# Homework Help: Relations between the four types of generating functions for a canonical transformati

1. Nov 19, 2009

### buffordboy23

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.