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Extended Canonical Transformation

  1. Jun 2, 2007 #1
    1. The problem statement, all variables and given/known data

    The transformation equations are:

    [tex]
    Q=q^\alpha cos(\beta p)
    [/tex]

    [tex]
    P=q^\alpha sin(\beta p)
    [/tex]

    For what values of [itex]\alpha[/itex] and [itex]\beta[/itex] do these equations represent an extended canonical transformation?

    3. The attempt at a solution
    Well, just for a start, what is the condition for a canonical transformation to be an extended canonical transformation?
     
  2. jcsd
  3. Jun 3, 2007 #2
    I got a solution but it doesn't seems very satisfactory

    I believe that the condition for a canonical transformation to be an extended canonical transformation is that

    [tex]
    PQ^\prime = \lambda p q^\prime
    [/tex]

    But I am not 100% sure.

    Then I do

    [tex]
    PQ^\prime = q^\alpha sen(\beta p)(\alpha q^{\alpha-1}q^\prime cos(\beta p)-q^\alpha \beta sen(\beta p) p^\prime)
    [/tex]

    Now I do the small angle approximation saying that [itex]\beta[/itex] is small. Is this point that I am not sure because the problem statement don't gives any information about this approximation.
    However, doing this I get:

    [tex]
    PQ^\prime \simeq \alpha q^{2\alpha -1} \beta p q^\prime - \beta^3 q^{2\alpha} p^2 p^\prime
    [/tex]

    Using

    [tex]
    \beta^3 \simeq 0
    [/tex]

    and

    [tex]
    \alpha=\frac{1}{2}
    [/tex]

    I get

    [tex]
    PQ^\prime \simeq \frac{\beta}{2} p q^\prime
    [/tex]

    At least I get the condition of an extended canonical transformation

    Am I thinking right?
    Thanks for any suggestion.
     
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