Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates $q,p$, Hamiltonian $H$, and new coordinates $Q(q,p),P(q,p)$, there exists a transformed Hamiltonian $K$ such that $\dot Q_i = \frac{\partial K}{ \partial P_i}$ and $\dot P_i = -\frac{\partial K}{Q_i}$ if and only if $MJM^T= J$ where $M$ is the Jacobian of $Q,P$ with respect to $q,p$ and $J = \begin{bmatrix} O&I\\\\ -I&O \end{bmatrix}$. I understood the book's proof that $MJM^T= J$ implies the existence of such $K$. However, the proof of the converse was not given and I do not know how to derive it myself despite the book saying (page 383)
That Eq. (9.55) is also a necessary condition for a restricted canonical transformation is easily shown directly by reversing the order of the steps of the proof
