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Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom(AMS-208)$
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Vadim Kaloshin and Ke Zhang

Print publication date: 2020

Print ISBN-13: 9780691202525

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691202525.001.0001

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Introduction

Introduction

Chapter:
(p.3) Chapter One Introduction
Source:
(p.iii) Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom
Author(s):

Kaloshin Vadim

Zhang Ke

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691202525.003.0001

This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

Keywords:   Arnold diffusion, ergodic hypothesis, KAM theory, Hamiltonian systems, time-periodic systems, KAM invariant tori, large-scale instability, Vladimir Arnold

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