A Hamiltonian model for the macroscopic Maxwell equations using exterior calculus

William Barham, Philip J. Morrison, Eric Sonnendrücker

Published 2021-08-17Version 1

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 variables 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. In addition to the three-dimensional macroscopic Maxwell equations, a one-dimensional variant, and several example polarizations are considered.