In the 1st page of Landau's "Mechanics", he used "known from experience" and "in principle" to state that a $N$-degree of freedom particle system is completely determined by $2N$ variables ($N$ generalized positions and $N$ generalized velocities) as its initial condition:
Is there a theoretical justification for the statement? i.e., why higher order derivatives ($>=2$) do not matter if $q$ and $q'$ are specified? Or other combinations of $2N$ variables (e.g., $N$ positions, $N/2$ velocities, $N/2$ accelerations) can also uniquely determine the system?