NONLINEAR DYNAMICS: QUANTUM CHAOS
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Georgia Tech PHYS 7224 Spring semester 2002
Predrag Cvitanovic'
Lecture 26 Tue, Apr 16 2002 12:05-13:25 in Howey S106
Collinear helium, periodic orbits
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This might help:
Cristel Chandre has contributed his implementation of
the periodic orbits routines for a Hamiltonian flow.
Carl Dettmann has contributed his notes on cycle search
on a Poincare section.
Igor Romanovski has contributed his 3-disk fortran code
that finds all cycles up to a given length.
Links to C, fortran code, notes:
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Classical and Quantum Chaos
--- www.nbi.dk/ChaosBook/extras/Welcome.html#chapterCycles
chapter 12 - Fixed points, and how to get them / extras /
-- Cristel Chandre's implementation of
sect. 12.3.1 "Newton method for flows",
for a 2-degree of freedom Hamiltonian flow,
periodic orbits of a forced pendulum.
Should be easily adoptable to other 2-degree
of freedom Hamiltonian systems, such as
the collinear helium: the overview, the C code.
Problem set
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was due Tue 16 2002:
Exer 24.5 Collinear helium cycles.
Guess some short cycles by requiring that topologically
they correspond to sequences of bounces either returning to
the same r_i axis or reflecting off the diagonal.
Compute them by some method or other.
(if you are quantizing 3-disks rather than helium, find all
cycles up in fundamental domain to length 5.)
Exer 24.6 Collinear helium cycle stabilities
Compute the stability eigenvalues for you collinear helium cycles.
Compare with Table 24.1
If you have developed some half-way decent programs,
comment them richly and contribute to the ChaosBook extras homepage.
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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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