Canonical transformation

  1. 1. The problem statement, all variables and given/known data

    Consider a canonical transformation with generating function

    [tex]F_2 (q,P) = qP + \epsilon G_2 (q,P)[/tex],

    where [tex]\epsilon[/tex] is a small parameter.
    Write down the explicit form of the transformation. Neglecting terms of order [tex]\epsilon^2[/tex] and higher,find a relation between this transformation and Hamilton's equations of motion, by setting [tex]G_2=H[/tex] (why is this allowed?) and [tex]\epsilon = dt[/tex].


    2. The attempt at a solution

    I think the transformation equations are

    [tex]\delta p = P - p = -\epsilon \frac{\partial G_2}{\partial q}[/tex]

    and

    [tex]\delta q =Q-q=\epsilon \frac{\partial G_2}{\partial q}[/tex]


    but how can I solve the last part? Can I just say that with the use of H and dt the equations can be written as

    [tex]\dot{p}=-\frac{\partial H}{\partial q}[/tex]

    and

    [tex]\dot{q}=\frac{\partial H}{\partial P}[/tex]

    which are the Hamiltonian equations of motion? And why is this allowed?

     
    Last edited by a moderator: Dec 17, 2006
  2. jcsd
  3. vanesch

    vanesch 6,236
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    Ooops ! :redface: :redface:

    I'm terribly sorry, instead of replying, I erroneously edited your post ! :bugeye:
     
  4. Yes it should be a P, not a q. I get the relations

    [tex]Q = q + \delta q[/tex]

    and

    [tex]P = p + \delta p[/tex]

    but why is it allowed to use G = H?
     
  5. vanesch

    vanesch 6,236
    Staff Emeritus
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    Gold Member

    You have the choice ! Every function G(q,P) (which is smooth enough) will generate a canonical transformation. So you may just as well use H(q,P), and then - that's the whole point - the transformation equations from (q,p) into (Q,P) give you simply the genuine time evolution where epsilon is the small time step. For an arbitrary G that isn't the case of course, you've just transformed your coordinates (q,p) in some other (Q,P). But for G = H, you've transformed the coordinates (q,p) in what they will be, a small moment later !
    The reason for that is that your transformation equations you've found for an arbitrary G are what they are, and become the Hamilton equations of motion when you pick G to be equal to H.

    Now, strictly speaking we should write H(q,P) instead of H(q,p), but we can replace the P by p here, because they are only a small amount different.
     
  6. and it also preserve the fundamental poisson brackets....

    its just a quick calcolus {Q,P}
     
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