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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 (there's a Wikipedia article) 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?

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This paper by Lieb might contain some interesting references. – Raskolnikov Aug 6 '11 at 9:28
Thanks, Raskolnikov, but the paper is paywalled for me. Can you tell me the journal reference? – Ben Crowell Aug 6 '11 at 16:45
@Ben Crowell The paper can be found here. – mmc Aug 7 '11 at 0:09
@mmc: Thanks! It's completely quantum-mechanical, though. – Ben Crowell Aug 7 '11 at 3:10
Tangentially related: You maybe interested in the various non-radiation conditions for classical particles. – Willie Wong Aug 17 '11 at 1:58

I don't think there's an easy answer to your question but as for some possible leads and maybe a setup of the problem:

I'd try to write down the Hamiltonian for your configuration of particles. (e.g chapter two in this fella's thesis in section 2.3 eq. 2.28 shows the Hamiltonian for two charged classical particles using center of mass coordinates. You have n charged particles but the generalization to n is pretty straight-forward and contained in most undergrad books on mechanics.)

Then from there you want to proceed in your analysis as you would with the classical N-body problem as outlined in something like Meyer and Hall Chapter 1. Specifically look at things like Ch. 1, Sec. 4 on 'Equilibria for the Restricted 3-body problem' and techniques for finding critical points of modified potentials.

I'm willing to bet that for certain initial conditions (perhaps as an initial condition all particles are contained in the same plane) you'll be able to make a statement about the final configuration in a limiting case. For the more general setup -- that's just GOTTA be an open problem. I'd be really surprised if that was actually known at this point in time.

Regarding the question "How would one go about stating the initial conditions on the radiation fields?".. I'm not sure how to interpret a radiation field in a classical unless you push off into derivations that you find in plasma physics. For a Hamiltonian setup tho this paper looks promising.. Truth is the more I think on it the more I think you might find an answer to your question in L&L's Physical Kinetics. Chapter IV on instabilities might have something to say on this.

Anyways.. good luck.

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Nice ideas, thanks! I don't think Adams' thesis helps, however, because he explicitly assumes no radiation (p. 10) -- which in my case is what I want to prove. Amazon doesn't let me view very much of Meyer and Hall's ch. 1, but from what I can see, it looks like they're talking about the Newtonian gravitational n-body problem, for which radiation is not an issue. – Ben Crowell Aug 16 '11 at 3:38

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