How the movies were made. Carles Simo implemented the following three main ideas, or methods, to find the orbits depicted in these movies: (1) gradient descent, (2) shooting (Poincare sections), and (3) the homotopy (orbit following) method.
1. Gradient descent. The gradient of the action functional yields a flow in path space. This flow is discretized, for example using Fourier modes, to yield a finite-dimensional optimization problem. The descent method works well to get the eight and Gerver's supereight. On the more complicated choreographies it converges too slowly to give good initial conditions to check with an ODE solver. See step 2. Slow convergence is in part because of near collisions and high instability.
2. Shooting. After time $T/N$ of a period $T$ choreography, the $k$th mass is where the $k-1$st mass started. In accordance with this, set up a time $T/N$ slice (Poincare section) to an initial guess orbit obtained via descent, thus obtaining an $i \mapsto i + 1$ type Poincare section. Look for fixed points to this map. (We are ``shooting'' from the $t = 0$ to the $t = T/N$ slice.)
The shooting just described often fails, due to dynamic instability. Multiple shooting, where we further divide $T/N$ into subintervals may help.
(3). (1) and (2) together still fail to find many of the orbits depicted. However if the Newtonian $1/r$ potential is replaced by $1/r^2$ then gradient descent works well enough to give initial conditions good enough for step 2. Better convergence occurs solutions cannot tend to collision: any curve with collision has infinite action for a $1/r^a$ potential, $a \ge 2$. Consider a family of problems indexed by $a$ where the potential is $1/r^a$. Implement descent for $a =2$. Use shooting to refine the result. Follow the orbit via implementing an implicit function theorem as $a$ is varied, trying to bring $a$ down from $2$ to $1$. This last step (3) is called the homotopy method by some. It allows us to follow periodic orbits for systems depending continuously on some real parameter.
The basic set up for choreographies and for (1) is described in our paper (referenced above, w/ Gerver et al).
Which choreography classes are realized? We have described the basic ideas Simo implements to find the orbits. For any fixed $N$ there are infinitely many topologically distinct choreography classes. All are realized by solutions when the potential is $1/r^2$. Probably some of these, maybe most, disappear upon bringing $a$ down to the the Newtonian case $a =1$. The subject is in a state of flux (as of May 7, 2001). Simo has recently found on the order of a million new 3-body choreographies the methods outlined above, combined with other ideas. Probably there are infinite families of choreographies for all $N$. Nothing is proved for the Newtonian case beyond the existence fof the eight.