The Electrodynamics of Material Media

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 $\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{D}=\ensuremath{\rho}, \ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\mathrm{H}\ensuremath{-}{\mathrm{D}}^{\ensuremath{'}}=\mathrm{J}$ as conditions auxiliary to the variation principle $\ensuremath{\delta}\ensuremath{\int}\ensuremath{\int}{L+\ensuremath{\Sigma}\stackrel{}{n}{\ensuremath{\theta}}_{n}[{{N}_{n}}^{\ensuremath{'}}+\ensuremath{\nabla}\ifmmode\cdot\else\textperiodcentered\fi{}({N}_{n}{\mathrm{V}}_{n})]dvdt}=0.$ The Lagrangian function, $L$, depends on D, H, ${N}_{n}$, ${\mathrm{V}}_{n}$, ${\ensuremath{\theta}}_{n}$ 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 ${\ensuremath{\theta}}_{n}$ 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.