Symplectic Techniques In Physics

Author : Victor Guillemin,Shlomo Sternberg
Publisher : Cambridge University Press
Page : 488 pages
File Size : 42,8 Mb
Release : 1990-05-25
Category : Mathematics
ISBN : 0521389909

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Symplectic Techniques in Physics by Victor Guillemin,Shlomo Sternberg Pdf

Symplectic geometry is very useful for formulating clearly and concisely problems in classical physics and also for understanding the link between classical problems and their quantum counterparts. It is thus a subject of interest to both mathematicians and physicists, though they have approached the subject from different viewpoints. This is the first book that attempts to reconcile these approaches. The authors use the uncluttered, coordinate-free approach to symplectic geometry and classical mechanics that has been developed by mathematicians over the course of the past thirty years, but at the same time apply the apparatus to a great number of concrete problems. Some of the themes emphasized in the book include the pivotal role of completely integrable systems, the importance of symmetries, analogies between classical dynamics and optics, the importance of symplectic tools in classical variational theory, symplectic features of classical field theories, and the principle of general covariance.

Author : Maurice A. de Gosson
Publisher : Springer Science & Business Media
Page : 351 pages
File Size : 43,7 Mb
Release : 2011-07-30
Category : Mathematics
ISBN : 9783764399924

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Symplectic Methods in Harmonic Analysis and in Mathematical Physics by Maurice A. de Gosson Pdf

The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Author : Victor Guillemin,Eugene Lerman,Shlomo Sternberg
Publisher : Cambridge University Press
Page : 238 pages
File Size : 47,9 Mb
Release : 1996-09-28
Category : Mathematics
ISBN : 9780521443234

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Symplectic Fibrations and Multiplicity Diagrams by Victor Guillemin,Eugene Lerman,Shlomo Sternberg Pdf

Applications of the techniques of symplectic geometry to describe 'symmetry breaking' in quantum physics.

Author : Maurice A. de Gosson
Publisher : Springer Science & Business Media
Page : 375 pages
File Size : 46,7 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 : Rolf Berndt
Publisher : American Mathematical Society
Page : 213 pages
File Size : 43,6 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 : Kang Feng,Mengzhao Qin
Publisher : Springer Science & Business Media
Page : 690 pages
File Size : 45,5 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 : Anonim
Publisher : Springer
Page : 364 pages
File Size : 50,7 Mb
Release : 2012-02-05
Category : Electronic
ISBN : 303460016X

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Symplectic Methods in Harmonic Analysis and in Mathematical Physics by Anonim Pdf

Author : Ana Cannas da Silva
Publisher : Springer
Page : 220 pages
File Size : 43,6 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 : V.I. Arnol'd
Publisher : Springer Science & Business Media
Page : 530 pages
File Size : 50,6 Mb
Release : 2013-04-09
Category : Mathematics
ISBN : 9781475720631

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Mathematical Methods of Classical Mechanics by V.I. Arnol'd Pdf

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

Author : Gerd Rudolph,Matthias Schmidt
Publisher : Springer Science & Business Media
Page : 766 pages
File Size : 42,8 Mb
Release : 2012-11-09
Category : Science
ISBN : 9789400753457

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Differential Geometry and Mathematical Physics by Gerd Rudolph,Matthias Schmidt Pdf

Starting from an undergraduate level, this book systematically develops the basics of • Calculus on manifolds, vector bundles, vector fields and differential forms, • Lie groups and Lie group actions, • Linear symplectic algebra and symplectic geometry, • Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous detailed examples, some of which are taken up several times for demonstrating how the methods evolve and interact.

Author : Maurice A. de Gosson
Publisher : Birkhäuser
Page : 368 pages
File Size : 46,6 Mb
Release : 2009-09-03
Category : Mathematics
ISBN : 3764391251

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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 : Vicente Cortés,Alexander S. Haupt
Publisher : Springer
Page : 99 pages
File Size : 54,6 Mb
Release : 2017-04-26
Category : Science
ISBN : 9783319564630

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Mathematical Methods of Classical Physics by Vicente Cortés,Alexander S. Haupt Pdf

This short primer, geared towards students with a strong interest in mathematically rigorous approaches, introduces the essentials of classical physics, briefly points out its place in the history of physics and its relation to modern physics, and explains what benefits can be gained from a mathematical perspective. As a starting point, Newtonian mechanics is introduced and its limitations are discussed. This leads to and motivates the study of different formulations of classical mechanics, such as Lagrangian and Hamiltonian mechanics, which are the subjects of later chapters. In the second part, a chapter on classical field theories introduces more advanced material. Numerous exercises are collected in the appendix.

Author : Robert Shar
Publisher : Elsevier
Page : 685 pages
File Size : 54,9 Mb
Release : 2012-12-02
Category : Science
ISBN : 9780323141529

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Group Theoretical Methods in Physics by Robert Shar Pdf

Group Theoretical Methods in Physics: Proceedings of the Fifth International Colloquium provides information pertinent to the fundamental aspects of group theoretical methods in physics. This book provides a variety of topics, including nuclear collective motion, complex Riemannian geometry, quantum mechanics, and relativistic symmetry. Organized into six parts encompassing 64 chapters, this book begins with an overview of the theories of nuclear quadrupole dynamics. This text then examines the conventional approach in the determination of superstructures. Other chapters consider the Hamiltonian formalism and how it is applied to the KdV equation and to a slight variant of the KdV equation. This book discusses as well the significant differential equations of mathematical physics that are integrable Hamiltonian systems, including the equations governing self-induced transparency and the motion of particles under an inverse square potential. The final chapter deals with the decomposition of the tensor product of two irreducible representations of the symmetric group into a direct sum of irreducible representations. This book is a valuable resource for physicists.

Author : Michael Spivak
Publisher : Unknown
Page : 733 pages
File Size : 40,9 Mb
Release : 2010
Category : Mechanics
ISBN : 0914098322

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Physics for Mathematicians by Michael Spivak Pdf

Author : John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce
Publisher : American Mathematical Soc.
Page : 300 pages
File Size : 53,5 Mb
Release : 2019-04-12
Category : Geometry, Differential
ISBN : 9781470450144

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Virtual Fundamental Cycles in Symplectic Topology by John W. Morgan,Dusa McDuff,Mohammad Tehrani,Kenji Fukaya,Dominic Joyce Pdf

The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov's original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.