Electromagnetic two-body problem and dynamics of nonlinear oscilator chains

Abstract

In this project we will study the dynamics of the electromagnetic two-body problem as the leading topic, see (1). As a secondary topic we shall study Hamiltonian dynamics of one-dimensional oscillator chains, see (2) below. (1) The equations of motion of the electromagnetic two-body problem are nontrivial because of the self-interaction, the delay, and the presence of singular denominators, as explained in the research project. Because of the delay, the dynamics in the neighborhood of circular orbits displays a motion with a fast timescale superposed to the slow circular motion. We already investigated some approximations for the balancing of the fast dynamics in the neighborhood of circular orbits in J. De Luca, Phys. Rev. E 71, 056219, (2005), e J. De Luca, Phys. Rev. E 73 026221 (2006).This balancing of the fast dynamics predicted results in the atomic scale and in qualitative and quantitative agreement with the results of quantum electrodynamics (QED). The orbits satisfying a perturbative non-radiation condition are parametrized by an integer and have angular momenta in approximate multiples of a fundamental angular momentum of the order of Planck´s constant. The main goal of this project is to develop a stable numerical integrator for the state-dependent delay equations of this two-body dynamics. To overcome the existence of denominators we shall construct a variational integrator for this problem in the action-at-a-distance electrodynamics. As explained in the project, there is a variational method to the equations of motion in this version of electrodynamics. The existence of an action integral allows for the construction of a variational integrator, which is the main topic of the project. In addition we are going to continue working with the numerical integration of the equations of motion of the other versions of the electromagnetic two-body problem using the modern integrators for delay equations (http://www.cs.kuleuven.ac.be/~twr/research/software/delay/software.shtml). (2) We shall continue our studies of the dynamics of unidimensional nonlinear oscillator chains. Unidimensional oscillator chains are a testbed for models of biological molecules and serve for several numerical experiments. As explained in the project, this is a continuation of the work initiated in Refs: --- J. De Luca, et. al, The Physical Review E 70, 026213 (2004) e A. Ponno, J. Ruggiero, E. Drigo e J. De Luca, The Physical Review E 73, 056609(2006). (AU)