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Poisson Bracket

  1. Oct 30, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that

    [itex]Q_{1}=\frac{1}{\sqrt{2}}(q_{1}+\frac{p_{2}}{mω})[/itex]
    [itex]Q_{2}=\frac{1}{\sqrt{2}}(q_{1}-\frac{p_{2}}{mω})[/itex]
    [itex]P_{1}=\frac{1}{\sqrt{2}}(p_{1}-mωq_{2})[/itex]
    [itex]P_{2}=\frac{1}{\sqrt{2}}(p_{1}+mωq_{2})[/itex]

    (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

    [itex][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}}][/itex]
    .
    .
    .
    etc

    Correct?
     
  2. jcsd
  3. Oct 30, 2013 #2

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    Yes,...
     
  4. Oct 30, 2013 #3
    Thank you!
     
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