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Relations between the four types of generating functions for a canonical transformati

  1. Nov 19, 2009 #1
    1. The problem statement, all variables and given/known data

    Problem 9.7(a) of Goldstein, 3rd edition: If each of the four types of generating functions exists for a given canonical transformation, use the Legendre transformations to derive the relations between them.

    2. Relevant equations

    1. F = F1(q,Q,t)
      p = partial(F1)/partial(q)
      P = - partial(F1)/partial(Q)
    2. F = F2(q,P,t) - QP
      p = partial(F2)/partial(q)
      Q = partial(F2)/partial(P)
    3. F = F3(p,Q,t) + qp
      q = - partial(F3)/partial(p)
      P = - partial(F3)/partial(Q)
    4. F = F4 (p,P,t) +qp - QP
      q = - partial(F4)/partial(p)
      Q = partial(F4)/partial(P)

    3. The attempt at a solution

    So this problem really isn't clear to me. What "relations" are they talking about? Goldstein gives a table on page 373 that F = F1(q,Q,t), F = F2(q,P,t) - QP, etc. Clearly, F1(q,Q,t) = F2(q,P,t) - QP, and so on for a total of six relations for the four types of generating functions. But, I wonder if this is what the question is really asking, b/c it's so simple.

    By hypothesis, F1, F2, F3, F4 exist. So, we can use a Legendre transformation to transform, say, (q,Q,t) --> (q,P,t), which just gives us F1(q,Q,t) = F2(q,P,t) - QP as already noted, plus

    partial(F1)/partial(q) = partial(F2)/partial(q)

    and

    partial(F1)/partial(t) = partial(F2)/partial(t).

    Any clarification would be appreciated thanks.
     
  2. jcsd
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