What methods have been used to rigorously prove that classical electrodynamics does not admit a robustly stable atom?
The conclusion is often stated and I am aware of the standard responses such as the sub-second collapse of the classical version of the Bohr atom. I am also aware of static stable configurations that are not stable under perturbations (the robust condition). This question is asking about broader techniques and techniques that are rigorous and not approximations. It is not a question about quantum mechanics and I have a moderately strong academic background in theoretical mechanics.
Answer
The argument about collapse of the atom in Rutherford's model is originally, I think, due to Niels Bohr. It is, originally, not "rigorous", if that word means a mathematical proof.
Bohr's himself formulated the argument in this way: (see N. Bohr, On the constitution of atoms and molecules, Philos. Mag. 26,1 (1913); http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf)
,,Let us now, however, take the effect of the energy radiation into account, calculated in the ordinary way from the acceleration of the electron. In this case the electron will no longer describe stationary orbits. $W$ will continuously increase, and the electron will approach the nucleus describing orbits of smaller and smaller dimensions, and with greater and greater frequency; the electron on the average gaining in kinetic energy at the same time as the whole system loses energy. This process will go on until the dimensions of the orbit are the same order of magnitude as the dimensions of the electron or those of the nucleus. A simple calculation shows that the energy radiated out during the process considered will be enormously great compared with that radiated out by ordinary molecular processes. ``
Calculating energy radiated using Larmor's formula is easy, but making this into mathematical proof of collapse is much harder. One would have to use strictly specified model of motion, for example, some specific differential equations, analyze them and find that typical evolution of initial circular orbit leads to a collapse. This is not easy to do, because the equations of motion for charged particle interacting with another charged particle, even if they are points, are very complicated to solve.
There are, however, some numerical calculations of this "two-body problem of electrodynamics":
J. L. Synge, On the electromagnetic two–body problem., Proc. Roy. Soc. A 177 118–39
Synge assumes purely retarded interactions, with no self forces. He starts with circular orbits of both electron and proton and he finds out that if both are allowed to move (the proton is not fixed), the orbit radii decrease in time, but much more slowly than the Larmor formula would suggest.
Both Bohr and Synge assumed that the fields are purely retarded just as in macroscopic physics, and they concluded the collapse. On the other hand, on the microscopic level, the fields may not be purely retarded but each particle field may contain some part of advanced field. L. Page and others showed this would suppress loss of energy by radiation from the atom:
L. Page, Advanced Potentials and their Application to Atomic Models, Phys. Rev. 24, 296 (1924)
Later people also realized that Rutherford's model ignores other EM forces acting on the electron and proton; for example those of other distant charged particles, or "background radiation". These forces are usually assumed to be very weak, but in fact they may compensate for the damping effect arising from the retarded character of mutual electron-proton forces. So, even if the Synge model leads to collapse for isolated system, in reality this collapse may be prevented due to presence of the background radiation, which there is always some.
There is some numerical evidence this may work, see
D.C. Cole, Yi Zou, Quantum mechanical ground state of hydrogen obtained from classical electrodynamics, Phys. Lett. A 317, p. 14-20 (2003) see https://doi.org/10.1016/j.physleta.2003.08.022
T.H. Boyer, Comments on Cole and Zou's Calculation of the Hydrogen Ground State in Classical Physics Found Phys Lett (2003) 16,p. 613,
see https://doi.org/10.1023/B:FOPL.0000012787.05764.4d
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