terminology - Sufficient conditions for a mapping to be canonical in Hamiltonian Mechanics

Friday, 10 July 2015

My professor mentioned: A simple way of testing whether a mapping $(q,p)$ to $(Q,P)$ is canonical is by examining:

$P · dQ − p · dq$

and if it equals to $dA$ (a differential) then it is canonical.

However, I'm wondering why is this the case, since the requirements for canonical map is that at first is

$P ·dQ − Kdt = p·dq − Hdt + dS$ (so that the closed contour integral of

$P ·dQ − Kdt$ to equal that of

$p·dq − Hdt$ . Then what about the

$Kdt$ and

$Hdt$ ?

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