This is probably a product of how I was taught, but I am unsure of the status of conservation of energy in classical EM with point particles. Here is the background: When I was taught in undergrad, general arguments were used to show that there needs to be a back-reaction on a point particle moving in electromagnetic fields due to coupling with the very fields it is producing (self-action), and that this can lead to difficulty including run-away solutions. Heck even the Griffith's textbook makes the following comments on these difficulties: "Perhaps they are telling us that there can be no such thing as a point charge in classical electrodynamics, or maybe they presage the onset of quantum mechanics." (pg 467, 3rd edition) We did not dwell on this in class. However in grad school, similar comments were made about the self-action, but the professor gave us two journal papers, one seeming to show quite generally that there was a problem with the self-action of point charges in classical electrodynamics, while the other claimed that careful analysis showed there was no problem, but went into enough mathematical detail for a case that it was unclear how general the 'solution' was. (Unfortunately, I didn't think I'd ever refer to my notes again, so I threw them out. I don't remember what papers they were.) Again we didn't dwell on it, and his attitude was 'be aware of the issue', and decide for yourself based on the math. Then, I saw electrodynamics rewritten in terms of a lagrangian and hamiltonian. With the lagrangian, since it is clearly time translation invariant, Noether's theorem should give us that there is a conserved energy. So at least in the Lagrangian formulation, it appears that any trace of 'run-away' solutions are gone ... forbidden by the very symmetries of the theory. We never discussed this in class, and I didn't realize this connection to the self-action 'problem' until a recent discussion. However, if the argument can be settled that simply, then why is there even still discussion of this? Am I glossing over important subtle details of Noether's theorem here that lend to problems? I've tried talking to several professors about this, and they all said they've never seen the self-action 'problem' in classical electrodynamics approached from the Lagrangian point of view so they were not sure ... and likewise were very hesitant to agree since if it was truly that simple, clearly that would be the preferred argument and everyone would use it. Help please? In particular, if anyone knows of textbooks or definitive journal review articles on this, that would be of tremendous help.