# Most General Canonical Transformation?

1. Feb 1, 2008

### strangerep

In classical Hamiltonian mechanics, the concept of a canonical transformation ("CT")
preserving the form of Hamilton's eqns is well known. Textbooks (e.g., Goldstein)
distinguish "restricted" CTs that just mix the q's and p's (generalized coordinates and
generalized momenta respectively). These form the usual group of symplectomorphisms
on Poisson manifolds discussed in the more high-brow books.
Textbooks also mention "extended" CTs that involve the time parameter.

But I can also imagine transformations that only mix H and t, but still preserve the
form of Hamilton's eqns (assuming the various functions and derivatives are
sufficiently well-behaved).

So... what are the most general CTs, and what group is formed from them?

TIA.