KAM theory: the legacy of Kolmogorov’s 1954 paper Henk W. Broer University of Groningen Department of Mathematics and Computing Science Blauwborgje 3 NL-9747 AC, Groningen, Email: broer@math.rug.nl December 30, 2003 Abstract Kolmogorov-Arnol’d-Moser (or KAM) theory was developed for conserva- tive dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori and KAM theory estab- lishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory which begins with Kolmogorov’s pioneering work [40, 41]. 1 Introduction At the International Mathematical Congress of 1954, held at Amsterdam, A.N. Kol- mogorov gave a closing lecture with title “The general theory of dynamical sys- tems and classical mechanics” [41], discussing the paper [40]. The event took place in the Amsterdam Concertgebouw and it has played a major role in the de- velopments of what is now called Kolmogorov-Arnold-Moser (or KAM) theory. In this lecture Kolmogorov discusses the occurrence of multi- or quasi-periodic motions, which in the phase space are confined to invariant tori. He restricts him- self to conservative (or Hamiltonian) dynamical systems, as these are generally used for modelling in classical mechanics. Invariant (Lagrangean) tori that carry quasi-periodic motions were well-known to occur in Liouville integrable systems 1