Moose352 said:
I'm not sure I understand the use of generating functions in canonical transformations. In particular, why are there four basic canonical transformations? It isn't true that any canonical transformation is one of the four basic types, so what makes them special over any other transformation.
Canonical transformations are smooth mappings from the (p,q) into the (P,Q) space which satisfy certain properties (the deeper reason behind is the "conservation of symplectic structure", but I only write that here to show off somewhat
). So not just all mappings from (p,q) to (P,Q) will do.
Now, of course, by the implicit function theorem, if you define a mapping from, say, (p,Q) into (P,q), you can partly inverse this relation, and you ALSO define a mapping from (p,q) into (P,Q).
So the "4 different types" are simply different ways of looking upon mappings from (p,q) into (P,Q) through the implicit-function theorem. The mapping itself is not different, we've just written it implicitly.
The nice thing about these 4 "types" is that they give us a simple way to generate canonical transformations, and we have 4 different ways of doing so. But the canonical transformations themselves couldn't care less of how they are written down, in a way.
Also, why is the generating fuction written in terms of both the new and the old coordinates? Since the old and new are related by the transformation, shouldn't it be possible to write the generating function solely in terms of the old or the new?
Doesn't work, unfortunately. You will have to inverse some equations in order to find the mapping (p,q) into (P,Q) in all 4 cases.
Also, is there any way to find a generating function given the transformation explicitly?
I'm not sure about this. I would think that if you write the new Hamiltonian as a function of, say, p and Q, that you obtain the generating function, but typing from the top of my head here, this might as well not be correct.