Symplectic Geometry

Author : Ana Cannas da Silva
Publisher : Springer
Page : 220 pages
File Size : 44,7 Mb
Release : 2004-10-27
Category : Mathematics
ISBN : 9783540453307

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Lectures on Symplectic Geometry by Ana Cannas da Silva Pdf

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Author : Jean-Louis Koszul,Yi Ming Zou
Publisher : Springer
Page : 121 pages
File Size : 50,8 Mb
Release : 2019-04-15
Category : Science
ISBN : 9789811339875

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Introduction to Symplectic Geometry by Jean-Louis Koszul,Yi Ming Zou Pdf

This introductory book offers a unique and unified overview of symplectic geometry, highlighting the differential properties of symplectic manifolds. It consists of six chapters: Some Algebra Basics, Symplectic Manifolds, Cotangent Bundles, Symplectic G-spaces, Poisson Manifolds, and A Graded Case, concluding with a discussion of the differential properties of graded symplectic manifolds of dimensions (0,n). It is a useful reference resource for students and researchers interested in geometry, group theory, analysis and differential equations.This book is also inspiring in the emerging field of Geometric Science of Information, in particular the chapter on Symplectic G-spaces, where Jean-Louis Koszul develops Jean-Marie Souriau's tools related to the non-equivariant case of co-adjoint action on Souriau’s moment map through Souriau’s Cocycle, opening the door to Lie Group Machine Learning with Souriau-Fisher metric.

Author : Rolf Berndt
Publisher : American Mathematical Society
Page : 213 pages
File Size : 45,6 Mb
Release : 2024-04-15
Category : Mathematics
ISBN : 9781470476885

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An Introduction to Symplectic Geometry by Rolf Berndt Pdf

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, Kähler 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.

Author : P. Libermann,Charles-Michel Marle
Publisher : Springer Science & Business Media
Page : 541 pages
File Size : 55,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400938076

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Symplectic Geometry and Analytical Mechanics by P. Libermann,Charles-Michel Marle Pdf

Approach your problems from the right end It isn't that they can't see the solution. and begin with the answers. Then one day, It is that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' Brown 'The point of a Pin'. in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thouglit to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci ences has changed drastically in recent years: measure theory is used (non-trivially) in re gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homo topy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Author : Maurice A. de Gosson
Publisher : Springer Science & Business Media
Page : 375 pages
File Size : 45,9 Mb
Release : 2006-08-06
Category : Mathematics
ISBN : 9783764375751

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Symplectic Geometry and Quantum Mechanics by Maurice A. de Gosson Pdf

This book offers a complete discussion of techniques and topics intervening in the mathematical treatment of quantum and semi-classical mechanics. It starts with a very readable introduction to symplectic geometry. Many topics are also of genuine interest for pure mathematicians working in geometry and topology.

Author : Pierre Dazord,Alan Weinstein
Publisher : Springer Science & Business Media
Page : 318 pages
File Size : 42,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461397199

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Symplectic Geometry, Groupoids, and Integrable Systems by Pierre Dazord,Alan Weinstein Pdf

The papers, some of which are in English, the rest in French, in this volume are based on lectures given during the meeting of the Seminare Sud Rhodanien de Geometrie (SSRG) organized at the Mathematical Sciences Research Institute in 1989. The SSRG was established in 1982 by geometers and mathematical physicists with the aim of developing and coordinating research in symplectic geometry and its applications to analysis and mathematical physics. Among the subjects discussed at the meeting, a special role was given to the theory of symplectic groupoids, the subject of fruitful collaboration involving geometers from Berkeley, Lyon, and Montpellier.

Author : Yakov Eliashberg,Lisa M. Traynor
Publisher : American Mathematical Soc.
Page : 452 pages
File Size : 43,6 Mb
Release : 2004
Category : Mathematics
ISBN : 0821886894

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Symplectic Geometry and Topology by Yakov Eliashberg,Lisa M. Traynor Pdf

Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Author : Kang Feng,Mengzhao Qin
Publisher : Springer Science & Business Media
Page : 676 pages
File Size : 49,5 Mb
Release : 2010-10-18
Category : Mathematics
ISBN : 9783642017773

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Symplectic Geometric Algorithms for Hamiltonian Systems by Kang Feng,Mengzhao Qin Pdf

"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.

Author : David E. Blair
Publisher : Springer Science & Business Media
Page : 263 pages
File Size : 54,6 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781475736045

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Riemannian Geometry of Contact and Symplectic Manifolds by David E. Blair Pdf

Book endorsed by the Sunyer Prize Committee (A. Weinstein, J. Oesterle et. al.).

