# 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 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 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 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 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 at 9:45

## 2 Answers

The Maxwell-Lorentz equations for point-like charged particles are meaningless. This is well-known since the beginnig of the 20th century. Older textbooks (like that of Becker) written between the two world wars discuss it in all details. The devil lies in the self-interaction. A hand-made correction, excluding from the Lorentz force the field produced by the particle itself has still survived. Ignoring all magnetic forces this approach leads to the Coulomb Hamiltonian used also in the non-relativistic quantum mechanics, where the Coulomb terms i=j are just omitted. Actually, one should not even teach the electrodynamics of point-like classical particles, since it is basically wrong. Has neither Lagrangian nor Hamiltonian formulation. A consistent formulation of the electrodynamics of charged particles may be formulated only in the frame of the field theory followed by a quantization. I recommend You the recent pedagogical arXiv preprint ( a future chapter of a textbook in preparation):

A field-theoretical approach to non-relativistic QED. by Ladislaus Alexander Bányai and Mircea Bundaru, arXiv:1907.13053v1 [quant-ph] 30 Jul 2019

• There are consistent theories of point charged particles with local conservation of energy due to Tetrode, Fokker, Frenkel, Feynman and Wheeler, with variations on whether fields are retarded, half retarded + half advanced, etc. See my answer here physics.stackexchange.com/questions/380741/… – Ján Lalinský Aug 28 at 19:05
• @Laci can you please give the precise references of the old textbooks? – aprendiz Aug 29 at 22:27
• Unfortunately, these old books are not listed in the web and therefore one should look for in the libraries. However, you may find a related nice discussion in Chapter 28 of the 6-th band of the Feynman Lectures of Physics. But he gives no bibliography. – Laci Aug 31 at 6:03

May be, one my construct such theories (the only ones I am aware, introduce a charge distribution for the classical particles) , but these are not the Lorentz-Maxwell theory. The whole development of quantum mechanics started from this one and has been developed along his lines to get finally the QED. Any other variants left no traces in the development of physics.