The 3D-Kepler 4-Body Problem of Classical Electrodynamics in Spherical Coordinates

This paper is a direct continuation of previous papers where the 4-body problem of classical electrodynamics is derived and solved in an internal frame of reference with rectangular coordinates. The equations of motion are 16 in number in the Minkowski space, but one can prove that only 12 of them are independent ones – as many as the unknown velocities (or trajectories). Here we consider the 3D-Kepler formulation in spherical coordinates setting the first particle (the nuclei) at the origin. Then we consider equations describing the motion of the last three particles orbiting the nucleus. We obtain 9 equations for three moving particles. The Kepler formulation leads to two groups of equations. The first one contains unknown functions on the initial interval. That is why we call them Initial equations. Their solutions become initial functions for the second group of equations (Basic equations). We look for periodic solutions of these equations on the interval to the right of the initial point. These equations are of neutral type and require prescribing of initial functions. We take initial functions to be the solutions of the Initial equations. However, here second derivatives appear due to the presence of the radiation terms. Therefore, we choose a space of infinitely differentiable periodic functions and operators whose fixed point is a periodic solution of the 3D Kepler 4-body problem. The method allows us to obtain estimates of the distances between moving particles of the Li-atom. Besides the Kepler formulation allows us to describe transitions from one stationary state to another.