Classically, a hydrogen atom should not be stable, since it should radiate away all its energy. I remember hearing from my favorite freshman physics prof ca. 1983 about a general theorem to the effect that all classical systems of charged particles were unstable, and I went so far as to contact him in 2009 and ask him if he remembered anything about it, but he didn't. There is something called Earnshaw's theorem that says that static equilibrium is impossible.
It certainly seems plausible that something similar holds for dynamical equilibrium, in some sense that I don't know how to define properly, much less prove rigorously. Intuitively, I would expect that any classical system of $m$ charged particles should end up having no "interesting structure," in the sense that the final state will consist of radiation plus $n \le m$ zero-size clusters whose charges $q_1,\ldots q_n$ all have the same sign (some possibly being zero); asymptotically, they all end up diverging radially from some central point.
Does anyone know of any formal proof along these lines?
Are there counterexamples, although possibly ones with initial conditions corresponding to zero volume in phase space?
How would one go about stating the initial conditions on the radiation fields?