How to Find a Specific Transformation for a Specific Hamiltonian?

[Cosmossos]

Homework Statement

Question 3b from the following file:
http://phstudy.technion.ac.il/~wn114101/hw/wn2010_hw07.pdf

I know I need to find a generating function for this spacific transformation. but I don't know how to find it, I mean , how I find a spacific transformation for a spacific hamiltonain?
thnaks

 

[gabbagabbahey]

Do you know what the Hamiltonian of a one dimensional harmonic oscillator looks like?

 

[Cosmossos]

It's the classic expression (H=p^2/2m+kx^2/2)

 

[gabbagabbahey]

It's the classic expression (H=p^2/2m+kx^2/2)

Right, so (using $P$ and $Q$ instead of 'p' and 'x'), you are looking for a canonical transformation $Q=Q(q,p)$ and $P=P(q,p)$, for which [tex]\frac{1}{2}\left(\frac{1}{q^2}+p^2q^4\right)\to \frac{P^2}{2m}+\frac{kQ^2}{2}[/itex] (give or take a constant)...what does the fact that the transformation is canonical tell you?[/tex]