Evolutionary equations and constraints: Maxwell equations

TL;DR

This paper presents a method to derive a covariant formulation of Maxwell's equations from constraint equations by fixing a reference frame and then covariantizing, bridging the gap between non-covariant and covariant electrodynamics.

Contribution

It introduces a covariantization procedure that reconstructs Maxwell's equations from constraints in specific reference frames, offering a new perspective on classical electrodynamics.

Findings

Covariant formulation of Maxwell's equations derived from constraints.

Reconstruction of full electrodynamics from Gauss' law and magnetic flux conservation.

Framework applicable in inertial reference frames on Minkowski spacetime.

Abstract

By fixing a reference frame in spacetime, it is possible to split the Euler-Lagrange equations associated with a degenerate Lagrangian into purely evolutionary equations and constraints on the allowed Cauchy data with respect to the notion of Space and Time associated with the given reference frame. In the context of Classical Electrodynamics, we introduce a "covariantization procedure" that allows to invert the perspective and to recover a full-fledged covariant formulation of Maxwell's equations starting only with constraint equations (i.e., Gauss' law and the local version of the law of conservation of magnetic flux) as perceived in a sufficient number of (inertial) reference frames on Minkowski spacetime.