V. I. Arnold's “Pointwise” KAM Theorem

We review V.I. Arnold's 1963 celebrated paper [1] Proof of A.N. Kolmogorov's Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising 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.