The Electrodynamics of Material Media

Abstract

The general equations of a gauge invariant, classical theory of the electrodynamics of material media are obtained. The gauge invariance is insured by taking the equations $\nabla ·\mathrm{D}=\rho , \nabla \times \mathrm{H}-{\mathrm{D}}^{\prime }=\mathrm{J}$ as conditions auxiliary to the variation principle $\delta \int \int \{L+\Sigma \stackrel{}{n}{\theta }_n[{N_n}^{\prime }+\nabla ·(N_n{\mathrm{V}}_n\left)\right]dvdt\}=0.\mathrm{}$ The Lagrangian function, $L$, depends on D, H, $N_n$, ${\mathrm{V}}_n$, ${\theta }_n$ and possibly their derivatives; here $N_n$ is the numerical density of atoms in the state $n$, ${\mathrm{V}}_n$ their macroscopic or average velocity, and ${\theta }_n$ is a variable that functions as the velocity potential in some cases and has the dimensions of action. The electromagnetic potentials enter the theory as Lagrangian multipliers only.