Axiomatics of classical electrodynamics and its relation to gauge field theory

TL;DR

This paper presents an axiomatic foundation of classical electrodynamics, linking it to gauge field theory, by deriving Maxwell's equations from fundamental conservation laws and constitutive relations.

Contribution

It provides a concise axiomatic derivation of Maxwell's equations and clarifies their relation to gauge field theory and fundamental conservation principles.

Findings

Maxwell's equations follow from charge and flux conservation.

The approach connects classical electrodynamics to gauge field theory.

Constitutive relations link field excitations to field strengths.

Abstract

We give a concise axiomatic introduction into the fundamental structure of classical electrodynamics: It is based on electric charge conservation, the Lorentz force, magnetic flux conservation, and the existence of local and linear constitutive relations. The {\it inhomogeneous} Maxwell equations, expressed in terms of $D^i$ and $H_i$, turn out to be a consequence of electric charge conservation, whereas the {\it homogeneous} Maxwell equations, expressed in terms of $E_i$ and $B^i$, are derived from magnetic flux conservation and special relativity theory. The excitations $D^i$ and $H_i$, by means of constitutive relations, are linked to the field strengths $E_i$ and $B^i$. Eventually, we point out how this axiomatic approach is related to the framework of gauge field theory.