The 4-body problem in space.
In the 1980s Rick Moeckel initiated and largely completed a program of extracting a symbolic dynamics out of the planar three-body problem, this dynamics arising out of solutions suffering repeated close approach to triple collision. McGehee's blow-up was an essential technique. We report on initial steps towards repeating Moeckel's program for the case of 4 bodies moving in 3-space. A key fact for us concerns central configurations, the only `road in' to total collision. The fact asserts that any total collision solution to this 4-body problem which tends to a planar central configuration is in fact planar: all 4 bodies lie on fixed plane throughout their motion. Some inspiration for our work also came from the paper ``oscilating about coplanarity' which uses the fact that when N = d+1 we find that the N-body problem in d-space describes a Newtonian evolution in the space of square d x d matrices. This is joint work with Guowei Yu (Chern Institute of Mathematics and LPMC, Nankai University).

