Integrals of motion in involution

Let us consider a Hamiltonian system with the -dimensional phase space , with the
Hamiltonian .

An integral of motion is a function in involution with the Hamiltonian, i.e.:

Suppose that there are integrals of motion all in involution with each other:

but at the same time functionally independent, i.e. the differentials
are linearly independent at each point. (The Hamiltonian can be one of them, or a function of them.)

For a list of values , consider the common level set:

Notice that has the following properties:

it is invariant under the Hamiltonian evolution, i.e. a trajectory starting at any point remains inside

it is a Lagrangian manifold, i.e. the restriction of on is zero

Suppose also that is compact and connected. Then, the following is true:

is a torus (this is the easy part)

- In the vicinity of exist functions such that for every the corresponding Hamiltonian vector field has periodic trajectories with the period :These functions are called action variables