Verifying a Canonical Transformation with Poisson Brackets

[darida]

Homework Statement

Show that

$Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω})$
$Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω})$
$P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2})$
$P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2})$

(where mω is a constant) is a canonical transformation by Poisson bracket test. This requires evaluating six simple Poisson brackets.

2. The attempt at a solution

$[Q_{1},P_{1}]=[\frac{∂Q_{1}}{∂q_{1}}\frac{∂P_{1}}{∂p_{1}}-\frac{∂Q_{1}}{∂p_{1}}\frac{∂P_{1}}{∂q_{1}}]+[\frac{∂Q_{1}}{∂q_{2}}\frac{∂P_{1}}{∂p_{2}}-\frac{∂Q_{1}}{∂p_{2}}\frac{∂P_{1}}{∂q_{2}}]$
.
.
.
etc

Correct?

 

[vanhees71]

Yes,...

 

[darida]

Thank you!