# Poisson brackets for simple harmonic oscillator

1. Jun 23, 2013

### Siberion

1. The problem statement, all variables and given/known data

Considering the Hamiltonian for a harmonic oscillator:

$H=\frac{p^2}{2m}+\frac{mw^2}{2}q^2$

We have seen that the equations of motion are significantly simplified using the canonical transformation defined by $F_1(q,Q)=\frac{m}{2}wq^2cot(Q)$

Show explicitly that between both coordinates, the following identities are satisfied

1) $[q,p]_{qp}=[Q,P]_{QP}$
2) $[q,p]_{QP}=[q,p]_{QP}$
3) $[Q,P]_{qp}=[Q,P]_{QP}$

where [ ] denote Poisson Brackets

2. Relevant equations

This is a well known Hamiltonian frequently used to introduce canonical transformations in many books (e.g. Goldstein 378). Given the generating function presented above, we find the following expresions for the new coordinates:

$p=\sqrt{2Pmw}cos(Q)$
$q=\sqrt{\frac{2P}{mw}}sin(Q)$

Also, from the definition of Poisson Brackets,
$[u,v]_{qp}=\frac{\partial{u}}{\partial{q}}\frac{\partial{v}}{\partial{p}}-\frac{\partial{u}}{\partial{p}}\frac{\partial{v}}{\partial{q}}$

3. The attempt at a solution

Well, by definition,

$[q,p]_{qp}=1=[Q,P]_{QP}$

Also, using the expressions for p and q presented above, we have

$[q,p]_{QP}=cos^2(Q)+sin^2(Q)=1=[q,p]_{qp}$

Am I missing some fundamental formalism in here? I'm far from confident about this topic, so I'm not sure I followed the right procedure.

Also, for 3) I was thinking about obtaining expressions for Q and P straight from the presented expressions for p and q, and then just apply the same principle. What do you think about this?

Thanks for your help and time!

Last edited: Jun 23, 2013
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