representation
is
and
in the new
representation the area is
University Physics Notes: Classical Mechanics – Generating Functions For Transformations of Coordinates
A general coordinate transformation requires two functions of two variables for it's specification, but a canonical transformation, being area preserving, can be specified in terms of a single function. Such a function is called a generating function.
The area of a closed phase curve in the originalrepresentation
isand
in the new
representation the area is
Since the transformation is canonical
(1).
We are using different coordinates for each integral. For a suitable
canonical transformation we may take
and
as
independent variables. We can no long take
and
as
independent, but must express them as functions of
and
(1)
becomes
Since this equation is true for all closed curves
the
integrand must be a derivative of some function
Equating coefficients of
and
gives
and
F is a generating function for the transformation
A necessary and suffient condition that a function F generate a
canonical transformation is that
Example: Show that F(Q,q) generates a canonical transformation and find the transformation.
so
the transformation is canonical.
We solve these to giveandin
terms ofandto
give
