Student Seminar: Classical and Quantum Integrable Systems

4. The equations of motion with Hamiltonian H can be integrated by quadratures.
Let us outline the proof. Consider the level set of the integrals
M f = {x ∈ M : F i = f i , i = 1, . . . , M} .
By assumptions, the n one-forms dF i are linearly independent at each point of M f ;
by the implicit function theorem, M f is an n-dimensional submanifold on the 2ndimensional
phase space M. Moreover, the n linearly-independent vector fields
ξ Fi = {F i , . . .}
are tangent to M f and commute with each other.
Let α = ∑ i p idq i be the canonical 1-form and ω = dα = ∑ i dp i ∧ dq i is the
symplectic form on the phase space M. Consider a canonical transformation
i.e.
(p i , q i ) → (F i , ψ i )
ω = ∑ i
dp i ∧ dq i = ∑ i
dF i ∧ dψ i
such that F i are treated as the new momenta. If we found this transformation then
equations of motion read as
˙ F j = {H, F j } = 0 ,
˙ψ j = {H, ψ j } = ∂H
∂F j
= ω j .
Thus, ω i are constant in time. In these coordinates equations of motion are solved
trivially
F j (t) = F j (0) , ψ j (t) = ψ j (0) + tω j .
Thus, we see that the basic problem is to construct a canonical transformation
(p i , q i ) → (F i , ψ i ). This is usually done with the help of the so-called generating
function S. Consider M f : F i (p, q) = f i and solve for p i : p i = p i (f, q). Consider the
function
We see that
and we further define
S(f, q) =
∫ m
m 0
α =
∫ q
p j = ∂S
∂q j
ψ j = ∂S
∂f j
q 0
∑
i
p i (f, ˜q)d˜q i
– 8 –