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Index Theory for Symplectic Paths with Applications

  • Book
  • © 2002

Overview

Authors:
  1. Yiming Long

    1. Nankai Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China

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

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Table of contents (15 chapters)

  1. Front Matter

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

    1. Front Matter

      Pages 1-1
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    2. Algebraic aspects

      • Yiming Long
      Pages 3-47

    3. Topological aspects

      • Yiming Long
      Pages 48-77

  3. The Variational Method

    1. Front Matter

      Pages 79-79
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    2. Hamiltonian systems and canonical transformations

      • Yiming Long
      Pages 81-90

    3. The variational functional

      • Yiming Long
      Pages 91-107

  4. Index Theory

    1. Front Matter

      Pages 109-109
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    2. Index functions for symplectic paths

      • Yiming Long
      Pages 111-131

    3. Properties of index functions

      • Yiming Long
      Pages 132-151

    4. Relations with other Morse indices

      • Yiming Long
      Pages 152-173

  5. Iteration Theory

    1. Front Matter

      Pages 175-175
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    2. Precise iteration formulae

      • Yiming Long
      Pages 177-189

    3. Bott-type iteration formulae

      • Yiming Long
      Pages 190-208

    4. Iteration inequalities

      • Yiming Long
      Pages 209-228

    5. The common index jump theorem

      • Yiming Long
      Pages 229-241

    6. Index iteration theory for closed geodesics

      • Yiming Long
      Pages 242-253

  6. Applications

    1. Front Matter

      Pages 255-255
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    2. The Rabinowitz conjecture

      • Yiming Long
      Pages 257-289

    3. Periodic Lagrangian orbits on tori

      • Yiming Long
      Pages 290-314
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Keywords

About this book

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.

Authors and Affiliations

  • Nankai Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China

    Yiming Long

Bibliographic Information

  • Book Title: Index Theory for Symplectic Paths with Applications

  • Authors: Yiming Long

  • Series Title: Progress in Mathematics

  • DOI: https://doi.org/10.1007/978-3-0348-8175-3

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 2002

  • Hardcover ISBN: 978-3-7643-6647-6Published: 25 March 2002

  • Softcover ISBN: 978-3-0348-9466-1Published: 23 October 2012

  • eBook ISBN: 978-3-0348-8175-3Published: 06 December 2012

  • Series ISSN: 0743-1643

  • Series E-ISSN: 2296-505X

  • Edition Number: 1

  • Number of Pages: XXIV, 380

  • Topics: Differential Geometry

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