An introduction to symplectic geometry / Rolf Berndt ; translated by Michael KlucznikType de document : MonographieCollection : Graduate studies in mathematics, 26Langue : anglais.Pays: Etats Unis.Éditeur : Providence : American Mathematical Society, 2001Description : 1 vol. (XVI-195 p.) : ill. ; 26 cmISBN: 0821820567.ISSN: 1065-7339.Bibliographie : Bibliogr. p. -187. Index.Sujet MSC : 53Dxx, Differential geometry - Symplectic geometry, contact geometry
53-01, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C15, Global differential geometry, General geometric structures on manifolds
70Hxx, Mechanics of particles and systems - Hamiltonian and Lagrangian mechanics
81S10, General quantum mechanics and problems of quantization, Geometry and quantization, symplectic methodsEn-ligne : Zentralblatt | AMS Item type: Monographie
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Bibliogr. p. -187. Index
This book is a standard introduction to symplectic geometry, with special emphasis in geometric quantization. To motivate the study of symplectic geometry, the author discusses in a preliminary section (Chapter 0) some rudiments of symplectic mechanics. So, he introduces the Euler-Lagrange equations, Hamilton equations (obtained using the Legendre transformation), Hamilton-Jacobi theory, and Poisson brackets. He also presents some preliminary aspects of quantization of classical mechanical systems. In the following chapters, the author presents symplectic algebra (Chapter 1), a good introduction to symplectic manifolds paying special attention to Kähler manifolds and including some initial results in symplectic invariants (Chapter 2), Hamiltonian vector fields including Poisson brackets and contact manifolds (Chapter 3), momentum maps and symplectic reduction (Chapter 4), and, finally, he studies the procedure of geometric quantization, including a proof of the Groenewold-van Hove theorem (Chapter 5). The book is completed with four appendices. Appendices A and B give a brief and clear introduction to differentiable manifolds, Lie groups and Lie algebras. Appendix C is devoted to study some cohomology theories in Lie groups, Lie algebras and manifolds, and Appendix D treats with the theory of representation of groups.
The book is very well-written, and contains the essential facts on symplectic geometry and symplectic mechanics. It is highly recommendable for graduate students in Mathematics and Physics. (Zentralblatt)
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