# Confusion about derivation of Abraham-Lorentz force

An accelerating charged particle produces EM radiation, so we conclude that there must be a reaction force acting on the particle, leading to the Abraham-Lorentz force: $${\bf F}_{rad}=\frac{q^2}{6\pi\epsilon_0 c^3}\dot{\bf a}$$ While this solution of the problem has known pathological behaviour, I'm unclear as to how we even got here in the first place. For example, Wikipedia says :

If we assume the motion of a charged particle is periodic, then the average work done on the particle by the Abraham–Lorentz force is the negative of the Larmor power integrated over one period.

This has me quite confused, so I hope the following brings my point across.

The basic intuition here seems to me to be that the particle is a kind of store of kinetic energy (whose change is just the work of the reaction force) while the EM field is a store of EM energy. The sum of these has to be constant, so we conclude that the work done by the reaction force on the particle has to compensate for the energy radiated into the field. Schematically, $$E_{kin} + E_{EM} = const.$$

However, it's not obvious to me that this is actually correct. For example, if we have a charged sphere instead of a particle, it has kinetic energy while the field has its own energy, but then there's also the potential energy of bits of the sphere in the field produced by the remainder of the sphere. Now granted, a particle is assumed to be without such internal structure, but if the whole point of this exercise is to figure out how the particle interacts with its own field, how can we be sure that we haven't missed some kind of intrinsic "potential self-interaction energy" that we would have to include into our tally? Then some of the radiated energy could come from the work done by the reaction force and the rest from this potential energy.

TL;DR: How do we know the radiated power has to take from the particle's kinetic energy instead of some kind of intrinsic self-potential energy?

As a short aside, I presume a derivation of the reaction force can also be carried out using the particle and field momenta instead of the energies, but I haven't seen this done anywhere. Are there any sources dealing with this approach? It might be easier for me to "get".

• I don't know if I would be comfortable outright restricting the self-energy to being only a function of the charge distribution, I was thinking more along the lines of extending energy conservation into $E_{kin} + E_{EM} + F({\bf v},\dot{\bf v}) = const.$ where $F$ is a kind of generalized (velocity-and-acceleration-dependent) self-potential. – J_P May 22 '19 at 13:46
• Regarding perpetual motion, since $F$ depends on ${\bf v}$ and $\dot{\bf v}$ and as these two are the same after one period, I'm not sure that there actually is any net gain of energy from $F$. $F$ would still add a force term, though, but it would apparently have to be such that its line integral over a period vanishes; I'm not sure what constraints that puts on $F$ though. I'll wait a bit to see if anyone else answers, otherwise I'll accept this. – J_P May 22 '19 at 13:47