Resources for the concept and discovery of Choreographies

The concept of Choreography emerged at the very end of the 20th century thanks to discoveries by Prof. Carles Simó, Richard Montgomery and others partially outlined below.

Choreography is a special kind of a periodic solution of the Newtonian n-body problem for equal masses such that...
• trajectories of all bodies are periodic (i.e. closed orbits), and...

• all these orbits are congruent, i.e. all n bodies move along one orbit in a formation one after the other.

That is the reason for the name "Choreography" or in Russian Хоровод (when dancers of a group follow each other along a curve).

For a long time it was believed that the only case of Choreography is the Lagrange formation of n bodies moving along a circumference (Set n body problem in the main menu). The first non-trivial choreography ("eight" shape, or rather ∞-shape) was discovered in 1993 by Cristopher Moore and subsequently proved to exist by Chenciner and Montgomery (Demo/3 bodies/Coreography/#1 "eight" and a few more sophisticated ones under the same menu).

Prof. Carles Simó kindly submitted his data file containing as many as 345 types of Choreographies, each of which can be loaded and played here. Here is how.

1) Go to Demo/3 bodies/Choreography/All 345 to integrate (not a script)

2) Enter a number of the desired sample between 1 and 345 into a small window (top left). The program loads the ODEs for the 3 body problem with the initial values corresponding to the selected choreography, compiles, and (blindly) integrates the problem until reaching the termination point – the period of this simulation entered from its file. (The period of the orbit is visible also in the Front panel in the Constant section as a comment line for constant  a).

3) When the integration reaches the termination point (the period), the program displays the message. As you click OK, the program opens the Graph window displaying the entire trajectory. You may wish to Play it dynamically (by default the duration is 25 s). Depending on the complexity of the curve, it may be something like 60-80 seconds. Enjoy the show, and then repeat everything from step 2 for another sample.

4) If you like the curves that you have obtained and played, you may wish to save this session as the script so that next time they integrate ready for playing automatically. In the File menu, Save Script into a location of your choice (available only in the licensed version). Next time you will be able to open
this script and play it.

Below are articles and sites covering the topic in-depth.

Dynamical properties of Hamiltonian Systems. Applications to Celestial Mechanics 1, by Carles Simó
Dances of n bodies - by Carles Simó
Simple Choreography Motions of N Bodies: A Preliminary Study

3D Platonic and Archimedean Solid symmetric
Minton's build-your-own