# 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).