Chemical Principle and PDE of Variational Electrodynamics

TL;DR

This paper develops a PDE framework for variational electrodynamics, introducing a Chemical Principle criterion to select physically relevant orbits and estimating quantum-like parameters such as the Bohr radius.

Contribution

It formulates a PDE-based approach to variational electrodynamics, incorporating a Chemical Principle criterion for orbit selection and connecting classical and quantum concepts.

Findings

Defined a synchronization function for periodic orbits.

Constructed oscillatory functions with continuous derivatives at breaking points.

Estimated the Bohr radius parameter from boundary-layer analysis.

Abstract

The two-body problem of variational electrodynamics possesses differential-delay equations of motion with state-dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets. From a periodic orbit possessing some mild properties at breaking points, we define a synchronization function in R x R3, which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points. The oscillatory functions are associated with a PDE identity in H2(R3), and we postulate ordering conditions for the PDE identity to define a Fredholm-Schroedinger operator with an O(1/r2) spin-orbit forcing term belonging to L2(R3). As an application, we introduce the Chemical Principle criterion to select orbits with asymptotically vanishing far-fields and estimate the Bohr radius parameter of the Fredholm-Schroedinger PDE using the boundary-layer of orbits chosen according to the Chemical Principle criterion. Last, working backward, we derive a Chemical Principle-like condition from the ordering conditions.