In the first sections of this article, we discuss two variations on Maxwell's
equations that have been introduced in earlier work--a class of nonlinear
Maxwell theories with well-defined Galilean limits (and correspondingly
generalized Yang-Mills equations), and a linear modification motivated by the
coupling of the electromagnetic potential with a certain nonlinear Schroedinger
equation. In the final section, revisiting an old idea of Lorentz, we write
Maxwell's equations for a theory in which the electrostatic force of repulsion
between like charges differs fundamentally in magnitude from the electrostatic
force of attraction between unlike charges. We elaborate on Lorentz'
description by means of electric and magnetic field strengths, whose governing
equations separate into two fully relativistic Maxwell systems--one describing
ordinary electromagnetism, and the other describing a universally attractive or
repulsive long-range force. If such a force cannot be ruled out {\it a priori}
by known physical principles, its magnitude should be determined or bounded
experimentally. Were it to exist, interesting possibilities go beyond Lorentz'
early conjecture of a relation to (Newtonian) gravity.Comment: 26 pages, submitted to a volume in preparation to honor Gerard Emch
v. 2: discussion revised, factors of 4\pi corrected in some equation
