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Poisson brackets for simple harmonic oscillator

  1. Jun 23, 2013 #1
    1. The problem statement, all variables and given/known data

    Considering the Hamiltonian for a harmonic oscillator:


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

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

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

    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:


    Also, from the definition of Poisson Brackets,

    3. The attempt at a solution

    Well, by definition,


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


    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! :smile:
    Last edited: Jun 23, 2013
  2. jcsd
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