Infinite-dimensional Lagrange-Dirac systems with boundary energy flow II: Field theories with bundle-valued forms
François Gay-Balmaz, Álvaro Rodríguez Abella, Hiroaki Yoshimura
TL;DR
The paper develops an infinite-dimensional Lagrange--Dirac framework for field theories that exchange energy with their surroundings through boundaries, extending Part I to bundle-valued forms and non-Abelian gauge theories. It introduces two equivalent restricted-dual realizations, $V^\star$ and $V^\dagger$, and constructs restricted Tulczyjew triples and canonical Dirac structures that incorporate boundary terms via boundary energy flow. The theory is applied to matter fields and to Yang--Mills fields, including their interactions (Yang--Mills–Higgs), deriving space–time decomposed equations, energy and charge balances, and boundary conditions in a unified geometric-variational setting. This approach yields consistent energy balance laws, boundary currents, and Poynting-type theorems, with potential applications to elasticity, field theories on bounded domains, and structure-preserving discretizations. The framework preserves the canonical symplectic/Dirac structures while accommodating boundary phenomena, enabling robust interconnections and symmetry reductions in future work.
Abstract
Part I of this paper introduced the infinite dimensional Lagrange-Dirac theory for physical systems on the space of differential forms over a smooth manifold with boundary. This approach is particularly well-suited for systems involving energy exchange through the boundary, as it is built upon a restricted dual space -- a vector subspace of the topological dual of the configuration space -- that captures information about both the interior dynamics and boundary interactions. Consequently, the resulting dynamical equations naturally incorporate boundary energy flow. In this second part, the theory is extended to encompass vector-bundle-valued differential forms and non-Abelian gauge theories. To account for two commonly used forms of energy flux and boundary power densities, we introduce two distinct but equivalent formulations of the restricted dual. The results are derived from both geometric and variational viewpoints and are illustrated through applications to matter and gauge field theories. The interaction between gauge and matter fields is also addressed, along with the associated boundary conditions, applied to the case of the Yang-Mills-Higgs equations.