  We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N.
Kolmogorov's theorem on the conservation of conditionally periodic motions with
a small variation in the Hamiltonian}, and prove that, optimizing Arnold's
scheme, one can get "sharp" asymptotic quantitative conditions (as
$\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All
constants involved are explicitly computed.
