In Goldstein's "Classical Mechanics", at page 384 it is claimed that given a point-transformation of phase space $$\underline{\zeta} = \underline{\zeta}(\underline{\eta}, t),\tag{9.59}$$ where $\underline{\eta}$ is a vector of canonical variables, then a necessary and sufficient condition for it to be canonical is the symplectic condition $$MJM^T = J\tag{9.55}$$ where $M$ is the Jacobian of the transformation (i.e. the Jacobian of $\underline{\zeta}(\underline{\eta}, t)$ restricted to the derivatives with respect to $\underline{\eta}$) and $J$ is the skew-symmetric matrix of the form:
$$ \begin{bmatrix} 0 & I_l \ -I_l & 0 \end{bmatrix}\tag{8.38a} $$
where $l$ denotes the number of degrees of freedom of the system.
Now, Goldstein first proves that this is true for restricted (i.e. time-independent) transformations, then he tries to do the same with a generic transformation but he just proves the symplectic condition to be a necessary condition and he does so by making use of infinitesimal canonical transformations. I have some problems with the latter derivation, especially when, at page 386, Goldstein claims that the transformation $\underline{\zeta}(t_0) \to \underline{\zeta}(t)$ can be written as "the succession of I.C.T.'s in steps of $dt$"; I tried making this statement rigorous by integrating but I had no success so far.
Therefore, I would like to know if anyone knows how to make the "necessary condition" part rigorous and how to prove the "sufficient condition" part.
As always any comment or answer is much appreciated and let me know if I can explain myself clearer!
