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008 081109s2001 riu |b 001 0 eng
020 _a0821820567 (alk. paper)
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_bVLOAD
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050 0 0 _aQA649
_b.B47 2001
100 1 _aBerndt, Rolf,
_d1940-
_925086
240 1 0 _aEnglish
245 1 3 _aAn Introduction to Symplectic Geometry /
_cRolf Berndt ; translated by Michael Klucznik.
260 _aProvidence, R.I. :
_bAmerican Mathematical Society,
_cc2001.
300 _axvi, 195 p. ;
_c26 cm.
440 0 _aGraduate studies in mathematics,
_vv. 26
_910389
504 _aIncludes bibliographical references (p. 185-187) and index.
505 _aSome aspects of theoretical mechanics Symplectic algebra Symplectic manifolds Hamiltonian vectorfields and the Poisson bracket The moment map Quantization Differentiable manifolds and vector bundles Lie groups and Lie algebras A little cohomology theory Representations of groups Bibliography Index Symbols.
520 _aSymplectic geometry is a central topic of current research in mathematics. Indeed, symplectic methods are key ingredients in the study of dynamical systems, differential equations, algebraic geometry, topology, mathematical physics and representations of Lie groups. This book is a true introduction to symplectic geometry, assuming only a general background in analysis and familiarity with linear algebra. It starts with the basics of the geometry of symplectic vector spaces. Then, symplectic manifolds are defined and explored. In addition to the essential classic results, such as Darboux's theorem, more recent results and ideas are also included here, such as symplectic capacity and pseudoholomorphic curves. These ideas have revolutionized the subject. The main examples of symplectic manifolds are given, including the cotangent bundle, Kahler manifolds, and coadjoint orbits.Further principal ideas are carefully examined, such as Hamiltonian vector fields, the Poisson bracket, and connections with contact manifolds. Berndt describes some of the close connections between symplectic geometry and mathematical physics in the last two chapters of the book. In particular, the moment map is defined and explored, both mathematically and in its relation to physics. He also introduces symplectic reduction, which is an important tool for reducing the number of variables in a physical system and for constructing new symplectic manifolds from old. The final chapter is on quantization, which uses symplectic methods to take classical mechanics to quantum mechanics.This section includes a discussion of the Heisenberg group and the Weil (or metaplectic) representation of the symplectic group. Several appendices provide background material on vector bundles, on cohomology, and on Lie groups and Lie algebras and their representations. Berndt's presentation of symplectic geometry is a clear and concise introduction to the major methods and applications of the subject, and requires only
650 0 _aSymplectic geometry.
_925087
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