Author : Michele Audin,Jacques Lafontaine
Publisher : Birkhäuser
Page : 333 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034885089

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Holomorphic Curves in Symplectic Geometry by Michele Audin,Jacques Lafontaine Pdf

This book is devoted to pseudo-holomorphic curve methods in symplectic geometry. It contains an introduction to symplectic geometry and relevant techniques of Riemannian geometry, proofs of Gromov's compactness theorem, an investigation of local properties of holomorphic curves, including positivity of intersections, and applications to Lagrangian embeddings problems. The chapters are based on a series of lectures given previously by the authors M. Audin, A. Banyaga, P. Gauduchon, F. Labourie, J. Lafontaine, F. Lalonde, Gang Liu, D. McDuff, M.-P. Muller, P. Pansu, L. Polterovich, J.C. Sikorav. In an attempt to make this book accessible also to graduate students, the authors provide the necessary examples and techniques needed to understand the applications of the theory. The exposition is essentially self-contained and includes numerous exercises.

Author : Leonid Polterovich
Publisher : Unknown
Page : 150 pages
File Size : 51,8 Mb
Release : 2001-03-01
Category : Electronic
ISBN : 3034883005

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The Geometry of the Group of Symplectic Diffeomorphism by Leonid Polterovich Pdf

Author : Michèle Audin,Ana Cannas da Silva,Eugene Lerman
Publisher : Birkhäuser
Page : 225 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034880718

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Symplectic Geometry of Integrable Hamiltonian Systems by Michèle Audin,Ana Cannas da Silva,Eugene Lerman Pdf

Among all the Hamiltonian systems, the integrable ones have special geometric properties; in particular, their solutions are very regular and quasi-periodic. This book serves as an introduction to symplectic and contact geometry for graduate students, exploring the underlying geometry of integrable Hamiltonian systems. Includes exercises designed to complement the expositiont, and up-to-date references.

Author : Franco Cardin
Publisher : Springer
Page : 222 pages
File Size : 54,9 Mb
Release : 2014-12-01
Category : Science
ISBN : 9783319110264

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Elementary Symplectic Topology and Mechanics by Franco Cardin Pdf

This is a short tract on the essentials of differential and symplectic geometry together with a basic introduction to several applications of this rich framework: analytical mechanics, the calculus of variations, conjugate points & Morse index, and other physical topics. A central feature is the systematic utilization of Lagrangian submanifolds and their Maslov-Hörmander generating functions. Following this line of thought, first introduced by Wlodemierz Tulczyjew, geometric solutions of Hamilton-Jacobi equations, Hamiltonian vector fields and canonical transformations are described by suitable Lagrangian submanifolds belonging to distinct well-defined symplectic structures. This unified point of view has been particularly fruitful in symplectic topology, which is the modern Hamiltonian environment for the calculus of variations, yielding sharp sufficient existence conditions. This line of investigation was initiated by Claude Viterbo in 1992; here, some primary consequences of this theory are exposed in Chapter 8: aspects of Poincaré's last geometric theorem and the Arnol'd conjecture are introduced. In Chapter 7 elements of the global asymptotic treatment of the highly oscillating integrals for the Schrödinger equation are discussed: as is well known, this eventually leads to the theory of Fourier Integral Operators. This short handbook is directed toward graduate students in Mathematics and Physics and to all those who desire a quick introduction to these beautiful subjects.

Author : A.T. Fomenko
Publisher : CRC Press
Page : 488 pages
File Size : 46,6 Mb
Release : 1995-11-30
Category : Mathematics
ISBN : 2881249019

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Symplectic Geometry by A.T. Fomenko Pdf

Author : Andrew McInerney
Publisher : Springer Science & Business Media
Page : 410 pages
File Size : 45,5 Mb
Release : 2013-07-09
Category : Mathematics
ISBN : 9781461477327

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First Steps in Differential Geometry by Andrew McInerney Pdf

Differential geometry arguably offers the smoothest transition from the standard university mathematics sequence of the first four semesters in calculus, linear algebra, and differential equations to the higher levels of abstraction and proof encountered at the upper division by mathematics majors. Today it is possible to describe differential geometry as "the study of structures on the tangent space," and this text develops this point of view. This book, unlike other introductory texts in differential geometry, develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. The main goal of this book is to bring the undergraduate student who already has a solid foundation in the standard mathematics curriculum into contact with the beauty of higher mathematics. In particular, the presentation here emphasizes the consequences of a definition and the careful use of examples and constructions in order to explore those consequences.