Canonical Transformations, Poisson Brackets

[UniPhysics90]

This isn't actually a homework problem, but a problem from a book, but as it's quite like a homework problem I thought this forum was probably the best place for it.

Homework Statement

Consider a system with one degree of freedom, described by the Hamiltonian formulation of classical mechanics in terms of the coordinate q, and the canonically conjugate momentum, p. A canonical transformation is applied, such that the transformed Hamiltonian is described in terms of the transformed coordinate Q and the transformed momentum P. Explain whether P will be canonically conjugate to Q, and how Poisson brackets may be used to check this.

Homework Equations

The Attempt at a Solution

P will be canonically conjugate to Q (it says so in a book!) and possibly that Poisson brackets will give a constant value (books seem to suggest either 0 or 1?). If anyone can explain this further I'd really appreciate it!

Thanks

 

[kloptok]

As far as I remember, we have for a canonical transformation
\{q,p\}=\{Q,P\}=0,
and
\{q,q\}=\{Q,Q\}=\{p,p\}=\{P,P\}=1,
i.e. the Poisson brackets are conserved in a canonical transformation.

Even if this property isn't directly mentioned, I found the wiki article on canonical transformations quite good: http://en.wikipedia.org/wiki/Canonical_transformation"

I guess the fact that the Poisson brackets are conserved is a consequence of the fact that the dynamics aren't changed in a canonical transformation.