Infinitesimal Canonical Transformations
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 parameterwhich 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 transformationswhich are close to the identity transformation:
This has the generating functionso a Hamiltonian is perturbed slightly:whereis a small parameter, we would expect a transformation close to the identity:to have a generating function of the formwhereis some well behaved function ofand
The canonical transformation produced by this generating function is(1)
Sincewe may substituteforon the right hand side of (2) to obtain
wherecalled the generator of the infinitesimal canonical transformation.
The transformation is canonical to first order since