Show
that the transformation
is
canonical and find a generating function F(Q,q).
University Physics Notes: Classical Mechanics – 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 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