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The Transition to Chaos

In Conservative Classical Systems: Quantum Manifestations

  • L. E. Reichl

Part of the Institute for Nonlinear Science book series (INLS)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Overview

    1. L. E. Reichl
      Pages 1-13
  3. Classical Systems

    1. L. E. Reichl
      Pages 14-65
    2. L. E. Reichl
      Pages 66-155
    3. L. E. Reichl
      Pages 156-221
  4. Quantum Systems

    1. L. E. Reichl
      Pages 222-247
    2. L. E. Reichl
      Pages 248-286
    3. L. E. Reichl
      Pages 287-317
    4. L. E. Reichl
      Pages 318-381
    5. L. E. Reichl
      Pages 382-444
  5. Stochastic Systems

    1. L. E. Reichl
      Pages 445-458
  6. Back Matter
    Pages 459-551

About this book

Introduction

resonances. Nonlinear resonances cause divergences in conventional perturbation expansions. This occurs because nonlinear resonances cause a topological change locally in the structure of the phase space and simple perturbation theory is not adequate to deal with such topological changes. In Sect. (2.3), we introduce the concept of integrability. A sys­ tem is integrable if it has as many global constants of the motion as degrees of freedom. The connection between global symmetries and global constants of motion was first proven for dynamical systems by Noether [Noether 1918]. We will give a simple derivation of Noether's theorem in Sect. (2.3). As we shall see in more detail in Chapter 5, are whole classes of systems which are now known to be inte­ there grable due to methods developed for soliton physics. In Sect. (2.3), we illustrate these methods for the simple three-body Toda lattice. It is usually impossible to tell if a system is integrable or not just by looking at the equations of motion. The Poincare surface of section provides a very useful numerical tool for testing for integrability and will be used throughout the remainder of this book. We will illustrate the use of the Poincare surface of section for classic model of Henon and Heiles [Henon and Heiles 1964].

Keywords

Evolution chaos chemistry path integral physics

Authors and affiliations

  • L. E. Reichl
    • 1
  1. 1.Center for Statistical Mechanics and Complex Systems, Department of PhysicsUniversity of Texas at AustinAustinUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-4352-4
  • Copyright Information Springer-Verlag New York 1992
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-4354-8
  • Online ISBN 978-1-4757-4352-4
  • Series Print ISSN 1431-4673
  • Buy this book on publisher's site