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Order of steps on Hamiltonian canonical transformations

  1. Feb 3, 2014 #1
    Just a little doubt.

    When we are performing a canonical transformation on a Hamiltonian and we have the equations of the new coordinates in terms of the old ones we have to find the Kamiltonian/new Hamiltonian using [itex]K = H +\frac{\partial G}{\partial t}[/itex]. My question is: do we have to derive the generating function before or after substituting the old coordinates for the new ones?

    For example, in a [itex](q, p)\rightarrow(Q, P)[/itex] that last derivative would be [itex]\frac{\partial G(q, p, t)}{\partial t}[/itex] or [itex]\frac{\partial G(q(Q, P, t), p(Q, P, t))}{\partial t} =\frac{\partial G(Q, P, t}{\partial t}[/itex]?
     
  2. jcsd
  3. Feb 3, 2014 #2

    TSny

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    The generating function is always a function of both old and new coordinates. So it would not be of the form ##G(q,p,t)## since that would be a function of only the old coordinates.

    There are four possible forms: ##G(q,Q,t)##, ##G(q,P,t)##, ##G(p,Q,t)##, and ##G(p,P,t)##.

    Suppose you have the form ##G(q,P,t)##. Then ##q## and ##P## would be considered independent variables in ##G##.

    When taking the partial derivative ##\frac{\partial G(q,P,t)}{\partial t}## you would hold ##q## and ##P## constant while taking the derivative with resepct to ##t##.

    Afterwards you could substitute for q in terms of ##Q## and ##P## in order to express ##\frac{\partial G(q,P,t)}{\partial t}## in terms of the new coordinates.
     
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