The Momentum-Energy Law in the Electrodynamics of Gustav Mie

Whereas the electron theory developed by H. A. Lorentz requires certain hypotheses about the structure of the electron (e.g. the hypothesis regarding the rigidity in the usual sense, or in the context of the theory of relativity), Gustav Mie¹ set himself the task of trying to modify Maxwell's equations in such a way that the existence of electrons (“nodes” of the field) and, even more generally, the existence of material atoms and molecules follows necessarily from the new equations. The fact that without the addition of new forces, stable accumulations of charge, as represented by electrons, are incompatible with the usual differential equations of the magnetic field is closely linked to the linearity of these equations. Therefore, it was first of all necessary to relinquish linearity. Mie carried out this idea in the most general and elegant manner which can be imagined in the framework of today's physics borne from Lagrange's analytical mechanics. To illustrate the type of generalization of the fundamental equations, it is perhaps | best to start with the equation of motion of a system of [24] masses with one degree of freedom q.