Evolution equation for N relativistic charged particles

The evolution equation, correct to second order in the charge, for N relativistic point charges in zero external field is derived. This is done by writing down the Liouville equation for the system of charges and fields, assuming appropriate initial conditions, and integrating the Liouville equation over the field variables. The initial conditions imply that the field has initially no statistical dispersion, and is the field of N stationary point charges. Renormalization is carried out, and the k role=presentation style=font-size: 90%; display: inline-block; position: relative;>$k$$k$ integrals are performed. The evolution equation exhibits an explicit time dependence. It reaches its asymptotic form in the time necessary for light signals to connect the mean positions of all the charges. The results are checked by deriving the asymptotic evolution, to order (v/c)², by a different method, related to the Lorentz gauge analogue of the Coulomb gauge Darwin Hamiltonian.