Canonical Transformation - Generating Function Approach

Generating Function Approach

To guarantee a valid transformation between and, we may resort to an indirect generating function approach. Both sets of variables must obey Hamilton's principle. That is the Action Integral over the Lagrangian and respectively, obtained by the Hamiltonian via ("inverse") Legendre transformation, both must be stationary (so that one can use the Euler-Lagrange equations to arrive at equations of the above-mentioned and designated form; as it is shown for example here):

To satisfy both variational integrals, we must have

This equation holds because the Lagrangian is not unique, one can always multiply by a constant and add a total time derivative and yield the same equations of motion (see for reference: http://en.wikibooks.org/wiki/Classical_Mechanics/Lagrange_Theory#Is_the_Lagrangian_unique.3F).

In general, the scaling factor is set equal to one; canonical transformations for which are called extended canonical transformations. is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.

Here is a generating function of one old canonical coordinate ( or ), one new canonical coordinate ( or ) and (possibly) the time . Thus, there are four basic types of generating functions, depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation is guaranteed to be canonical.