Achim Feldmeier – författare

INTRODUCTION TO ARNOLD’S PROOF OF THE KOLMOGOROV–ARNOLD–MOSER THEOREM

This book provides an accessible step-by-step account of Arnold’s classical proof of the Kolmogorov–Arnold–Moser (KAM) Theorem. It begins with a general background of the theorem, proves the famous Liouville–Arnold theorem for integrable systems and introduces Kneser’s tori in four-dimensional phase space. It then introduces and discusses the ideas and techniques used in Arnold’s proof, before the second half of the book walks the reader through a detailed account of Arnold’s proof with all the required steps. It will be a useful guide for advanced students of mathematical physics, in addition to researchers and professionals.

Features

• Applies concepts and theorems from real and complex analysis (e.g., Fourier series and implicit function theorem) and topology in the framework of this key theorem from mathematical physics.

• Covers all aspects of Arnold’s proof, including those often left out in more general or simplifi ed presentations.

• Discusses in detail the ideas used in the proof of the KAM theorem and puts them in historical context (e.g., mapping degree from algebraic topology).

INTRODUCTION TO ARNOLD’S PROOF OF THE KOLMOGOROV–ARNOLD–MOSER THEOREM

This book provides an accessible step-by-step account of Arnold’s classical proof of the Kolmogorov–Arnold–Moser (KAM) Theorem. It begins with a general background of the theorem, proves the famous Liouville–Arnold theorem for integrable systems and introduces Kneser’s tori in four-dimensional phase space. It then introduces and discusses the ideas and techniques used in Arnold’s proof, before the second half of the book walks the reader through a detailed account of Arnold’s proof with all the required steps. It will be a useful guide for advanced students of mathematical physics, in addition to researchers and professionals.

Features

• Applies concepts and theorems from real and complex analysis (e.g., Fourier series and implicit function theorem) and topology in the framework of this key theorem from mathematical physics.

• Covers all aspects of Arnold’s proof, including those often left out in more general or simplifi ed presentations.

• Discusses in detail the ideas used in the proof of the KAM theorem and puts them in historical context (e.g., mapping degree from algebraic topology).

This textbook gives an introduction to fluid dynamics based on flows for which analytical solutions exist, like individual vortices, vortex streets, vortex sheets, accretions disks, wakes, jets, cavities, shallow water waves, bores, tides, linear and non-linear free-surface waves, capillary waves, internal gravity waves and shocks.

Advanced mathematical techniques ("calculus") are introduced and applied to obtain these solutions, mostly from complex function theory (Schwarz-Christoffel theorem and Wiener-Hopf technique), exterior calculus, singularity theory, asymptotic analysis, the theory of linear and nonlinear integral equations and the theory of characteristics.

Many of the derivations, so far contained only in research journals, are made available here to a wider public.