The figure eight orbit
initial conditions: (x1,y1) = (-0.97000436, 0.24308753), (x2,y2) = (-x1, -y1), (x3,y3) = (0,0)
initial velocites (vx1,vy1) = (vx2, vy2) = -(vx3, vy3)/2; where (vx3,vy3) = (0.93240737, 0.86473146)
masses: 1. gravitational constant: 1.
thanks to Carles Simo numerical wizard. (The initial positions are chosen to be collinear, with center of mass zero. The initial velocities are chosen so that the total linear and angular momenta are zero, and so that the total moment of inertia $\Sigma m_a (x_a ^2 + y_a ^2)$ is extremized. This leaves two free parameters represented by the velocity of mass 3.)
A bit of history. The unpublished paper mentioned above predated the Annals paper with Chenciner. The `eight' of its title was in shape space not in inertial space. and was a different orbit from the eventual eight, an orbit whose existence is still in question. Chenciner and Venturelli discovered an error in the proof of `theorem 1' which claimed the existence of this orbit. Theorem 2 led to my paper with Chenciner. Venturelli, in his thesis, made use of some ideas here concerning using local perturbation analysis to delete collisions and thereby decreasing action, and concerning the constancy of energy along collision minimizers.
A popular accounting of the eight and more new orbits can be found in the Notices article plus Casselman's commentary and pictures
Choreographies, are orbits in which N planets chase each other around the same planar curve. The eight and Lagrange's orbit are choreographies. Soon after we rediscovered the eight solution, a horde of new solutions were discovered using the same ideas: variational methods plus symmetry. For the Newtonian potential case we lack existence proofs for all new solutions except the eight. In 2003 Ferrario and Terrracini proved the existence of some infinite families of choreographies. The breakthrough was Marchall's averaging of perturbations idea, as exposed by Chenciner's ICM notes All of these new variational minimizers EXCEPT the eight are unstable dynamically.