Student Seminar: Classical and Quantum Integrable Systems

7.4 Seminar 4
Exercise 1 (K. Bohlin). Consider the Kepler problem. Let x, y be the Cartesian
coordinates on the plane of motion. Introduce a complex variable z = x + iy and
show that the non-linear change of variables z → u 2 , t → τ given by
z = u 2 dt
,
dτ = 4|u2 | = 4|z|
maps the Kepler orbits with the constant energy E < 0 into the ones of the harmonic
oscillator with the complex amplitude u (a two-dimensional oscillator). Find the
period of oscillations.
Exercise 2 (Lissajous figures). Consider the two-dimensional harmonic oscillator.
Show that if
ω 1
ω 2
= r s ,
where r, s are relatively prime integers then there is a new additional integral of
motion
F = 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 ) .
The corresponding closed trajectories of the two-dimensional harmonic oscillator
are called the Lissajous figures. Find the Poisson brackets between F and F i =
1
2 (p2 i + ωi 2 qi 2 ), i = 1, 2.
Exercise 3. Consider the Kepler problem. Show that the components of the angular
momentum J i and the components of the Runge-Lenz vector R i form w.r.t. the
Poisson bracket the Lie algebra so(4). Recall that a 4 × 4 matrix X belongs to the
Lie algebra so(4) if it is skew-symmetric, i.e. X t + X = 0. Express the Kepler
Hamiltonian in terms of the conserved quantities J i and R i .
Exercise 4. Prove that the Poisson bracket
{L 1 , L 2 } = [r 12 , L 1 ] − [r 21 , L 2 ]
between the components of the matrix L implies that the quantities I k = trL k are
in involution, i.e. that {I k , I m } = 0.
Exercise 5 (Calogero model). Consider a dynamical system of n particles with
the coordinates q j and momenta p j , where j = 1, . . . n. The Hamiltonian of the
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