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Confusion about canonical transformations

  1. Jul 23, 2009 #1
    I have read that "any" function of new and old coordinates may be used to generate a canonical transformation. However, it seems there must be some restriction on the generating function.

    For example, if you try to define a generating function for a system with 2 degrees of freedom:

    G(x,y,X,Y) = sin(X+Y) + sin(x+y)

    where x,y are the old coordinates and X,Y the new, and take the appropriate partial derivatives, you seem to end up with:

    px = cos(x+y)
    py = cos(x+y)

    Px = cos(X+Y)
    Py = cos(X+Y) (Px,Py are the new momenta and px,py the old)

    Not only does this seem to tell you nothing about a "transformation", but it seems to imply px = py, Py = Px.

    What is going on here?
  2. jcsd
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