Why do the expressions for Feynman-Heaviside fields look completely different from the Lienard-Wiechart fields though both of which are fields due to a point charge moving along a specified trajectory? I don't understand this. For a reference, refer to eq. (23.31), (23.32), (23.49) and (23.50) of Andrew Zangwill's Modern Electrodynamics.
I give them below for ready reference:
Firstly: they are equivalent. Feynman claims in the Feynman Lectures that you can easily check this by doing the differentiation in his formula and comparing with the fields derived from the L-W potentials. It's taken me a lifetime of trying and giving up because it's so difficult before I worked out how to do it reasonably efficiently. I ought probably to write this up and publish it!
So secondly, I think the real reason is that Feynman was very proud of his formula, which does indeed give real insight into what's going on and how it relates to the static charge case, but wanted to cover his tracks so everybody would think 'Wow how ever did he find that?' He references the formula very early in Chapter 1 of Volume 1, and again in its more natural position in the elctrodynamics volume, II, and then purports to derive it without actually doing so. There has to be an explanation, and I think this is the most natural one.
