2001/01/01 by Jürgen Pöschel · 7 citations
Mathematics · Computer Science · #Markov Chains and Monte Carlo Methods #Bayesian Methods and Mixture Models #Algorithms and Data Compression
paper · doi:10.1090/pspum/069/1858551
openalex publication_date 2001/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
The Classical KAM-Theorem a.The purpose of this lecture is to describe the Kam theorem in its most basic form and to give a complete and detailed proof.This proof essentially follows the traditional lines laid out by the inventors of this theory, Kolmogorov, Arnold and Moser (whence the acronym 'Kam'), and the emphasis is more on the underlying ideas than on the sharpness of the arguments.After all, Kam theory is not only a collection of specific theorems, but rather a methodology, a collection of ideas of how to approach certain problems in perturbation theory connected with 'small divisors'.b.The classical Kam theorem is concerned with the stability of motions in hamiltonian systems, that are small perturbations of integrable hamiltonian systems.These integrable systems are characterized by the existence of action angle coordinates such that the hamiltonian depends on the action variable alone -see [2,14] for details.Thus we are going to consider hamiltonians of the formfor small , where p = (p 1 , . . ., p n ) are the action variables varying over some domain D ⊂ R n , while q = (q 1 , . . ., q n ) are the conjugate angular variables, whose domain is the usual n-torus T n obtained from R n by identifying points whose components