Canonical transformations - sufficient & neccessary argument

Question

I see in many textbooks that for a transformation of coordinates  $P=P(q,p,t), Q=Q(q,p,t)$ it is sufficient & neccessary to check: $$[Q_i,Q_j]_{q,p} = 0$$ $$[P_i,P_j]_{q,p} = 0 $$ $$[Q_i,P_j]_{q,p}=delta_{ij}$$
for the transformation to be canonical, as follows from:
$$[f,g]_{q,p} = [f,g]_{Q,P}.$$
It seems to be trivial and it is always given without any proof, but why is that? I don't see why it is sufficient.

Young Kindaichi · Answer

The fundamental Poisson brackets are invariant under canonical transformation. It follows from here that
$$[zeta,zeta]_eta=tilde{M}JM$$
the invariance is a necessary and sufficient condition for the transformation matrix to be symplectic. The invariance of the fundamental Poisson brackets is thus in all ways equivalent to the symplectic condition for a canonical transformation.
I haven't get into much detail, As you can find the details in Goldstein's book. :)

Canonical transformations - sufficient & neccessary argument

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