An Introduction to Symplectic Geometry
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.
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Some Aspects of Theoretical Mechanics
Hamiltonian Vector Fields and the Poisson Bracket
The Moment Map
Abraham-Marsden Aebischer associated basis bilinear form called canonical chapter chart coadjoint orbit coadjoint representation cohomology groups commutative compatible complex structure concept construction contact manifold coordinates cotangent bundle covariant defined Definition denote diffeomorphism differentiable function differentiable manifold differentiable map differentiable vector differential equations differential forms dimension equivalent equivariant example Exercise fixed form uj formula G operates given gives Guillemin-Sternberg GS Hamiltonian vector field Heisenberg group Hilbert space infinitesimal invariant irreducible isomorphism Kirillov Lagrangian subspace Lemma Lie algebra Lie bracket Lie derivative Lie group Lie group G linear map F matrix moment map multiplication neighborhood notation orthogonal plectic Poisson bracket proof quantization Remark representation of G Riemannian satisfies scalar product Section A.4 Sp(V statement subgroup symmetric symplectic form symplectic geometry symplectic manifold symplectic space tangent spaces tangent vector tensor Theorem theory transformation unitary vector bundle vector space