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General Form of Canonical Transformations

  1. Apr 12, 2016 #1
    1. The problem statement, all variables and given/known data
    How do I go about finding the most general form of the canonical transformation of the form
    Q = f(q) + g(p)
    P = c[f(q) + h(p)]
    where f,g and h are differential functions and c is a constant not equal to zero. Where (Q,P) and (q,p) represent the generalised cordinates and conjugate momentum in the new and old system

    2. Relevant equations
    {Q,Q}={P,P}=0 {Q,P}=1

    3. The attempt at a solution
    I arrived at a function

    c.f'(q)[h'(p)-g'(p)]=1

    I don't know how to get further to prove canonicity
     
  2. jcsd
  3. Apr 12, 2016 #2

    andrewkirk

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    There are two independent variables in this equation: p and q, and we require it to hold for all values of p and q. That should enable us to radically narrow down the possibilities for functions f, g and h.

    Try partial differentiating both sides of the equation with respect to q to get one equation and then wrt p to get another.
    What do those two equations tell you about the functions f and (h-g)?
     
  4. Apr 13, 2016 #3
    I did that and got to "see the attachment"
    But I am not so sure if I did it right
    The rest of the question see solve the inverse of the canonical transformation: express q, p in terms of Q and P. The actual question is attached see attachment(it is the second question)
     

    Attached Files:

  5. Apr 13, 2016 #4

    andrewkirk

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    I'm sorry but that photo is way too hard to read. Try typing it in using latex. This post is a primer to get you started. If you're studying physics then any time spent learning latex is a very good investment.
     
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