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Birkhäuser

Momentum Maps and Hamiltonian Reduction

  • Textbook
  • © 2004

Overview

Authors:
  1. Juan-Pablo Ortega

    1. CNRS-Laboratoire de Mathématiques de Bensançon, Université de Franche-Comté, UFR des Sciences et Techniques, Bensançon Cedex, France

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  2. Tudor S. Ratiu

    1. Départment de Mathématiques, Lausanne, Switzerland

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  • Winner of the Ferran Sunyer i Balaguer Prize in 2000
  • Reviews the necessary prerequisites, beginning with an introduction to Lie symmetries on Poisson and symplectic manifolds
  • Currently in classroom use in Europe
  • Can serve as a resource for graduate courses and seminars in Hamiltonian mechanics and symmetry, symplectic and Poisson geometry, Lie theory, mathematical physics, and as a comprehensive reference resource for researchers

Part of the book series: Progress in Mathematics (PM, volume 222)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xxxiv
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  2. Manifolds and Smooth Structures

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 1-35

  3. Lie Group Actions

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 37-91

  4. Pseudogroups and Groupoids

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 93-119

  5. The Standard Momentum Map

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 121-169

  6. Generalizations of the Momentum Map

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 171-204

  7. Regular Symplectic Reduction Theory

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 205-269

  8. The Symplectic Slice Theorem

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 271-300

  9. Singular Reduction and the Stratification Theorem

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 301-330

  10. Optimal Reduction

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 331-362

  11. Poisson Reduction

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 363-399

  12. Dual Pairs

    • Juan-Pablo Ortega, Tudor S. Ratiu
    Pages 401-441

  13. Back Matter

    Pages 443-501
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Keywords

About this book

The use of the symmetries of a physical system in the study of its dynamics has a long history that goes back to the founders of c1assical mechanics. Symmetry-based tech­ niques are often implemented by using the integrals 01 motion that one can sometimes associate to these symmetries. The integrals of motion of a dynamical system are quan­ tities that are conserved along the fiow of that system. In c1assieal mechanics symme­ tries are usually induced by point transformations, that is, they come exc1usively from symmetries of the configuration space; the intimate connection between integrals of motion and symmetries was formalized in this context by NOETHER (1918). This idea can be generalized to many symmetries of the entire phase space of a given system, by associating to the Lie algebra action encoding the symmetry, a function from the phase space to the dual of the Lie algebra. This map, whose level sets are preserved by the dynamics of any symmetrie system, is referred to in modern terms as a momentum map of the symmetry, a construction already present in the work of LIE (1890). Its remarkable properties were rediscovered by KOSTANT (1965) and SOURlAU (1966, 1969) in the general case and by SMALE (1970) for the lifted action to the co tangent bundle of a configuration space. For the his tory of the momentum map we refer to WEINSTEIN (1983b) and MARSDEN AND RATIU (1999), §11. 2.

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

Authors and Affiliations

  • CNRS-Laboratoire de Mathématiques de Bensançon, Université de Franche-Comté, UFR des Sciences et Techniques, Bensançon Cedex, France

    Juan-Pablo Ortega

  • Départment de Mathématiques, Lausanne, Switzerland

    Tudor S. Ratiu

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