Introduction To Classical Integrable Systems

Author : Olivier Babelon,Denis Bernard,Michel Talon
Publisher : Cambridge University Press
Page : 622 pages
File Size : 45,7 Mb
Release : 2003-04-17
Category : Mathematics
ISBN : 052182267X

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Introduction to Classical Integrable Systems by Olivier Babelon,Denis Bernard,Michel Talon Pdf

This book provides a thorough introduction to the theory of classical integrable systems, discussing the various approaches to the subject and explaining their interrelations. The book begins by introducing the central ideas of the theory of integrable systems, based on Lax representations, loop groups and Riemann surfaces. These ideas are then illustrated with detailed studies of model systems. The connection between isomonodromic deformation and integrability is discussed, and integrable field theories are covered in detail. The KP, KdV and Toda hierarchies are explained using the notion of Grassmannian, vertex operators and pseudo-differential operators. A chapter is devoted to the inverse scattering method and three complementary chapters cover the necessary mathematical tools from symplectic geometry, Riemann surfaces and Lie algebras. The book contains many worked examples and is suitable for use as a textbook on graduate courses. It also provides a comprehensive reference for researchers already working in the field.

Author : Olivier Babelon,Denis Bernard,Michel Talon
Publisher : Unknown
Page : 602 pages
File Size : 52,8 Mb
Release : 2003
Category : Dynamics
ISBN : 7510004578

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Introduction to Classical Integrable Systems by Olivier Babelon,Denis Bernard,Michel Talon Pdf

Author : Gleb Arutyunov
Publisher : Springer
Page : 414 pages
File Size : 55,9 Mb
Release : 2019-07-23
Category : Science
ISBN : 9783030241988

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Elements of Classical and Quantum Integrable Systems by Gleb Arutyunov Pdf

Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.

Author : Fabio Franchini
Publisher : Springer
Page : 180 pages
File Size : 47,5 Mb
Release : 2017-05-25
Category : Science
ISBN : 9783319484877

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An Introduction to Integrable Techniques for One-Dimensional Quantum Systems by Fabio Franchini Pdf

This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.

Author : Asesh Roy Chowdhury,Aninlya Ghose Choudhury
Publisher : CRC Press
Page : 425 pages
File Size : 55,8 Mb
Release : 2004-01-28
Category : Science
ISBN : 9780203498019

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Quantum Integrable Systems by Asesh Roy Chowdhury,Aninlya Ghose Choudhury Pdf

The study of integrable systems has opened new horizons in classical physics over the past few decades, particularly in the subatomic world. Yet despite the field now having reached a level of maturity, very few books provide an introduction to the field accessible to specialists and nonspecialists alike, and none offer a systematic survey of the m

Author : Richard H. Cushman,Larry M. Bates
Publisher : Birkhäuser
Page : 477 pages
File Size : 43,9 Mb
Release : 2015-06-01
Category : Science
ISBN : 9783034809184

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Global Aspects of Classical Integrable Systems by Richard H. Cushman,Larry M. Bates Pdf

This book gives a uniquely complete description of the geometry of the energy momentum mapping of five classical integrable systems: the 2-dimensional harmonic oscillator, the geodesic flow on the 3-sphere, the Euler top, the spherical pendulum and the Lagrange top. It presents for the first time in book form a general theory of symmetry reduction which allows one to reduce the symmetries in the spherical pendulum and the Lagrange top. Also the monodromy obstruction to the existence of global action angle coordinates is calculated for the spherical pendulum and the Lagrange top. The book addresses professional mathematicians and graduate students and can be used as a textbook on advanced classical mechanics or global analysis.

Author : N.J. Hitchin,G. B. Segal,R.S. Ward
Publisher : Oxford University Press, USA
Page : 148 pages
File Size : 45,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9780199676774

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Integrable Systems by N.J. Hitchin,G. B. Segal,R.S. Ward Pdf

Designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors, this book has its origins in a lecture series given by the internationally renowned authors. Written in an accessible, informal style, it fills a gap in the existing literature.

