Existence of action variables
We have therefore a family of tori, with the following additional structure. On each torus, we have a notion of “straight line”. Indeed, these “straight lines” are orbits of the action of linear combinations of . Locally, this is the same as to say that we are given the action on on each . In particular, suppose that form a basis in . We will now prove that for every , is a Hamiltonian vector field (that is, ).

Let us denote the Hamiltonian vector field corresponding to :

There exists periodic combinations:

where are constant on the tori, i.e. depend only on . This immediately implies:

In particular, we see that:

This immediately implies (by taking ):

(2)

This, plus periodicity, implies:

Indeed, (2) with implies that is linear in :

This could only be periodic in when .

We conclude that is, in fact, a Hamiltonian vector field. What is the corresponding Hamiltonian? Being constant on , it is equal to its average over the period:

After these observations, to calculate , it is enough to calculate its variation from one torus to another. Which is equal to the variation of:

This proves that the action variable is given by the formula: