NONLINEAR DYNAMICS: QUANTUM CHAOS
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Georgia Tech PHYS 7224 Spring semester 2002
Predrag Cvitanovic'
Problem set: Exer 24.5 Collinear helium cycles.
Collinear helium, periodic orbits
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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
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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.
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