Earnshaw's Theorem Generalised

[Eynstone]

Consider a system of N (>1) point charges inside a box with a totally reflective inner wall.Assume that there are no interactions other than electromagnetic interactions.
Earnshaw's theorem implies that the system can't stay in equilibrium. Can the system have a periodic solution for an ascribed initial configuration? If not, what's the asymptotic behaviour of the system?
Even the case when N=2 is not so easy . I guess that if the charges were like, the particles would stay near the opposite corners .If the charges were opposite, the particles
would approach . However, the system could be ergodic / chaotic in either cases.
The static version of Earnshaw's theorem rests on that potential has no extrema in free space. The difficulty is that this doesn't hold for solutions of hyperbolic equations (the
4-potential being of this type).

 

[Valerie Park]

Thus, the asymptotic behaviour of the system is not determined by Earnshaw's theorem. However, it is possible that the system could have periodic solutions for an ascribed initial configuration, depending on the details of the initial configuration.