Deriving Relations Between Generating Functions via Legendre Transformations

[buffordboy23]

Homework Statement

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.

Homework 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)

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.

 

[Jhero]

Thank you for your question on the relations between the four types of generating functions in Goldstein's book. The question is asking for a more detailed derivation of the relations between the four types of generating functions, using Legendre transformations. The six relations that you have mentioned are correct, but the question is asking for a more comprehensive explanation of how these relations are derived.

To start, we can define the four types of generating functions as follows:

F1(q,Q,t) = F(q,Q,t)
F2(q,P,t) = F(q,P,t) - QP
F3(p,Q,t) = F(p,Q,t) + qp
F4(p,P,t) = F(p,P,t) + qp - QP

Now, using the Legendre transformation, we can transform the variables (q,Q,t) to (q,P,t) as follows:

F1(q,Q,t) = F2(q,P,t) - QP
p = partial(F1)/partial(q)
P = - partial(F1)/partial(Q)

Similarly, we can transform the variables (q,Q,t) to (p,Q,t) as follows:

F1(q,Q,t) = F3(p,Q,t) + qp
q = - partial(F1)/partial(p)
P = - partial(F1)/partial(Q)

And lastly, we can transform the variables (q,Q,t) to (p,P,t) as follows:

F1(q,Q,t) = F4(p,P,t) + qp - QP
q = - partial(F1)/partial(p)
Q = partial(F1)/partial(P)

Using these transformations, we can derive the six relations mentioned earlier. However, we can also derive additional relations by using the properties of Legendre transformations, such as:

1. Inverse transformations: If we transform from (q,Q,t) to (q,P,t) and then back to (q,Q,t), we should end up with the original function F1(q,Q,t). Similarly, if we transform from (q,Q,t) to (p,Q,t) and then back to (q,Q,t), we should also end up with the original function F1(q,Q,t).

2. Chain rule: If we transform from (q,Q,t) to (q,P,t) and then to (p,P,t), we should get the same result as directly transforming from (q,Q,t) to (p,P