A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus

TL;DR

This paper develops a Hamiltonian field theory for macroscopic Maxwell equations using exterior calculus, emphasizing differential forms and duality to ensure metric and orientation independence, with applications to various constitutive models.

Contribution

It introduces a novel Hamiltonian formulation of Maxwell's equations in differential forms, ensuring metric and orientation independence and detailed translation from vector calculus.

Findings

Hamiltonian structure for Maxwell equations established

Differential forms and duality ensure coordinate independence

Examples demonstrate applicability to different constitutive models

Abstract

A Hamiltonian field theory for the macroscopic Maxwell equations with fully general polarization and magnetization is stated in the language of differential forms. The precise procedure for translating the vector calculus formulation into differential forms is discussed in detail. We choose to distinguish between straight and twisted differential forms so that all integrals be taken over densities (i.e. twisted top forms). This ensures that the duality pairings, which are stated as integrals over densities, are orientation independent. The relationship between functional differentiation with respect to vector fields and with respect to differential forms is established using the chain rule. The theory is developed such that the Poisson bracket is metric and orientation independent with all metric dependence contained in the Hamiltonian. As is typically seen in the exterior calculus formulation of Maxwell's equations, the Hodge star operator plays a key role in modeling the constitutive relations. As a demonstration of the kind of constitutive models this theory accommodates, the paper concludes with several examples.