Kolmogorov’s The general theory of dynamical systems and classical mechanics (1957): Encountering a new metric and spectral grammar

This article offers a historical and mathematical reading of Andrej N. Kolmogorov’s essay “The General Theory of Dynamical Systems and Classical Mechanics”, based on a close textual analysis. The essay is reinterpreted as a scientific manifesto that reframes classical mechanics through a combination of measure-theoretic and spectral perspectives, with particular attention to its rhetorical architecture. The analysis brings into view the intersections between classical mechanics, ergodic theory, within the theory of dynamical systems, and operator theory through a network of references spanning four decades (1917-1954) and different national traditions. It also highlights the dramatic structure of the text, intended to draw attention to “old-fashioned” open problems in classical mechanics and to a new approach – imbuing it with a sense of pathos likely derived from its origin as a lecture presented during the 1954 Amsterdam International Congress of Mathematicians. In this sense, the scope of Kolmogorov’s text goes far beyond the presentation of early instances of what would later become KAM theorems – thus establishing the essay as a landmark writing in modern mathematics. The rhetorical architecture culminates in the recognition of two paradigmatic regimes of conservative dynamics: global transitivity with continuous spectrum, versus invariant tori and quasi‑periodic motions with discrete spectrum.