دانلود کتاب نظریه KAM ضعیف گسسته

The aim of these notes is to present a self contained account of discrete weak KAM theory. Put aside the intrinsic elegance of this theory, it is also a toy model for classical weak KAM theory, where many technical difficulties disappear, but where central ideas and results persist. It can therefore serve as a good introduction to (continuous) weak KAM theory. After a general exposition of the general abstract theory, several examples are studied. The last section is devoted to the historical problem of conservative twist maps of the annulus. At the end of the first three Chapters, the relations between the results proved in the discrete setting and the analogous theorems of classical weak KAM theory are discussed. Some key differences are also highlighted between the discrete and classical theory. An appendix briefly comes back to the etymology of weak KAM theory and non exhaustively presents some of its recent striking developments. The present text initially emerged from lecture notes of a course given in Cortona in 2015 at the INDAM meeting entitled The Hamilton–Jacobi Equation: At the crossroads of PDE, dynamical systems and geometry. The goal of the lectures was to give a complete and thorough introduction to weak KAM theory through the prism of its discrete pendant. It culminated with the proof of convergence of the solutions of the discounted equation, which was new at the time.
The pedagogical motivation is that discrete weak KAM theory is peculiarly elementary. Basic topology is the only prerequisite and the most advanced tools are the Arzelà–Ascoli Theorem and weak compactness of probability measures on a compact metric space. However, all important results of weak KAM theory find their analogue in discrete weak KAM theory and the proofs being stripped of technicalities reveal the key ideas more clearly. After studying this toy model, the interested reader can then go on to learn more on the major theories, as Calculus of Variations, Viscosity solutions of Hamilton–Jacobi equations, and Tonelli Hamiltonian Dynamical Systems. The latter are at the core of classical weak KAM theory.