NONLINEAR DYNAMICS: QUANTUM CHAOS ----------------------------------------------------------- Georgia Tech PHYS 7224 Spring semester 2002 Predrag Cvitanovic' Problem set: Exer 24.5 Collinear helium cycles. Collinear helium, periodic orbits --------------------------------- was due Tue 16 2002: Exer 24.5 Collinear helium cycles, comment on symbolic dynamics: In order to find longer cycles, good starting guesses would be useful. With this in mind, I have suggested to Kapil and Andreas to shoot out from the triple collision point (r_1,r_2) at various angles, in order to trace out the symbolic dynamics partition in the (r_1,p_1) Poincare section. My suggestion was wrong in an interesting way: Rytis finds that in the KS-regularized coordinates one can start out from the (Q_1,Q_2), but that all such trajectories escape to infinity (what happens is one is arbitrarily close to 45 deg, shooting close to the symmetry axis? I do not know). So the way to get a sense for the symbolic dynamics partitoning of the (r_1,p_1) Poincare section is to sketch the first few folds of the stable/unstable manifolds of the 1^{\infty} fixed point. If you have a sketch of such partitoning, let me know. Predrag -------------------------------------------------------------- The solution of problems starting with these three and ending with the helium spectrum form the final exam in the course. If you prefer, you can compute the 3-disk resonances instead, the determination of perioidc orbits is easier in billards than it is in the flows. -------------------------------------------------------------------