Chemical Principle and PDE of Variational Electrodynamics.

We formulate the Chemical Principle condition of variational electrodynamics to select globally-bounded two-body orbits possessing vanishing far-fields. For periodic orbits further satisfying some mild conditions at breaking points we construct a synchronization function in an infinite-dimensional Hilbert space. The partial differential equation (PDE) naturally associated with the Chemical Principle is in the Sobolev space $ \mathbb{W}^{2,2}(\mathbb{R}^3)$ and takes the form of a Fredholm-Schroedinger PDE problem with spin-orbit terms. We construct a perturbation scheme about an orbit with a high-frequency standing shock controlled by Weierstrass-Erdmann collisions-at-a-distance which trap the proton at the center position while the atomic far-fields vanish by destructive interference. Last, we formulate the conditions to vanish the far-fields with the perturbative scheme and discuss some simple estimates of orbital magnitudes.