Finding a Generating Function From a Given Canonical Transformation
Generating functions have many uses and it is useful to be able to construct one from a given transformation. We can do this from the defining relationship between the generating function and the transformation.
ExampleShow that the transformationis canonical and find a generating function F(Q,q).
We use the fact thatandFor the transformation to be canonical it suffices to show, using the equivalence of mixed partial derivativesthat(1)
Henceandso (1) is satisfied and the equation is canonical.
To find the generating functionwe use the relationships andto give
whereis an arbitrary function of
Differentiating this expressing with respect togives the relationship above forso we may setand