Connection between the adiabatic hypothesis of old quantum theory and classical electrodynamics with classical electromagnetic zero-point radiation

The adiabatic hypothesis of old quantum theory which formed the basis for Bohr-Sommerfeld quantization assumed that the adiabatic invariants $J_l=\oint \int p_{l}d{q}_l$ of multiply periodic mechanical systems should be quantized with the discrete values $J_l=nh$ or sometimes $J_l=(n+\frac{1}{2})h$, where $h$ is Planck's constant. The hypothesis did not follow from any more general assumption. In the present paper we derive some related results within the theory of classical electrodynamics with classical electromagnetic zero-point radiation. We derive the fact that the adiabatic invariants of a charged nonrelativistic periodic mechanical system with no harmonics remain adiabatic invariants when placed in the classical electromagnetic zero-point radiation spectrum and only in this spectrum. Furthermore the phase-space distribution for the three-dimensional mechanical system in zero-point radiation becomes $P(J_1, J_2, J_3)=\mathrm{const}\times \exp [-2\mathrm{}\frac{(J_1+J_2+J_3)}{h}]$ leading to the average value $〈J_l〉=\frac{1}{2h}$ for each of the adiabatic invariants. Here $h$, which is naturally chosen as Planck's constant, is just the scale factor appearing in the classical zero-point radiation. For each of the systems in equilibrium with zero-point radiation already treated in the literature, we sketch the connection with the unifying formalism of action-angle variables.