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 ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.

中文翻译：

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VI阿诺德的“ Pointwise” KAM定理

我们回顾VI阿诺德1963年著名论文[1]关于哈密顿量有小变化的条件周期运动守恒的Kolmogorov定理的证明，并证明，优化阿诺德方案，可以得到“尖锐”的渐近定量条件（如ε →0，ε为扰动强度）。所涉及的所有常量均已明确计算。