Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field

Minkowski’s classical work underlying modern electrodynamics is described. Primary attention is given to the mathematical refinements that are required if the parameters $\varepsilon$ and $\mu$ depend on the properties of the dielectric fluid, i.e., the medium carrying charges in the field under study. It is shown that the motion of the medium and the accompanying evolution of the electromagnetic field are described by differential equations that are symmetric and hyperbolic in the sense of Friedrichs. This property guarantees their well-posedness. Note that this class of equations was not known in Minkowski’s time. At present, it plays an important role in the mathematical simulation of nonstationary processes and in the design of numerical algorithms. The author’s view of the mathematical foundations of Minkowski’s work is presented, which relates the latter to present-day insights into the theory of differential equations. This paper can possibly be of interest to physicists.