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.
Example
Show that the transformation
is canonical and find a generating function F(Q,q).
We use the fact that
and
For the transformation to be canonical it suffices to show, using the equivalence of mixed partial derivatives
that
(1)
Hence
and
so (1) is satisfied and the equation is canonical.
To find the generating function
we use the relationships
and
to give
where
is an arbitrary function of
Differentiating this expressing with respect to
gives the relationship above for
so we may set
and