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Classical Mechanics: Canonical transformation problem

  1. Nov 30, 2006 #1
    1. The problem statement, all variables and given/known data

    Show directly that the transformation; Q=log(1/q*sinp), P=q*cotp is canonical.


    2. Relevant equations

    Since these equations have no time dependence, the equations are canonical if (with d denoting a partial derivative)

    dQ_i/dq_j = dp_j/dP_i, and dQ_i/dp_j = -dq_j/dP_i

    3. The attempt at a solution

    With

    Q=log(1/q*sinp), dQ/dq = -1/q

    P=q*cotp => p=tan^-1(q/P), dp/dP = -q/(p^2+q^2).

    The first problem I encounter is that -1/q not= -q/(p^2+q^2).

    With dQ/dp = cotp, and -dq/dP = -1/(dP/dq) = -cotp

    so, cotp not= -cotp.

    :mad: :mad:
     
    Last edited: Nov 30, 2006
  2. jcsd
  3. Dec 1, 2006 #2

    OlderDan

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    Science Advisor
    Homework Helper

    It worked for me. I solved for

    cosp = Pe^Q

    q = sinp/e^Q = sqrt[1 - (Pe^Q)²]/e^Q

    and took the derivatives.
     
  4. Dec 2, 2006 #3
    I know what I did now. For partial derivatives, dx/dy not= 1/(dy/dx). I falsely made that assumption.
     
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