  I investigate the properties of forces on bodies in theories governed by the
generalized Poisson equation div[mu(abs(grad_phi))grad_phi]=G rho, for the
potential phi produced by a distribution of sources rho. This equation
describes, inter alia, media with a response coefficient, mu, that depends on
the field strength, such as in nonlinear, dielectric, or diamagnetic, media;
nonlinear transport problems with field-strength dependent conductivity or
diffusion coefficient; nonlinear electrostatics, as in the Born-Infeld theory;
certain stationary potential flows in compressible fluids, in which case the
forces act on sources or obstacles in the flow. The expressions for the force
on a point charge is derived exactly for the limits of very low and very high
charge. The force on an arbitrary body in an external field of asymptotically
constant gradient, E, is shown to be F=QE, where Q is the total effective
charge of the body. The corollary Q=0 implies F=0 is a generalization of
d'Aembert's paradox. I show that for G>0 (as in Newtonian gravity) two point
charges of the same (opposite) sign still attract (repel) each other. The
opposite is true for G<0. I discuss the generalization of this to extended
bodies, and derive virial relations.
