Electromagnetic two-body problem and nonlinear dynamics of oscillator chains

Abstract

In this project we will study the dynamics of the electromagnetic two-body problem as the leading topic, see (1) and (2) below. We shall also study the Hamiltonian dynamics of one-dimensional oscillator chains, see (3) below, and some optimization in quasi-electrostatics, ítem (4). In the year of 2005 I have already published 4 papers on the above topics (see currículum Lattes). (1) A satisfactory description of the dynamics of a point charge in Maxwell's electrodynamics appeared only in 1938. This equation of motion for the limit case of a charged particle of zero radius was given by Dirac in 1938, see reference in the project. These equations of motion are the differential expression of the laws of conservation of energy and momentum. The electromagnetic two-body problem in Dirac's theory has a nontrivial dynamics because of the self-interaction and the delay in the equations of motion. Because of the delay, the dynamics in the neighborhood of circular orbits displays a motion in a fast timescale superposed to the slow circular motion. We have started to investigate the consequences of balancing this fast dynamics in the neighborhood of circular orbits. This balancing of the fast dynamics predicts results in the atomic scale and in qualitative and quantitative agreement with the results of quantum electrodynamics (QED); The stable orbits are parametrized by an integer and the angular momenta are approximate multiples of a fundamental angular momentum of the order of Planck's constant. In this project I am going to study the complete stability of circular orbits and develop a detailed multiscale method that starts from the fast dynamics... (AU)