Infinite-dimensional Lagrange-Dirac systems with boundary energy flow II: Field theories with bundle-valued forms

TL;DR

This paper extends the infinite-dimensional Lagrange-Dirac framework to include vector-bundle-valued forms and non-Abelian gauge theories, effectively modeling boundary energy flow in complex field systems with applications to gauge-matter interactions.

Contribution

It introduces a novel extension of the Lagrange-Dirac theory to non-Abelian gauge fields and bundle-valued forms, incorporating boundary energy exchange in a unified geometric and variational setting.

Findings

Extended the theory to vector-bundle-valued forms and gauge theories.

Derived boundary conditions for gauge-matter interactions.

Applied the framework to Yang-Mills-Higgs equations.

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.