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
That is the reason for the name "Choreography" or in Russian
Хоровод (when dancers of a
follow each other along a curve).
trajectories of all bodies are periodic (i.e. closed
all these orbits are congruent, i.e. all n bodies move along one orbit in a
formation one after the other.
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
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).
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
Simple Choreography Motions of N Bodies: A
Platonic and Archimedean Solid symmetric