We study the dynamics of N charges interacting with the Maxwell field. If t
heir initial velocities are small compared to the velocity of light, c, the
n in lowest order their motion is governed by the static Coulomb Lagrangian
. We investigate higher-order corrections with an explicit control on the e
rror terms. The Darwin correction, order \ upsilon /c \ (2), has been prove
d previously. In this contribution we obtain the dissipative corrections du
e to radiation damping, which are of order \ upsilon /c \ (3) relative to t
he Coulomb dynamics. If all particles have the same charge-to-mass ratio, t
he dissipation would vanish at that order.