(I'm considering classical relativistic mechanics and classical electrodynamics throughout this post.)

Usually, you calculate the motion of charges due to an external electromagnetic field, or you calculate the external electromagnetic field due to point charges. Once you start to try to couple the two, you have to deal with the issues of infinite self-energy and self-force. If I search for the theory of the problem , there's an endless stream of papers with different formulations. It seems like the microscopic point-particle theory is ill-defined and borderline inconsistent*.

So solving for the motion of a coupled particle+field is hard. How, then, do you physically solve these problems when you want to figure out laser wakefield acceleration, gamma ray bursts, or other highly coupled particle-field situations? This question is not really about the simulation aspects. Rather, in light of the previous paragraph, I want to know how to even write down a physical system of ordinary or partial differential equations to be solved.

As an aside, in Griffiths Introduction to Electrodynamics the self-force is said to pointedly not depend on the power/momentum radiated out to infinity. Instead, he writes, "as the particle accelerates and decelerates, energy is exchanged between it and the velocity fields, at the same time as energy is irretrievably radiated away by the acceleration fields." It is less practical than the research applications I listed above, but this makes it sound like the problem of a coupled electron+self-force+field should be solvable. Not analytically, but by computer. Again, even writing down any equations is difficult!

*It comes to my attention (from Ján Lalinský) that the self force is a tiny effect, with a value for the electron of $6\cdot 10^{-24}s$ (seconds is equal to units acceleration per unit jerk), and so does not matter for practical physical observations. While this means we can ignore self-force, the $\frac{q}{r^2}$ field still leaves $\vec{E}$ undefined on top of a point charge, rendering $F=q\vec{E}+q\vec{v}\times\vec{B}$ useless, so that one cannot easily figure out the motion of a point charge in its own field.

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    $\begingroup$ There are numerous methods/algorithms that PIC (particle-in-cell) simulations use to attempt to self-consistently model particle and field dynamics, where the particles can affect/modify the field. Generally there are two classes where one solves the Lorentz force and the other uses Lagrangians/Hamiltonians, depending on the problem to be solved. The subtlety is how to update the fields and particles (i.e., which is updated first before advancing the other)... I forget which is the current approach but could look it up. $\endgroup$ – honeste_vivere Sep 5 '16 at 18:51

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