FPU model and Toda model: a survey, a view

A huge amount of papers investigated, over more than 65 years, the Fermi–Pasta–Ulam problem. One of the leading ideas, present already in the early literature, is that the unexpected regular behavior observed by the authors, quite different from the expected ergodicity, could be explained by the presence of a close hidden nonlinear integrable dynamics. This was initially searched among nonlinear wave equations, but rather soon, after the discovery of the integrability of the Toda model, it was progressively understood that Toda provides the natural integrable approximation to FPU. The aim of this paper is to provide a short updated review of the relation between the FPU dynamics and the Toda dynamics. Updated means it also includes new results, see Sect. 3. The paper is ideally addressed to the wide—very wide!—community of people who feel The legacy of Carlo Cercignani, so it includes a few introductory comments.