Student Seminar: Classical and Quantum Integrable Systems

We find
α = ∑ i
p i dx i = p r dr + p θ dθ + p φ dφ ,
where the original momenta are expressed as
p 1 = 1 (
sin φ
)
rp r cos φ sin θ + p θ cos θ cos φ − p φ ,
r
sin θ
p 2 = 1 (
cos φ
)
rp r sin φ sin θ + p θ cos θ sin φ + p φ ,
r
sin θ
p 3 = p r cos θ − 1 r p θ sin θ .
Conserved quantities
(
p 2 r + 1 r 2 p2 θ +
H = 1 2
J 2 = p 2 θ + 1
sin 2 θ p2 φ
J 3 = p φ
1
)
r 2 sin 2 θ p2 φ + V (r)
To better understand the physics we note that the motion happens in the plane
orthogonal to the vector J. ⃗ Without loss of generality we can rotate our coordinate
system such that in a new system J ⃗ has only the third component: J ⃗ = (0, 0, J3 ).
This simply accounts in putting in our previous formulae θ = π . Then we note that
2
p 2 φ
˙φ = {H, φ} = {
2r 2 sin 2 θ , φ} =
that for θ = π 2 expresses the integral of motion p φ as
p φ = r 2 ˙φ.
p φ
r 2 sin 2 θ
This is the conservation law of angular momentum discovered by Kepler through
observations of the motion of Mars. The quantity p φ = J has a simple geometric
meaning. Kepler introduced the sectorial velocity C:
C = dS
dt ,
where ∆S is an area of the infinitezimal sector swept by the radius-vector ⃗r for time
∆t:
∆S = 1 2 r · r ˙φ∆t + O(∆t 2 ) ≈ 1 2 r2 ˙φ∆t .
This is the (second) law discovered by Kepler: in equal times the radius vector sweeps
out equal areas, so the sectorial velocity is constant. This is one of the formulations
of the conservation law of angular momentum. 1
1 Some satellites have very elongated orbits. According to Kepler’s law such a satellite spends
most of its time in the distant part of the orbit where the velocity ˙φ is small.
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