An Introduction to Symplectic Geometry
Rolf Berndt
ααααΆ 2001 Β· American Mathematical Soc.
ααααα
βα’αα‘α·α
ααααΌαα·α
195
ααααα
reportααΆαααΆαααααα αα·αααα·ααΆαααααααα·αααααΌαααΆααααααααααΆαααα αααααααααααααα
α’αααΈααααα βα’αα‘α·α ααααΌαα·αααα
Symplectic 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 a minimum of prerequisites. This book would be an excellent text for a graduate course or as a source for anyone who wishes to learn about symplectic geometry.
ααΆααααααααααα βα’αα‘α·α ααααΌαα·αααα
ααααΆααααΎαα’αααΈααΆααααααΎαααααα’αααα
α’αΆαβααααααΆα
ααΌαααααααααΆααα αα·αβααααααα
ααα‘αΎααααααα·ααΈ Google Play Books αααααΆαα Android αα·α iPad/iPhone α ααΆβααααΎααααΆαααααβαααααααααααααααα·ααΆαα½αβααααΈβααααα’αααβ αα·αβα’αα»ααααΆαα±ααβα’αααα’αΆααααβααΆαα’ααΈαααΊαα·α α¬ααααΆαβα’ααΈαααΊαα·αβαα
αααααααΈααααααα
αα»αααααΌαααβαα½ααα αα·ααα»αααααΌααα
α’αααα’αΆα
ααααΆααααααα
ααΆααα‘αααααααΆααα·ααα
αααα»α Google Play αααααααΎαααααα·ααΈαα»αααααΆαα’ααΈαααΊαα·ααααα»ααα»αααααΌαααααααα’αααα
eReaders αα·αβα§αααααβαααααβααα
ααΎααααΈα’αΆααα
ααΎβα§ααααα e-ink ααΌα
ααΆβα§αααααα’αΆαβααααα
α’αα‘α·α
ααααΌαα·α Kobo α’αααααΉαααααΌαβααΆαααβα―αααΆα α αΎαβαααααααΆαα
βα§αααααβααααα’αααα ααΌαα’αα»ααααααΆαβααΆαααααΆααααα’α·αααααααααααααααααα½α ααΎααααΈαααααα―αααΆαβαα
α§αααααα’αΆαααααα
βα’αα‘α·α
ααααΌαα·ααααααααΆααα
