Modern Birkhäuser Classics

Symplectic Invariants and Hamiltonian Dynamics

Authors: Hofer, Helmut, Zehnder, Eduard

  • Opened new perspectives for the old field of Hamiltonian systems and let to the creation of a new field: symplectic topology
  • Invariants which grew out of lectures given by the authors
  • Selection by a single principle: the action principle of mechanics
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About this book

The discoveries of the last decades have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic mappings is very different from that of volume preserving mappings. This raises new questions, many of them still unanswered. On the other hand, analysis of an old variational principle in classical mechanics has established global periodic phenomena in Hamiltonian systems. As it turns out, these seemingly different phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book, which includes such topics as basic symplectic geometry, symplectic capacities and rigidity, periodic orbits for Hamiltonian systems and the action principle, a bi-invariant metric on the symplectic diffeomorphism group and its geometry, symplectic fixed point theory, the Arnold conjectures and first order elliptic systems, and finally a survey on Floer homology and symplectic homology.

The exposition is self-contained and addressed to researchers and students from the graduate level onwards.

 

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All the chapters have a nice introduction with the historic development of the subject and with a perfect description of the state of the art. The main ideas are brightly exposed throughout the book. (…) This book, written by two experienced researchers, will certainly fill in a gap in the theory of symplectic topology. The authors have taken part in the development of such a theory by themselves or by their collaboration with other outstanding people in the area.

(Zentralblatt MATH)

 

This book is a beautiful introduction to one outlook on the exciting new developments of the last ten to fifteen years in symplectic geometry, or symplectic topology, as certain aspects of the subject are lately called. (…) The authors are obvious masters of the field, and their reflections here and there throughout the book on the ambient literature and open problems are perhaps the most interesting parts of the volume.

(Matematica)

About the authors

Helmut Hofer is Professor at the Institute for Advanced Study in Princeton, New Jersey, USA.

Eduard Zehnder is emeritus Professor of ETH Zürich, Switzerland.

Reviews

From the reviews:

“Symplectic Invariants and Hamiltonian Dynamics is obviously a work of central importance in the field and is required reading for all would-be players in this game. Happily, it is very well written and sports a lot of very useful commentary by the authors; the sections introducing the individual chapters are particularly well done … . It is all fine scholarship in an exciting and fertile area.” (Michael Berg, The Mathematical Association of America, June, 2011)


Table of contents (6 chapters)

  • Introduction

    Hofer, Helmut (et al.)

    Pages 1-50

  • Symplectic capacities

    Hofer, Helmut (et al.)

    Pages 51-67

  • Existence of a capacity

    Hofer, Helmut (et al.)

    Pages 69-103

  • Existence of closed characteristics

    Hofer, Helmut (et al.)

    Pages 105-142

  • Geometry of compactly supported symplectic mappings in ℝ2n

    Hofer, Helmut (et al.)

    Pages 143-192

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-0348-0104-1
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.95
price for USA
  • ISBN 978-3-0348-0103-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Symplectic Invariants and Hamiltonian Dynamics
Authors
Series Title
Modern Birkhäuser Classics
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-0104-1
DOI
10.1007/978-3-0348-0104-1
Softcover ISBN
978-3-0348-0103-4
Series ISSN
2197-1803
Edition Number
1
Number of Pages
XIV, 341
Number of Illustrations and Tables
49 b/w illustrations
Topics