Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom

Vadim Kaloshin, Ke Zhang

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English

Princeton University Press
11 January 2021

Mathematics & Sciences; Classical mechanics; Dynamics & statics; Mathematical physics

Series: Annals of Mathematics Studies

The first complete proof of Arnold diffusion-one of the most important problems in dynamical systems and mathematical physicsArnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. In this groundbreaking book, Vadim Kaloshin and Ke Zhang provide the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two-and-a-half degrees of freedom).

This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. Kaloshin and Zhang follow Mather's strategy but emphasize a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, this book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.

By: Vadim Kaloshin, Ke Zhang
Imprint: Princeton University Press
Country of Publication: United States
Dimensions: Height: 235mm, Width: 156mm,
ISBN: 9780691202532
ISBN 10: 0691202532
Series: Annals of Mathematics Studies
Pages: 224
Publication Date: 11 January 2021
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Hardback
Publisher's Status: Active

Vadim Kaloshin is the Michael Brin Chair in Mathematics at the University of Maryland, College Park. He was a student of John Mather at Princeton University. Ke Zhang is associate professor of mathematics at the University of Toronto.