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

Abstract

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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.