Georgia Institute of Technology College of Sciences

We develop a method to construct solutions of some (retarded or advanced) equations. A prime example could be the motion of point charges interacting via the fully relativistic Lienard-Wiechert potentials (as suggested by J.A. Wheeler and R.P. Feynman in the 1940's). These are retarded equations, but the delay depends implicitly on the trajectory. We assume that the equations considered are formally close to an ODE and that the ODE admits hyperbolic solutions (that is, perturbations transversal to trajectory grow exponentially either in the past or in the future) and we show that there are solutions of the functional equation close to the hyperbolic solutions of the ODE. The method of proof does not require to formulate the delayed problem as an evolution for a class of initial data. The main result is formulated in an "a-posteriori" format and allows to show that solutions obtained by non-rigorous approximations are close (in some precise sense) to true solutions. In the electrodynamics (or gravitational) case, this allows to compare the hyperbolic solutions of several post-newtonian approximations or numerical approximations with the solutions of the Lienard-Weichert interaction. This is a joint work with R. de la Llave and J. Yang.