Independence of Maxwell's equations: A B\"acklund-transformation view

TL;DR

This paper presents a novel perspective on Maxwell's equations by viewing them as a Bäcklund transformation, suggesting that the conservation of charge and wave equations are integrability conditions, thus exploring their independence.

Contribution

It introduces a Bäcklund-transformation framework to analyze the independence of Maxwell's equations, offering a new theoretical insight into their foundational structure.

Findings

Maxwell's equations can be viewed as a Bäcklund transformation.

Conservation of charge and wave equations are integrability conditions.

This approach questions the redundancy of Gauss's laws.

Abstract

It is now widely accepted that the Maxwell equations of Electrodynamics constitute a self-consistent set of four independent partial differential equations. According to a certain school of thought, however, half of these equations - namely, those expressing the two Gauss' laws for the electric and the magnetic field - are redundant since they can be "derived" from the remaining two laws and the principle of conservation of charge. The status of the latter principle is thus elevated to a law of Nature more fundamental than, say, Coulomb's law. In this note we examine this line of reasoning and we propose an approach according to which the Maxwell equations may be viewed as a Backlund transformation relating fields and sources. The conservation of charge and the electromagnetic wave equations then simply express the integrability conditions of this transformation.