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Canonical Transform proof

  1. Mar 3, 2012 #1
    Consider the following change of variables in phase space f: maps the reals and is smooth and invertable change of coordinates Q=f(q), q = f-1(Q). Given f, define a change of variables on phase space (q,p) -> (Q,P) by the pair of relations

    Pj = (∇f-1)Tjk(f(q))pk

    q runs from 1 to n

    show that its canonical.

    I know that for this to be canonical

    (dQi/dqj)(q,p) = (dpj/dPi)(Q,P)

    (dQi/dpj)(q,p) = -(dqj/dPi)(Q,P)

    i'm having a couple problems, is f the generating function that i have to find explicitly?

    Can i use the sympletic method such that MJM^T = J

    what is the point of the transpose for the (∇f-1)Tjk part for? i thought f had to be symmetric.
  2. jcsd
  3. Mar 4, 2012 #2
    for Pj it can equal


    =(∇f)T *-1jkqpk

    but for that to be true then (∇f-1)Tjk has to be symmetric therefore the transpose dissapears

    looking at my notes, i think this is supposed to be the associated point transformation in phase space
    Last edited: Mar 4, 2012
  4. Mar 5, 2012 #3
    Guess i can answer my own question... i knew how to do it but i had an error in my notes

    [dQ, dP)T = [{dQ/dq, dQ/dp}, {dP/dq, dP/dp}]*[dq, dp] = Mij*[dq, dp]

    to prove its a canonical transformation

    MJMT = J where J = [{0,I},{-I,0}] and T represents the transpose

    If i do the matrix multiplication and say:

    now i will let {a,b} where a and b are some arbitrary coordinates be the poisson brackets

    MJMT = [(0,{Q,P}), (-{Q,P},0)]

    It is known that {P,P} = 0 = {Q,Q} and {Q,P} = δij

    Using the following vvvv

    and there you have it
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