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Momentum maps and hamiltonian reduction
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Momentum maps and hamiltonian reduction

Momentum maps and hamiltonian reduction

Juan-pablo Ortega, Tudor Stefan Ratiu - Collection Progress in Mathematics

500 pages, parution le 29/03/2004

Résumé

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.

Sommaire

  • Manifolds and smooth structures
  • Lie group actions
  • Pseudogroups and groupoids
  • The standard momentum map
  • Generalizations of the momentum map
  • Regular sympletic reduction theory
  • The symplectic slice theorem
  • Singular reduction and the stratification theorem
  • Optimal reduction
  • Poisson reduction
  • Dual Pairs
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Caractéristiques techniques

  PAPIER
Éditeur(s) Birkhäuser
Auteur(s) Juan-pablo Ortega, Tudor Stefan Ratiu
Collection Progress in Mathematics
Parution 29/03/2004
Nb. de pages 500
Format 16 x 24
Couverture Relié
Poids 888g
Intérieur Noir et Blanc
EAN13 9780817643072
ISBN13 978-0-8176-4307-2

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