Given a canonical transformation, how does one find its type?

[mjordan2nd]

I'm given the following transformation

$$X=x \cos \alpha - \frac{p_y}{\beta} \sin \alpha$$
$$Y=y \cos \alpha - \frac{p_x}{\beta} \sin \alpha$$
$$P_X=\beta y \sin \alpha + p_x \cos \alpha$$
$$P_Y=\beta x \sin \alpha + p_y \cos \alpha$$

and I'm asked to find what type(s) of transformation it is. I'm not sure how to go about doing this without being given a generator. Do I basically try and find a generator? If so, how do I do this. Do I start from Hamilton's principle and try the definition of each type of generator?

 

[antibrane]

Do I basically try and find a generator?

Not quite, that is a lot more work. You don't have to actually find the generator to determine what type it should be.

If so, how do I do this. Do I start from Hamilton's principle and try the definition of each type of generator?

Say, for the sake of argument, the generator $F$ is a function of $p_x,y,X,Y$ (it's not) so that $F=F(p_x,y,X,Y)$. This means that you would be able to write the other coordinates (namely $p_y,x,P_x,P_y$) in terms of these ones. I can see that this type of generating function won't work since when I try to write $P_y$ I find (you'll need to invert the transformation to see this):
$$P_y = x\beta \sin\alpha + p_y\cos\alpha = \left(X\cos\alpha + \frac{\sin\alpha}{\beta}P_y\right) \beta\sin\alpha + p_y\cos\alpha$$
and that simplifies to $P_y=X\beta\tan\alpha + p_y\sec\alpha$. Therefore, I cannot write $P_y$ as a function of only those chosen coordinates. Try something else and see if you can find a coordinate set that works---then you will know what type of generating function you can have.