Student Seminar: Classical and Quantum Integrable Systems

Then using the eoms ˙q = p and ṗ = −ω 2 q we find
Thus,
˙ā 1 = iω 1 ā 1 , ȧ 2 = −iω 2 a 2 ,
˙ F 3 = iā s 1a r 2(
sω1 − rω 2
)
= 0 .
This integral is homogenous function of degree r + s both over the coordinates and
momenta. The trajectories are closed. They are the so-called Lissajous figures. Find
the Poisson brackets between F and F i = 1 2 (p2 i + ω 2 i q 2 i ).
The Kepler problem. We know that the orbits in the Keplerian problem are closed
for E < 0. There exists an additional conserved Runge-Lenz vector:
⃗R = ⃗v × ⃗ J − k ⃗r r .
This vector is othogonal to the angular momentum:
( ⃗ J, ⃗ R) = ( ⃗ J, ⃗v × ⃗ J) − k r ( ⃗ J, ⃗r) = 0 − 0 = 0 .
Thus, there are five independent integrals of motion in the system with six phasespace
degrees of freedom. The Kepler Hamiltonian can be expressed via these five
quantities. Thus, the motion is completely degenerate.
The Euler top. The phase space has dimension six. We found four globally defined
conserved quantities: the Hamiltonian and three components of the angular momentum.
That is the reason why the Liouville torus has dimension two instead of three.
Since 6 − 4 = 2 ≠ 1 the motion is partially, but not completely degenerate.
3. Lax pairs and classical r-matrix
In this section we will study the cornerstone concepts of the modern theory of integrable
systems: the Lax pairs and classical r-matrix.
3.1 Lax representation
Let L, M be two matrices which are also functions on the phase space, i.e. L ≡ L(p, q)
and M = M(p, q), such that the Hamiltonian equations of motion can be written in
the form
˙L = [M, L] .
This is the Lax representation (the Lax pair) of the Hamiltonian equations. The
importance of this representation lies in the fact that it provides a straightforward
construction of the conserved quantities:
I k = trL k .
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