Time Dependent Canonical Transformations
Time dependent canonical transformations are very similar to the time independent case, and as with the time independent case, the preservation of area is a part of the analysis. Consider the transformation
The condition for such a transformation to preserve Hamiltonian form is that the Jacobian is a non zero constant.
Suppose therefore that
not necessarily Hamiltonian, is transformed to
where
The Jacobian will depend explicitly on the time in general and in terms of the new coordinates
we have
If in the original coordinates
the system is Hamiltonian then
and
From this it can be deduced that for Hamiltonian form to be preserved we must have
The properties of generating functions are carried over from the time independent case, though now the generating function are now also functions of time, with a generating function being determined up to the addition of a function of time alone. The relations between the generating functions and the old and new variables remain the same. For a generating function
For all the generating functions,
up to the addition of a function of time,
For example,
But
and
so