Retarded Central Force Problem in Classical Physics

1. Apr 14, 2012

fermi

There are many good treatments of the classical central force problem in many undergraduate and graduate text books. But I was unable to find a similar treatment of the retarded central force problem. I am looking for the classical treatment of the potentials of type:
$$\delta(t'-t + |\mathbf{x}-\mathbf{x}'|/c) V({|\mathbf{x}-\mathbf{x}'|})$$
I will be also happy with the treatment of a special case with:
$$V({|\mathbf{x}-\mathbf{x}'|}) = \frac{1}{|\mathbf{x}-\mathbf{x}'|}$$
Can anybody recommend a good book or a published article?

Thank you.

PS: Please do not refer me to the Lienard-Wiechert potentials in classical electromagnetism. They only treat the case when one of the particle's position (path) is given (and unaltered by the other particle.) I am looking for the dynamic interaction of a two-body-system.

2. Apr 14, 2012

jjustinn

I just finished Spivak's Mechanics for Mathematicians, and there was a good treatment of central force / two-body problems, and of wave propagation (presumably you mean this central "force" is derivable from a central potential that propagates from sources as a wave?)...but as far as the full retarded two-body problem goes, I don't think I have seen any treatments of that (even neglecting energy loss through radiation)...my understanding is it would be a highly non-linear equation, if an analytic solution could even be derived.

In fact, I don't think I've ever even seen a full treatment of the retarded *one* body problem; when researching another question here I did track down a long-OOP text book on Google books that gave a pictorial analysis of what it would look like, and showed how the self-field forces would cancel for a rigid sphere with zero jerk, but nothing like an equation of motion.

I would be really interested in hearing if you find anything, though (provided I properly understand the question).