- Canonical Transformations
It seems to me that this section contains two things, the basic idea--transform to K(P) alone, and some details of the tranformation machinery. I think it would be easier to understand the point if you separated the first from the second. A new section before Canonical transformations, perhaps containing the material up to (3). The Canonical transformation section could be a subsection.
- Overall structure
I think the article should have more initial motivation. The statement that "here is the H-J" equation doesn't give much help as to why one would want to study it. Talk about how it is one way to attempt to solve a system? You mention that there are applications but don't have any details or references to these. It would be nice to have a section on applications.
I would like to see at least one example. There are simple cases where the H-J equation can be solved exactly, why not give one of these? Perhaps the only easy case is the Harmonic oscillator?
You could talk a bit more about the integrable case and show how the solution works easily there. Or more generally that the solution, if it exists, implies integrability. The scholarpedia article on integrability has not yet been written, so you shouldn't invent this wheel here, but a bit more discussion would be useful.
First response to Reviewer A
I also felt the lack of examples, but was concerned about the article getting too long. I have added a section giving the example of planar motion in a central potential, solved by separation of variables. This is more informative than the harmonic oscillator and not much harder. I put it in the context of the general method of separation of variables, and use it to illustrate the abstract treatment of preceeding sections.
As for applications, I did mention some and gave some references, but in a rather scattered manner. I can see that it would be better to list and describe applications in a more orderly manner, and to give more thorough references. To this end I expect to add material on the WKB method and the eikonal equation, two of the chief applications. I am still working on the organization and bibliography.
In the meantime, I would appreciate the reviewer's comments on the new section "Solution of classical problems by separation of variables".
I like the new section very much. Let me know when you consider the remainder complete so I can finish the review.