  This article is about the proof of the celebrated KAM theorem as sketched out
in \cite{KOL} Kolmogorov's original presentation to the ICM. The proof
presented here has been detailed as an effort to clarify if Kolmogorov's
argument can be made rigorous. This is a legitimate question because the
published proof \cite{ARN} a few years later by his then student, the eminent
mathematician V.I Arnold, did not follow the same plan
