Student Seminar: Classical and Quantum Integrable Systems

Commuting integrals
F i = 1 2 (p2 i + ω 2 i q 2 i ) .
Define the common level manifold
M f = {x ∈ M : F i = f i , i = 1, . . . , M}
This manifold is isomorphic to n-dimensional real torus which is a cartesian product
of n topological circles. These tori foliate the phase space and can be parametrized
with n angle variables θ i which evolve linearly in time with frequencies ω i . This
motion is conditionally periodic: if all the periods T i = 2π
ω i
are rationally dependent:
T i
T j
= rational number
the motion is periodic, otherwise the flow is dense on the torus.
1.3 The Liouville theorem
The system is Liouville integrable if it possesses n independent conserved quantities
F i , i = 1, . . . , n, {H, F i } which are in involution
{F i , F j } = 0 .
The Liouville theorem. Suppose that we are given n functions in involution on a
symplectic 2n-dimensional manifold
Consider a level set of the functions F i :
F 1 , . . . , F n , {F i , F j } = 0 .
M f = {x ∈ M : F i = f i , i = 1, . . . , n}
Assume that the n functions F i are independent on M f . In other words, the n-forms
dF i are linearly independent at each point of M f . Then
1. M f is a smooth manifold, invariant under the flow with H = H(F i ).
2. If the manifold M is compact and connected then it is diffeomorphic to the
n-dimensional torus
T n = {(ψ 1 , . . . , ψ n ) mod 2π}
3. The phase flow with the Hamiltonian function H determines a conditionally
periodic motion on M f , i.e. in angular variables
dψ i
dt = ω i , ω i = ω i (F j ) .
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