1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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)


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

    Any clarification would be appreciated thanks.
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted