An exceptional orbit was recently discovered by, among others, Chenciner and Montgomery who provided proof of its existence2 using the shape sphere. Carles Simó used numerical simulation (perhaps similar to the Java applet that accompanies this document) to determine appropriate initial positions and velocities for the three bodies to realize this orbit.
To describe this orbit, first imagine a two-body problem exhibiting Keplerian motion, that is, both bodies orbit in a circular fashion around the 2-body center of mass. The exceptional orbit described by Chenciner and Montgomery resembles a coupled Keplerian system where one body is traded in every rotation. The result is a periodic orbit which traces out a figure-eight.
The figure-eight orbit has constant moment of inertia and potential energy, and zero total angular momentum. This orbit is significant since it represents the first orbit since the Euler and Lagrange central configurations which lies completely on a fixed curve in phase space.
The projection of this orbit onto the shape sphere is represented by the free homotopy class in the braid group , in Montgomery's notation. The following is Chenciner and Montgomery's depiction of the figure-eight orbit's projection onto the shape sphere.
The accompanying Java applet initializes to Simó's initial conditions; if the user opts not to modify them, the resulting trajectories will trace out this figure-eight orbit.