Student Seminar: Classical and Quantum Integrable Systems

We can now see how the solution can be found by using the general approach
based on the Liouville theorem. The expressions for the momenta on the surface of
constant energy and J = J 3 are
√
p r = 2(H − V ) − J 2
r , p 2 φ = J 3 = J .
We can thus construct the generating function of the canonical transformation from
from the Liouville theorem
∫ r
S =
√2(H − V ) − J ∫ 2 φ
r + Jdφ
2
and the associated angle variables
We have eoms
Integrating the first one we obtain
ψ H = ∂S
∂H ,
ψ J = ∂S
∂J
˙ψ H = 1 , ˙ψ J = 0 .
and, therefore,
The equation for ψ J gives
ψ J = −
t − t 0 =
∫ r
ψ H = t − t 0
∫ r
dr
√
.
2(H − V ) − J2
r 2
Jdr
√
+ φ = 0 ,
r 2 2(H − V ) − J2
r 2
so that
φ =
∫ r
Jdr
√
r 2 2 ( ) .
E − V (r) − J 2
2r 2
Generically, equation which defines the values of r at which ṙ = 0:
E − V (r) − J 2
2r 2 = 0
has two solutions: r min and r max , they are called pericentum and apocentrum respectively
2 . When ṙ = 0, ˙φ ≠ 0. The r oscillates monotonically between rmin and
2 If the earth is the center then r min and r max are called perigee and apogee, if the sun – perihelion
and apohelion, if the moon – perilune and apolune.
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