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Index Theory for Symplectic Paths with Applications

  • Yiming Long

Part of the Progress in Mathematics book series (PM, volume 207)

Table of contents

  1. Front Matter
    Pages i-xxiv
  2. The Symplectic Group Sp(2n)

    1. Front Matter
      Pages 1-1
    2. Yiming Long
      Pages 3-47
    3. Yiming Long
      Pages 48-77
  3. The Variational Method

    1. Front Matter
      Pages 79-79
    2. Yiming Long
      Pages 91-107
  4. Index Theory

    1. Front Matter
      Pages 109-109
    2. Yiming Long
      Pages 111-131
    3. Yiming Long
      Pages 132-151
    4. Yiming Long
      Pages 152-173
  5. Iteration Theory

    1. Front Matter
      Pages 175-175
    2. Yiming Long
      Pages 177-189
    3. Yiming Long
      Pages 190-208
    4. Yiming Long
      Pages 209-228
    5. Yiming Long
      Pages 229-241
  6. Applications

    1. Front Matter
      Pages 255-255
    2. Yiming Long
      Pages 257-289
    3. Yiming Long
      Pages 290-314
  7. Back Matter
    Pages 361-380

About this book

Introduction

This book is based upon my monograph Index Theory for Hamiltonian Systems with Applications published in 1993 in Chinese, and my notes for lectures and courses given at Nankai University, Brigham Young University, ICTP-Trieste, and the Institute of Mathematics of Academia Sinica during the last ten years. The aim of this book is twofold: (1) to give an introduction to the index theory for symplectic matrix paths and its iteration theory, which form a basis for the Morse theoretical study on Hamilto­ nian systems, and to give applications of this theory to periodic boundary value problems of nonlinear Hamiltonian systems. Here the iteration theory means the index theory of iterations of periodic solutions and symplectic matrix paths. (2) to serve as a reference book on these topics. There are many different ways to introduce the index theory for symplectic paths in order to establish Morse type index theory of Hamiltonian systems. In this book, I have chosen a relatively elementary way, i.e., the homotopy classification method of symplectic matrix paths. It depends only on linear algebra, point set topology, and certain basic parts of linear functional analysis. I have tried to make this part of the book self-contained and at the same time include all of the major results on these topics so that researchers and students interested in them can read it without substantial difficulties, and can learn the main results in this area for their possible applications.

Keywords

Boundary value problem functional analysis index theory perturbation perturbation theory symplectic geometry

Authors and affiliations

  • Yiming Long
    • 1
  1. 1.Nankai Institute of MathematicsNankai UniversityTianjinPeople’s Republic of China

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-8175-3
  • Copyright Information Birkhäuser Verlag 2002
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-0348-9466-1
  • Online ISBN 978-3-0348-8175-3
  • Series Print ISSN 0743-1643
  • Series Online ISSN 2296-505X
  • Buy this book on publisher's site