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. The program loads the ODEs for the 3 body problem with the initial values corresponding to the selected choreography, compiles, and opens the Graph window for visual integration. At that length of the period of this sample is entered into the termination condition and the program is set to integration until this termination point. (The period is visible also in the Front panel in the Constant section as a comment line for first c1.)

3) In the Graph window you will see already a result of integration of 10 steps (by default). 10 steps may not be enough for reaching the terminal point. You may need to click the button More several times changing the number of steps to something bigger, say 100. For some samples with long periods (over 80) you may need over 1000 steps. (By default the storage is reserved for 1000 points, therefore when the program asks you to append the storage, do it). You may notice that graph has not enough room (in default setting). Any moment you can adjust the room clicking at the button Adjust.

4) Finally when you obtained the full graph of this choreography, you may wish to Play it dynamically. The default time 5 sec is too short. Enter a longer interval. Depending on the complexity of the curve, it may be something like 60-80 seconds. Enjoy the show, and then repeat everything from step 1 for another sample.

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