A free homotopy class (mod rotations) for the planar three body problem is equivalent to a list of syzygies: which body goes between the other two at each collinearity (each eclipse), along with the cancellation rules: 11 = 22 = 33 = empty. (For example: 121123=1223=13.) For nearly 17 years I tried using variational methods to prove every free homotopy class is realized by some reduced periodic solution to the Newtonian problem. (It is dead easy to do this for a 1/r^2 -type potential). Two weeks ago, Rick Moeckel and I solved the problem using symbolic dynamics established by Rick in the early 1980s. Rick’s methods center on classical heteroclinic-tangle horseshoe type pictures for stable and unstable manifolds of the five central configurations.
These objects only exist AFTER you McGehee blow-up the planar three body problem. Consequently our solutions repeatedly come very close to triple collision, although they never collide. I will do my best to explain all the words above and to motivate the problem and its solution, and the limitations of our solution.
Audience participation may be required.