Juan-Pablo Ortega, Tudor S. Ratiu

Momentum Maps and Hamiltonian Reduction

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Progress in Mathematics .222
• Verlag: Birkhäuser / Birkhäuser Boston / Springer, Basel
• Artikelnr. des Verlages: 978-1-4757-3813-1
• Softcover reprint of the original 1st ed. 2004
• Seitenzahl: 536
• Erscheinungstermin: 14. Februar 2013
• Englisch
• Abmessung: 235mm x 155mm x 29mm
• Gewicht: 813g
• ISBN-13: 9781475738131
• ISBN-10: 1475738137
• Artikelnr.: 40775375

Autorenporträt

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. 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. 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.

Inhaltsangabe

1 Manifolds and Smooth Structures.- 2 Lie Group Actions.- 3 Pseudogroups and Groupoids.- 4 The Standard Momentum Map.- 5 Generalizations of the Momentum Map.- 6 Regular Symplectic Reduction Theory.- 7 The Symplectic Slice Theorem.- 8 Singular Reduction and the Stratification Theorem.- 9 Optimal Reduction.- 10 Poisson Reduction.- 11 Dual Pairs.

Rezensionen

"...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