I am trying, without success, to find an example (preferably simple) of solution for the Maxwell-Lorentz equations, i.e., the coupled system of Maxwell equations + dynamics of a charged particle given by Lorentz force. Say we have a (for simplicity, non relativistic) particle of mass m, charge q, position $\vec x$ and velocity $\vec v$, then the Lorentz force will give
$$m \vec x''(t) = q ( \vec E (\vec x(t),t) + \vec v(t) \times \vec B (\vec x(t),t ))$$
Is there any system for which we can exhibit at some instant $t_0$ the 'state' of the system $( \vec E(\vec r,t_0),\vec B(\vec r,t_0) ,\vec x(t_0),\vec v(t_0))$?
Standard textbooks seems not to consider solutions of coupled Maxwell-Lorentz equations, the only one I didn't check is Jackson's, because I don't have a copy with me.