# Solution to Maxwell-Lorentz equations

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.

• Am I correct to assume that you want no external fields? For example, you want to know how two charged particles move under the influence of their own fields? BTW, I have never seen such a solution. – G. Smith Aug 27 '19 at 0:19
• :-) I was going to suggest Jackson, Chapter 16. If I remember rightly, the message was that apart from few special cases, or cases where either charge motion or the field evolution is approximattely unaffected, trying to solve the full problem of moving charges and fields where both interact and are affected by each other, leads to anomalous solutions. – Cryo Aug 27 '19 at 0:23
• You can calculate EM fields produced by a moving charge from the Liénard-Wiechert formulas. Then you can use the Lorentz force law to calculate the force on the second particle from the retarded fields of the first particle (and vice-versa). The result is a system of two "delay differential equations". I asked a similar question previously. One answer cited this paper studying numerical solutions. – Tob Ernack Aug 27 '19 at 0:43
• @G.Smith There could be external fields, or it could be self interacting problems. I just would like to see if there is any complete solution of ML equations. – aprendiz Aug 27 '19 at 9:45
• @Cryo thanks for suggestion, I took a look at J's cap 16 and indeed it describes this problem there. I am sad that no simple solution (even some toy model) seems to be known. – aprendiz Aug 27 '19 at 9:45