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. The
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
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,
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
Simple Choreography Motions of N Bodies: A
Platonic and Archimedean Solid symmetric