Student Seminar: Classical and Quantum Integrable Systems

2.4 Systems with closed trajectories
The Liouville integrable systems of phase space dimension 2n are characterized by the
requirement to have n globally defined integrals of motion F j (p, q) Poisson commuting
with each other. Taking the level set
M f = {F j = f j , j = 1, . . . n}
we obtain (in the compact case) the n-dimensional torus. In general frequencies of
motion ω j on the Liouville torus are not rationally comparable and, as the result,
the corresponding trajectories are not closed.
A special situation arises if at least two frequencies become rationally comparable.
Such a motion is called degenerate. Here we will be interested in the situation
of the completely degenerate motion, i.e. when all n frequencies ω j are comparable.
In this case the classical trajectory is a closed curve and the number of global integrals
raises to 2n − 1. 5 They cannot Poisson-commute with each other because the
maximal possible number of commuting integrals can be n only. Below we will give
already accounted examples of degenerate motion.
Two-dimensional harmonic oscillator. The Hamiltonian is
H = 1 2 (p2 1 + p 2 2) + 1 2 (ω2 1q 2 1 + ω 2 2q 2 2) .
There are two independent and mutually commuting integrals
F 1 = 1 2 p2 1 + 1 2 ω2 1q 2 1 , F 2 = 1 2 p2 2 + 1 2 ω2 2q 2 2 ,
such that H = F 1 +F 2 . If the ratio ω 1 /ω 2 is irrational the trajectories are everywhere
dense on the Liouville torus. However, if
ω 1
ω 2
= r s ,
where r, s are relatively prime integers then there is a new additional integral of
motion
F 3 = ā s 1a r 2 ,
where
ā 1 = 1 √ 2ω1
(p 1 + iω 1 q 1 ) , a 2 = 1 √ 2ω2
(p 2 − iω 2 q 2 ) .
Indeed, we have
( )
F˙
3 = ā s−1
1 a r−1
2 sa2 ˙ā 1 + rā 1 ȧ 2 .
5 In quantum mechanics we have in this case the degenerate levels.
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