# V.I. Arnold's "pointwise" KAM Theorem

## TL;DR

This paper reviews Arnold's 1963 KAM theorem proof, optimizing the scheme to derive sharp asymptotic conditions with explicit constants as perturbation strength approaches zero.

## Contribution

It provides an optimized version of Arnold's scheme with explicit constants, leading to sharper asymptotic conditions in KAM theory.

## Key findings

- Explicit constants are computed for the theorem.
- Optimized scheme yields sharper asymptotic conditions.
- Results improve understanding of small perturbations in Hamiltonian systems.

## Abstract

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.