Cris: If I understand you correctly, you posit the existence of periodic orbits which (therefore) have Fourier series, and then follow these constraints until everything matches ? At 08:06 PM 3/9/2013, Cris Moore wrote:
One nice fact is that if the force is 1/r^3, or more generally 1/r^a for a >= 3, then any braid at all corresponds to a valid trajectory of n bodies in the plane: you can get masses 1 and 3 to do-si-do 17 times counterclockwise, then 3 and 4 to twirl 6 times clockwise, and so on.
The reason is that the action of any colliding path is infinite, so if you start off with a fictional trajectory that has the topology you want, you can relax it until it is actually a solution of the equations of motion, and it will keep the same topology throughout the relaxation.
For 1/r^2 forces, on the other hand, some braids are allowed and others aren't. You can see some of these results here: http://tuvalu.santafe.edu/~moore/braids-prl.pdf
- Cris