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Progress in Mathematics

Momentum Maps and Hamiltonian Reduction

Authors: Ortega, Juan-Pablo, Ratiu, Tudor

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About this Textbook

The use of symmetries and conservation laws in the qualitative description of dynamics has a long history going back to the founders of classical mechanics. In some instances, the symmetries in a dynamical system can be used to simplify its kinematical description via an important procedure that has evolved over the years and is known generically as reduction. The focus of this work is a comprehensive and self-contained presentation of the intimate connection between symmetries, conservation laws, and reduction, treating the singular case in detail.

The exposition reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds. This is followed by a discussion of momentum maps and the geometry of conservation laws that are used in the development of symplectic reduction. The Symplectic Slice Theorem, an important tool that gave rise to the first description of symplectic singular reduced spaces, is also treated in detail, as well as the Reconstruction Equations that have been crucial in applications to the study of symmetric mechanical systems. The last part of the book contains more advanced topics, such as symplectic stratifications, optimal and Poisson reduction, singular reduction by stages, bifoliations and dual pairs. Various possible research directions are pointed out in the introduction and throughout the text. An extensive bibliography and a detailed index round out the work.

This Ferran Sunyer i Balaguer Prize-winning monograph is the first self-contained and thorough presentation of the theory of Hamiltonian reduction in the presence of singularities. It can serve as a resource for graduate courses and seminars in symplectic and Poisson geometry, mechanics, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers.

Reviews

"…The present book offers a thorough description of [reduction] theory and a unified treatment of most of its developments and generalizations, with a particular emphasis on those due to the authors. It contains many important results which cannot be found in other books, and covers a large part of the recent developments related to momentum maps and reduction. This book fills a need and will be appreciated by specialists as well as by persons new to the field…."

—MATHEMATICAL REVIEWS


Table of contents (11 chapters)

  • Manifolds and Smooth Structures

    Ortega, Juan-Pablo (et al.)

    Pages 1-35

  • Lie Group Actions

    Ortega, Juan-Pablo (et al.)

    Pages 37-91

  • Pseudogroups and Groupoids

    Ortega, Juan-Pablo (et al.)

    Pages 93-119

  • The Standard Momentum Map

    Ortega, Juan-Pablo (et al.)

    Pages 121-169

  • Generalizations of the Momentum Map

    Ortega, Juan-Pablo (et al.)

    Pages 171-204

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-1-4757-3811-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA
  • ISBN 978-0-8176-4307-2
  • Free shipping for individuals worldwide
  • This title is currently reprinting. You can pre-order your copy now.
Softcover $99.00
price for USA
  • ISBN 978-1-4757-3813-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Momentum Maps and Hamiltonian Reduction
Authors
Series Title
Progress in Mathematics
Series Volume
222
Copyright
2004
Publisher
Birkhäuser Basel
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4757-3811-7
DOI
10.1007/978-1-4757-3811-7
Hardcover ISBN
978-0-8176-4307-2
Softcover ISBN
978-1-4757-3813-1
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XXXIV, 501
Number of Illustrations and Tables
2 b/w illustrations
Topics