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I have a problem understanding the criteria of an canonical transformation. I am preparing for my exam and came across this question:

For which $A$, $B$, $C$, $D$ is: $Q = A q^2+B p^2, \quad P = Cq^2+Dp^2$ a canonical transformation of $(q,p) \to (Q,P) $?

I know that it must hold: $\{Q,P\} = 1$

My Calculation results in $AD-BC = \frac{1}{4}pq$. But now i wonder if this is enough. Intuitively i think it should further hold that the mapping $(q,p) \to (Q,P) $ is invertible which is not the case, due to the the four Solutions $ q = \pm 2\sqrt{DQ-BP} $ and $p = \pm 2\sqrt{PA-QC}$.

I am currently looking thorugh my lecture notes and the books. Still i cannot find anything regarding if the transformation must be bijective in order to be a canonical transformation.

My Question: Which assumption(s) must $(q,p) \to (Q,P)$ fulfill in order to be a canonical transformation.

EDIT: I was just looking at my calculation. I did a stupid mistake. The calculation does not result in: $AD-BC = \frac{1}{4}$ it should be $$AD-BC = \frac{1}{4}pq$$ and therefore this transformation can't be canonical. I corrected it above. However this mistake made me question about the assumptions for $(q,p) \to (Q,P)$ in order to be canonical. I finally got my answer in the Book "Theoretische Physik" written by W. Nolting. It's basically the same as in the accepted answer, Nolting further provides a clear proof of criteria for canonical transformations. So, whoever has the same question as i did, might find an answer there.

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One condition is just the one you wrote, the other three are that the remaining Poisson commutators vanish. These conditions also imply that the Jacobian determinant of the transformation is $\pm 1$ and thus, in particular, the transformation il locally a diffeomorphism. In the definition of canonical transformation there is the requirement that it must be a diffeomorphism between two open sets of the space of phases. So you must restrict the domain and/or the codomain in order to have a bijective map. In your case it means that you have to (arbitrarily) chose a sign in your equations. However you must also impose the further requirements I pointed out above.

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Show that transformation $Q=1/p\,$ and $P=qp^2$ is canonical using invariance of bilinear symplectic form.

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    $\begingroup$ Hi Vinay. Welcome to Phys.SE. This post seems to be a new question rather than an answer. $\endgroup$ – Qmechanic Aug 13 '19 at 15:04

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