A variational formulation of electrodynamics with external sources

TL;DR

This paper develops a comprehensive variational framework for electrodynamics using differential forms, enabling unified treatment of boundary problems and derivation of constitutive relations with external sources.

Contribution

It introduces a more complete variational principle for electrodynamics with differential forms, allowing for a unified approach to boundary problems and external sources.

Findings

Derivation of field equations with external sources

Unified treatment of boundary problems

Smooth transition to infinitesimal versions

Abstract

We present a variational formulation of electrodynamics using de Rham even and odd differential forms. Our formulation relies on a variational principle more complete than the Hamilton principle and thus leads to field equations with external sources and permits the derivation of the constitutive relations. We interpret a domain in space-time as an odd de Rham 4-current. This permits a treatment of different types of boundary problems in an unified way. In particular we obtain a smooth transition to the infinitesimal version by using a current with a one point support.