Problema relativístico de dois corpos - Hollander, Efrain Buksman (2003)
We study the relativistic two-body problem of the action-at-a-distance electrodynamics. This dynamical system appeared 100 years ago as a time-symmetric relativistic motion and acquired the status of electrodynamics in the 1940 s by the works of Dirac, Wheeler and Feynman. The equations of motion for this problem are delay equations involving retarded and advanced arguments symmetrically. We outline our dynamical studies with an emphasis on the physics of this complex conservative dynamical system. We study the following versions of this electromagnetic two-body problem: (i) For the case of two arbitrary masses with attractive interaction (hydrogen atom), we develop a numerical method to integrate the three-dimensional motion. This method has a very limited applicability and could not answer several dynamical questions. We calculated numerically some orbits. The diﬃculties of this complex case suggested that we should restrict the study to the simpler problem of straight-line orbits and equal masses ( (ii) and (iii) ). (ii) We study the colinear orbits of the repulsive problem of two electrons (two electrons moving on the same line). We obtain an analytical approximation for the low-energy colinear orbits. We also develop a stable numerical method based on steepest-descent minimization. Using this method we calculated the orbits numerically for several energies. We also found a two-degree-of-freedom implicit Hamiltonian formalism to describe this colinear motion. (iii) For the attractive problem with equal masses, we derive an equation of motion that is regular at the collision. Our method uses the energy constant related to the Poincaré invariance of the theory to motivate the regularizing coordinate transformation and to remove infinities from the equation of motion. The collision orbits are calculated numerically using the regular equation adapted in a self-consistent minimization method (a stable numerical method that chooses only nonrunaway orbits). We compare our regularization of this Poincaré-invariant case to the Levi-Civita regularization of the Galilei-invariant Kepler problem.