In my 1993 paper, I naively discretized the orbit, and adjusted each point until the curvature of the path matches the acceleration it ought to feel. This is equivalent to gradient descent in the action. In the joint paper with Nauenberg, we used the first 10 or 20 Fourier coefficients, and did gradient descent in the action (evaluating the action by numerical integration). Of course, neither of these techniques proves rigorously that an orbit exists. That was done for the figure-8 by Chenciner and Montgomery in 2001, and for the Henon "criss-cross" orbit by Chen. Cris On Mar 10, 2013, at 10:59 AM, Henry Baker wrote:
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
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Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/