The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity of solving the effect of three bodies which all pull on each other while moving, a total of six interactions. Mathematician Richard Arenstorf while at NASA solved a special case of this problem, by simplifying the interactions to four, because, the effect of the spacecraft's gravity upon the motion of the vastly more massive Earth and Moon is practically non-existent. Arenstorf found a stable orbit for a spacecraft orbiting between the Earth and Moon, shaped like an '8'
Was Arenstorf's solution purely analytical, or did he use numerical mechanisms? Is the '8' shape an optimal path, meaning the route on which the spacecraft would expand the least amount of energy? If yes, how was this requirement included in the derivation in mathematical form?
If anyone has a clean derivation for this problem, that would be great, or any links to books, other papers, etc.
Note: Apparently there was an earlier related mathoverflow question on this as well:
Arenstorf's technical report is here
The most clean, succint description of the equations I could find were here
Restricted 3 Body Problem mentioned in this book (chapter 8)
Some vpython (visual python) code can simulate Apollo 13 trip,
the result is the 8-orbit as seen here
Some explanation on the math, and superposition of gravities is also mentioned in this link.
I have no idea yet how this would relate to Arenstorf's method, I believe in this code Earth and Moon are not moving, but I thought it'd be nice to share.