Perturbed and unperturbed phase curves of a Hamiltonian system can be connected via a canonical transformation. This suggest that perturbation theory for Hamiltonian systems is really the study of canonical transformation which depend upon a parameter
which form a family of transformations which reduce to the identity when
Generating functions are really not suitable here because they are functions of both new and old coordinates and give implicit expressions for the coordinates. For most purposes we need the old coordinates as functions of the new coordinates.
Consider those canonical transformations
which are close to the identity transformation:
This has the generating function
so a Hamiltonian is perturbed slightly:
where
is a small parameter, we would expect a transformation close to the identity:
to have a generating function of the form
where
is some well behaved function of
and 
The canonical transformation produced by this generating function is
(1)
(2)
Since
we may substitute
for
on the right hand side of (2) to obtain
where
called the generator of the infinitesimal canonical transformation.
The transformation is canonical to first order since
