Student Seminar: Classical and Quantum Integrable Systems

Thus, we have
Since d 2 S = 0 we get
i.e. the transformation is canonical.
dS = ∂S
∂q j
dq j + ∂S
∂f j
df j = p j dq j + ψ j df j
∑
dp j ∧ dq j = ∑
j
j
df j ∧ dψ j ,
The next point is to show that S exists, i.e. it does not depend on the path. If
we have a closed path from m 0 to m and from m to m 0 and assume that M f does
not have non-trivial cycles then by the Stokes theorem we get
∫ m0
∫ ∫
∆S = α = dα = ω = 0
m 0
because the form ω vanishes on M f :
ω(ξ Fi , ξ Fj ) = {F i , F j } = 0 .
In case the manifold M f has non-trivial cycles the situation changes and one gets
the change of S given by integral of α over a cycle
∫
∆ cycle S = α
which is a function of F i only! This tells us that in this case the variables ψ j are
multi-valued.
cycle
Mention Darboux
1.4 Action-angle variables
As follows from the Liouville theorem under suitable assumptions of compactness
and connectedness motion of a dynamical system in the 2n-dimensional phase space
happens on a n-dimensional torus T n being a common level of n commuting integrals
of motion. The torus has n fundamental cycles C j which allow to introduce the
“normalized” action variables
I j = 1
2π
∮
C j
p i (q, f)dq i ≡ 1
2π
∮
C j
α ,
where f i define the common level T n of the commuting integrals F i . The variables
I j are functions of f i only and therefore they are constants of motion. The angle
variables are introduced as independent angle coordinates on the cycles
1
2π
∮
C j
dθ i = δ ij .
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