On the incompatibility of Maxwell's addition and the Lorentz force in non-relativistic electrodynamics

This article addresses whether the Lorentz force formula, as used in electrostatics and magnetostatics, is also applicable to non-relativistic electrodynamics.To answer this question, the article uses the Liénard-Wiechert potentials to calculate the general analytical solution of Maxwell's equations for arbitrarily moving, non-relativistic point charges.It is then shown that the obtained solution only produces the correct force of a direct current on a moving test charge under illogical assumptions.This problem does not arise from the Maxwell equations but is probably related to the fact that the Lorentz force is already contained in Maxwell's equations due to Maxwell's addition.The Lorentz force should therefore not be added a second time in the form of an explicit supplementary formula.Instead, it is reasonable to integrate an old hypothesis by Carl Friedrich Gauss from 1835 into Maxwell's electrodynamics.This integration turns Maxwell's electrodynamics into Weber-Maxwell electrodynamics and resolves the contradictions.