Suspension and levitation in nonlinear theories

TL;DR

This paper explores stable equilibrium configurations in nonlinear potential fields across various physical systems, demonstrating conditions under which suspension is possible, thus extending classical results like Earnshaw's theorem.

Contribution

It generalizes Earnshaw's theorem to nonlinear theories, showing stable suspensions of finite bodies are feasible where linear theory forbids them.

Findings

Stable equilibria exist for extended bodies in nonlinear fields.

Suspension of point charges remains impossible in these theories.

Conditions for magnetic trapping of neutral particles are identified.

Abstract

I investigate stable equilibria of bodies in potential fields satisfying a generalized Poisson equation: divergence[m(grad phi) grad phi]= source density. This describes diverse systems such as nonlinear dielectrics, certain flow problems, magnets, and superconductors in nonlinear magnetic media; equilibria of forced soap films; and equilibria in certain nonlinear field theories such as Born-Infeld electromagnetism. Earnshaw's theorem, totally barring stable equilibria in the linear case, breaks down. While it is still impossible to suspend a test, point charge or dipole, one can suspend point bodies of finite charge, or extended test-charge bodies. I examine circumstances under which this can be done, using limits and special cases. I also consider the analogue of magnetic trapping of neutral (dipolar) particles.