Lectures on the Geometry of Poisson Manifolds

Izu Vaisman

ISBN 10: 3764350164 / ISBN 13: 9783764350161

Published by Springer, Basel, Birkhäuser Basel, Birkhäuser, 1994

About this Item

Neuware - Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L. ~[ji - [ji~ , ;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie].

Seller Inventory # 9783764350161

Bibliographic Details

• Title: Lectures on the Geometry of Poisson Manifolds
• Publisher: Springer, Basel, Birkhäuser Basel, Birkhäuser
• Publication Date: 1994
• Language: English
• ISBN 10: 3764350164
• ISBN 13: 9783764350161
• Binding: Buch
• Condition: Neu

About this title

Synopsis

Everybody having even the slightest interest in analytical mechanics remembers having met there the Poisson bracket of two functions of 2n variables (pi, qi) f g ~(8f8g 8 8 ) (0.1) {f,g} = L...~[ji - [ji~ ,;=1 p, q q p, and the fundamental role it plays in that field. In modern works, this bracket is derived from a symplectic structure, and it appears as one of the main in- gredients of symplectic manifolds. In fact, it can even be taken as the defining clement of the structure (e.g., [TIl]). But, the study of some mechanical sys- tems, particularly systems with symmetry groups or constraints, may lead to more general Poisson brackets. Therefore, it was natural to define a mathematical structure where the notion of a Poisson bracket would be the primary notion of the theory, and, from this viewpoint, such a theory has been developed since the early 19708, by A. Lichnerowicz, A. Weinstein, and many other authors (see the references at the end of the book). But, it has been remarked by Weinstein [We3] that, in fact, the theory can be traced back to S. Lie himself [Lie].

Review

    "The book serves well as an introduction and an overview of the subject and a long list of references helps with further study."   
  -- Zbl. Math.   

    "The book is well done...should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research."  --   Mathematical Reviews   

    "The importance and actuality of the subject, as well as the very rigorous and didactic presentation of the content, make out of this book a valuable contribution to current mathematics. The book is intended first of all to mathematicians, but it can be interesting also for a wide circle of readers, including mechanicists and physicists."    -- Mathematica