ , y A , p µ A , v A µ ) = π + p µ A v A µ − L(x µ , y A , v A µ ) d n+1 x, it verifies the condition (5.2). In this case, (5.3) recovers the covariant Hamilton-Pontryagin variational principle developed in [40]. Since e loc = p µ A v A µ −L(x µ , y A , v A µ ), the stationarity conditions (5.4) read Recovering classical Pontryagin principles in optimal control and mechanics.…”

“…In this section we develop a general covariant variational principle that generalizes the covariant Clebsch variational principle and reproduces, as general case, the classical and covariant Hamilton-Pontryagin variational principles of [39] and [40], respectively, together with variational principles arising in geometric optimal control problems, e.g. [29].…”

Tensor products of Dirac structures and interconnection in Lagrangian mechanics

“…• Interconnection of multi-Dirac structures and Lagrange-Dirac field systems: In conjunction with classical field theories or infinite dimensional dynamical systems, the notion of multi-Dirac structures have been developed by Vankerschaver, Yoshimura, and Leok [2012], which may be useful for the analysis of fluids, continuums as well as electromagnetic fields. The present work of the interconnection of Dirac structures and the associated Lagrange-Dirac systems may be extended to the case of classical fields or infinite dimensional dynamical systems.…”

On the geometry of multi-Dirac structures and Gerstenhaber algebras

“…In this section, we show that the graph (in some suitable sense) of a differential form determines a multi-Dirac structure. This multi-Dirac structure was already defined in Vankerschaver, Yoshimura, Leok, and Marsden [2011]. We refer to that paper for an overview of applications in classical field theory, and for proofs of the basic theorems.…”

“…In the current paper, which should be seen as a companion paper to Vankerschaver, Yoshimura, Leok, and M [2011], we describe the mathematical properties of multi-Dirac structures in somewhat greater detail. We recall the definition of a multi-Dirac structure D and show that there exists a graded multiplication and a graded bracket (referred to as the multi-Courant bracket) on the space of sections of D and that the latter is endowed with the structure of a Gerstenhaber algebra with respect to these two operations.…”