LAUSR

We study the dynamics of many charges interacting with the Maxwell field. The particles are modeled by means of nonnegative distribution functions $$f^+$$ f + and $$f^-$$ f - representing two species of charged matter with positive and negative charge, respectively. If their initial velocities are small compared to the speed of light, $$\mathrm{c}$$ c , then in lowest order, the Newtonian or classical limit, their motion is governed by the Vlasov–Poisson system. We investigate higher-order corrections with an explicit control on the error terms. The Darwin order correction, order $$|\bar{\mathrm{v}}/\mathrm{c}|^2$$ | v ¯ / c | 2 , has been proved previously. In this contribution, we obtain the dissipative corrections due to radiation damping, which are of order $$|\bar{\mathrm{v}}/\mathrm{c}|^3$$ | v ¯ / c | 3 relative to the Newtonian limit. If all particles have the same charge-to-mass ratio, the dissipation would vanish at that order.