# Non-linear dynamics of classical hydrogen atom

I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom.

Taking into account Newton equations for the electron and the proton and Maxwell equations for the electromagnetic field produced by these charges one obtains a higly non-linear set of coupled equations. In such a nonlinear system could some feedback effects between proton and electron take place so to make possible a stable dynamics (or at least a dynamics unstable on such long time scales longer that we can consider hydrogen to be stable)? In this way the system's stability already obtained through quantum mechanics could be reproduced by a full classical approach!

P.s.: Please, as I know of the great successes that quantum theory has had since its birth, try not to answer the question telling how quantum mechanics wonderfully solves the problem.

P.p.s.: I'm also aware of the fact that electron should lose energy and that this should cause it to fall on the proton in a very short time, so please try to avoid also this argument. I asked this question to understand if the oversymplifing hypothesis', which are fundamental in solving this problem (neglect proton's motion and, as a consequence, magnetic effects) and are quiet ubiquitous in physics, wouldn't mask the potential richness that could arise from mathematical complexity.

• I vaguely remember some efforts by André Julg (not really followed on by others) in considering a non-isolated H atom in some stochastic field allegedly produced by the remaining atoms in the Universe, that could provoke the desired stability and mimic the Heisenberg uncertainty relation for this case. – perplexity Oct 29 '12 at 10:38
• Also, I already answered your question completely--- there is no dimensional parameter to allow a stable orbit--- you can't make a size using electromagnetic constants, the charge of the electron, and the mass of the electron/proton. So assuming a stable orbit, you get a whole family which are rescalings of each other. – Ron Maimon Oct 30 '12 at 5:15
• derbucher, please stop creating new accounts and posting inappropriate answers with them. You should reuse your existing account and respond using comments. – David Z Nov 2 '12 at 16:38

In order to get a stable equilibrium, you need a constant with dimensions length, and using $\hbar$, you get such a constant. It is not possible to do this without introducing something like $\hbar$, just from dimensional analysis, independent of the quantum formalism.