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Calculating generator function in canonical transformation

  1. Apr 16, 2008 #1
    I'm searching for an example of how to find out generator function for a canonical transformation, when new canonical variables are given in terms of old variables. Any help is greatly appreciated.
     
  2. jcsd
  3. Apr 17, 2008 #2
    See any classical mechanics textbook, such as Goldstein.
     
  4. Aug 30, 2008 #3
    Let us take, for example, a generator function of type 2 (please see goldstein), i.e , F2(q,P,t) function of the old coordinates (q) and the new momenta (P), and consider the following non-relativistic transformation:

    x=x'+Vt , x'- new (spacial) coordinate
    t=t' , (time remains the same)

    as you can see, you will also have to know how the momenta transforms (in order to determine F2)

    px=mV+px' , px' new momentum (px'=P, if you prefer)

    For the type 2 generator function,

    p=dF2/dq (partial derivative)
    Q=dF2/dP (partial derivative)

    So, all you have to do is integrate, i.e

    px is your old momentum, therefore F2 = (mV+px')x + A
    now lets determine A,

    "A" is not a constant because if you take dF2/dP it won't equal the new momenta. There's still a " - Vtpx' "
    lacking in the equation (with the minus sign included).
    Well if that quatity is missing , all we have to do is add it to the equation (A=-Vtpx').

    So here it is, the F2(q,P,t) generator function for this non-relativistic transformation is:

    F2 = (mV+px')x - Vtpx'

    I down know if there is another (better) way to do it, but I hope it helps.

    Best regards

    Rico B.
     
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