Continuity equations, 0 electrodynamics

The Lehnert equations are consistent [10] with the continuity equation (428) of U(l) electrodynamics. Using the vacuum continuity equation in Lehnert s vacuum Coulomb law, we find... 

This interaction Lagrangian density may depend explicitly on the space-time coordinates x and the 4-velocity u via the charge-current density T. However, as far as only the equation of motion for the electrodynamic field is concerned they do not represent dynamical variables. Lorentz invariance of this interaction term is obvious, and gauge invariance of the corresponding action is a direct consequence of the continuity equation for the charge-current density f, cf. Eq. (3.162),... 

Landau, Lifshitz and Pitaevskii [14] derive from a combination of thermodynamics and the electrodynamics of continuous media an important equation ... 

As the primary objective of continuum physics is to determine the fields of density, motion, and temperature, field equations are needed. It is customary to base such field equations upon the equations of balance of mechanics, thermodynamics, and electrodynamics. These are the equations of balance of mass, momentum, moment of momentum, and energy or, in other terms, the continuity... 

We neglect electrodynamic effects, so the Poisson equation continues to hold. The kinetic equations just derived, along with the Poisson equation and the necessary boundary and initial conditions, can be solved numerically (a similar set of equations is solved in ref. [24]). 

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