Gravitational n-body choreographies are periodic orbits for n equal-mass bodies in which each planet traces out the same curve. Interest in choreographies was piqued by the figure-eight orbit discovered numerically by Cris Moore and later proven to exist by Alain Chenciner and Richard Montgomery. Since then many more choreographies have been found by Carles Simó.
This site also has a
lighter introduction
to choreographies and an
interactive program
for finding them.
Using the principle of least action, choreographies can be found by starting with a curve and numerically minimizing the action functional. More specifically, the software shown starts with a crude sketch, takes a Fourier transform, and uses the conjugate gradient method to minimize the action as a function of the Fourier modes.
This technique, based on Lagrangian mechanics, is well-suited to finding choreographies. However, since most of us are more familiar with Newtonian physics, it is useful to also see a simulation. The video on the left was made by sampling the solution curve at a more-or-less random time and using the resulting position/velocity (phase space) initial conditions to start a dynamics simulation. When used for the figure-eight, this both experimentally confirms that it is a choreography and provides evidence for its stability.
Instead of minimizing the action, one could directly look for stationary points. The version of the software shown here does this by using a damped Newton's method algorithm to solve for zeros of the gradient of the action.
This is particularly useful for generalizing to 3-space because all of the nontrivial (i.e., not a circle) choreographies I have experimentally observed are saddle points when considered with three-dimensional degrees of freedom. With the second order Newton's method approach we can find truly three-dimensional choreographies.
In addition to writing software for numerically finding choreographies, I developed a method of computer-assisted proof for proving their existence. This approach complements a previous approach to this problem; that method focuses on the (finite-dimensional) dynamics problem and relies on algorithms for rigorous solution of ODEs, while my method directly addresses the infinite-dimensional action extremization problem.
This method will be discussed in an upcoming paper.