# Derive the generating function for canonical transformation of type $F_3$

I'm working on some practice questions and I am a bit confused with this one:

Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition derive the analogous result for the generating functions of type $F_3(q,P)$.

I must admit, I am a little confused with the Legendre transform and how the other generating functions are derived.

Here is what I have tried by following my textbook,

Applying the Legendre transform with respect to the new coordinates and momenta to $F_1$:

$$d(\sum QP + F_1) = PdQ + QdP + pdq-PdQ = QdP + pdq = dF_2$$

Which seems correct for $F_2$. (I'm not even sure if I have done this correctly, I'm pretty much just following my textbook). To get $dF_3$, my textbook just says perform the Legendre transform with respect to the 'other' variables?

However, I'm not sure what this means?. I know I am expecting to get $-qdp-PdQ$, but just not too sure about what I am doing.

I think I may have gotten a little further with my understanding. If I take the legendre transform of $F_1$ with respect to the old coordinate $q$ and it's conjugate momenta, then:

$$F_1 - pq = F_3$$ so, $$d(F_1 - pq) = pdq-PdQ-qdp-pdq = -qdp-PdQ = dF_3$$

Which seems to be correct. But then isn't

$$d(F_1 - QP) = pdq-PdQ-QdP-PdQ$$ which is not $dF_2$. Where am I going wrong? How would I say, derive $dF_4$? By definition of the legendre transform, is it true that for instance if $G$ is the legendre transform of $F$, i.e. $G=F-wy$, the pair $w,y$ must always be a conjugate pair (i.e $w=\frac{\partial f}{\partial y})$?

• Hi JackReacher. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. – Qmechanic Oct 13 '14 at 11:17

Intuitively, Legendre transform is just "integrate by parts". So, from $pdq - PdQ =F_1$, we have $pdq -d(PQ)+QdP =dF_1$, i.e., $pdq +QdP =d(F_1+PQ) \equiv dF_2$. The "Transform" means after "integration by parts", we need to change the independent variables. For example, in $F_1$, $p=p(q,Q)$ but in $F_2$, $p=p(q,P)$. So we need to solve $Q = Q(q,P)$ and insert in to $p=p(q,Q(q,P))$ to get $p(q,P)$.

• Thanks for your help, but I'm still confused, and to be honest still confused about the definition of a legendre transform. My understanding so far is that the definition of the Legendre transform of a function $f(x,y)$ is $G = f-wy$ where $w= \frac{\partial f}{\partial y}$ - and then this can be done with a few combinations of variables. I'm a bit confused as to why differentials come into play? – JackReacher Oct 13 '14 at 11:24
• we have $df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy \equiv u dx + wdy$ – Deliang Zhong Oct 13 '14 at 11:46
• Thanks, I think I have gotten a little further with my understanding, could you look at my edit in my question, if you are able to help me? – JackReacher Oct 13 '14 at 12:09
• $F_2 = F_1 +PQ$ rather than minus... – Deliang Zhong Oct 13 '14 at 20:08
• So for a Legendre transform, does the sign not matter? In that sense, how is one supposed to know how to derive the other generating functions? Just by guess/memorization? For example, if I wanted to derive $F_4$, I have already done $F_2 = F_1 + PQ$, $F_3 = F_1 - qp$, aren't these the only conjugate variables? Can you do also $pP$? But then aren't these not conjugate to each other? Thanks so much for your help. – JackReacher Oct 13 '14 at 20:57

This here is not exactly an answer, but a clarification of why Legendre transforms are not needed to get this result. Using exterior derivatives is much easier.

If you have $pdq - PdQ = dF$, take the exterior derivative on both sides to get $dp \wedge dq - dP \wedge dQ = 0$ (the exterior derivative of a differential is zero). The wedge product is antisymmetric, so $dp \wedge dq - dP \wedge dQ = dp \wedge dq + dQ \wedge dP =0$. Take now the 1-form whose exterior derivative gives us $dp \wedge dq + dQ \wedge dP =0$, that is, $pdq + QdP = dG$. This means that

$$\partial_q G(q,P) = p \\ \partial_P G(q,P) = Q ,$$

and that's the answer. The Legendre transform is just a cumbersome way of transforming variables in 1-forms instead of using the much simpler structure of the 2-forms.