Generating function for canonical transformation

[Loxias]

Homework Statement

Given the transformation

$$Q = p+iaq, P = \frac{p-iaq}{2ia}$$

Homework Equations

find the generating function

The Attempt at a Solution

As far as I know, one needs to find two independent variables and try to solve. I couldn't find such to variables.

I've tried expressing it in terms of F(Q,P), and F(q,p) but always had one more term in the equation that prevented me from getting to $$H(q,p) = -H(Q,P) + \frac{\partial F}{\partial t}$$

I'm pretty clueless as to what is needed here. Can someone help me get started?

Thanks.

 

[Loxias]

Ok, this is what I did :

$$Q = 2ia(P + 2q) , P = \frac{p-iaq}{2ia}$$
which means that Q and p are independent coordintes, which means the generating function will be of the third kind, $$F_3(Q,p)$$.

for the third kind,
$$q = -\frac{\partial F_3}{\partial p} = \frac{p}{ia}-2P$$
$$P = -\frac{\partial F_3}{\partial Q} = \frac{Q}{2ia}-2q$$

from the first equation we get
$$F_3 = 2pP - \frac{p^2}{2ia} + F(Q)$$
and from the second
$$F_3 = 2Qq - \frac{Q^2}{4ia} + F(p)$$

summing both I get
$$F_3 = 2Qq + 2pP - \frac{1}{2ia} (p^2 + \frac{Q^2}{2})$$

Does this seem right??