On Scalar and Vector Potentials for the Nonlinear Electromagnetic Forces

TL;DR

This paper investigates the applicability of potential functions to nonlinear electromagnetic forces, revealing limitations of Helmholtz's theorem and coordinate-dependent issues in alternative methods, impacting the understanding of electromagnetic potentials.

Contribution

It demonstrates the failure of Helmholtz's theorem for nonlinear electromagnetic forces and shows that existing methods produce coordinate-dependent potentials, questioning their physical invariance.

Findings

Helmholtz's theorem often fails for nonlinear electromagnetic forces.

Alternative potential methods depend on coordinate choices.

Invariance of potentials in nonlinear electromagnetism is problematic.

Abstract

The potential concept that is successful in classical electrodynamics should also be applicable to the nonlinear electromagnetic forces acting on matter. The obvious method of determining these potentials should be provided by Helmholtz's theorem. It is found, however, that the theorem fails in most practical instances. Other methods to find the potentials - as pursued in plasma physics - are examined and found to yield functions which depend on the chosen coordinate system. Thus they cannot be considered as invariant potentials from which physical forces may be derived. Practical consequences of these mathematical findings are discussed.