Author : Matteo Petrera
Publisher : Unknown
Page : 0 pages
File Size : 52,5 Mb
Release : 2015
Category : Differential equations, Nonlinear
ISBN : 3832539506

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Mathematical Physics III - Integrable Systems of Classical Mechanics by Matteo Petrera Pdf

These Lecture Notes provide an introduction to the modern theory of classical finite-dimensional integrable systems. The first chapter focuses on some classical topics of differential geometry. This should help the reader to get acquainted with the required language of smooth manifolds, Lie groups and Lie algebras. The second chapter is devoted to Poisson and symplectic geometry with special emphasis on the construction of finite-dimensional Hamiltonian systems. Multi-Hamiltonian systems are also considered. In the third chapter the classical theory of Arnold-Liouville integrability is presented, while chapter four is devoted to a general overview of the modern theory of integrability. Among the topics covered are: Lie-Poisson structures, Lax formalism, double Lie algebras, R-brackets, Adler-Kostant-Symes scheme, Lie bialgebras, r-brackets. Some examples (Toda system, Garnier system, Gaudin system, Lagrange top) are presented in chapter five. They provide a concrete illustration of the theoretical part. Finally, the last chapter is devoted to a short overview of the problem of integrable discretization.

Author : Andreas Knauf
Publisher : Springer
Page : 683 pages
File Size : 51,5 Mb
Release : 2018-02-24
Category : Science
ISBN : 9783662557747

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Mathematical Physics: Classical Mechanics by Andreas Knauf Pdf

As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics. The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.

Author : A. M. Perelomov,Askolʹd Mikhaĭlovich Perelomov
Publisher : Springer
Page : 328 pages
File Size : 40,7 Mb
Release : 1990
Category : Electronic books
ISBN : UOM:39015017745830

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Integrable Systems of Classical Mechanics and Lie Algebras by A. M. Perelomov,Askolʹd Mikhaĭlovich Perelomov Pdf

This book offers a systematic presentation of a variety of methods and results concerning integrable systems of classical mechanics. The investigation of integrable systems was an important line of study in the last century, but up until recently only a small number of examples with two or more degrees of freedom were known. In the last fifteen years however, remarkable progress has been made in this field via the so-called isospectral deformation method which makes extensive use of group-theoretical concepts. The book focuses mainly on the development and applications of this new method, and also gives a fairly complete survey of the older classic results. Chapter 1 contains the necessary background material and outlines the isospectral deformation method in a Lie-algebraic form. Chapter 2 gives an account of numerous previously known integrable systems. Chapter 3 deals with many-body systems of generalized Calogero-Moser type, related to root systems of simple Lie algebras. Chapter 4 is devoted to the Toda lattice and its various modifications seen from the group-theoretic point of view. Chapter 5 investigates some additional topics related to many-body systems. The book will be valuable to students as well as researchers.

Author : Valeriĭ Viktorovich Kozlov
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 45,6 Mb
Release : 1995
Category : Mathematics
ISBN : 0821804278

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Dynamical Systems in Classical Mechanics by Valeriĭ Viktorovich Kozlov Pdf

This book shows that the phenomenon of integrability is related not only to Hamiltonian systems, but also to a wider variety of systems having invariant measures that often arise in nonholonomic mechanics. Each paper presents unique ideas and original approaches to various mathematical problems related to integrability, stability, and chaos in classical dynamics. Topics include... the inverse Lyapunov theorem on stability of equilibria geometrical aspects of Hamiltonian mechanics from a hydrodynamic perspective current unsolved problems in the dynamical systems approach to classical mechanics.

Author : A.V. Bolsinov,A.T. Fomenko
Publisher : CRC Press
Page : 752 pages
File Size : 45,6 Mb
Release : 2004-02-25
Category : Mathematics
ISBN : 9780203643426

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Integrable Hamiltonian Systems by A.V. Bolsinov,A.T. Fomenko Pdf

Integrable Hamiltonian systems have been of growing interest over the past 30 years and represent one of the most intriguing and mysterious classes of dynamical systems. This book explores the topology of integrable systems and the general theory underlying their qualitative properties, singularites, and topological invariants. The authors,

Author : Sacha Friedli,Yvan Velenik
Publisher : Cambridge University Press
Page : 643 pages
File Size : 48,9 Mb
Release : 2017-11-23
Category : Mathematics
ISBN : 9781107184824

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Statistical Mechanics of Lattice Systems by Sacha Friedli,Yvan Velenik Pdf

A self-contained, mathematical introduction to the driving ideas in equilibrium statistical mechanics, studying important models in detail.

Author : M. Audin
Publisher : Cambridge University Press
Page : 156 pages
File Size : 46,7 Mb
Release : 1999-11-13
Category : Mathematics
ISBN : 0521779197

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Spinning Tops by M. Audin Pdf

Since the time of Lagrange and Euler, it has been well known that an understanding of algebraic curves can illuminate the picture of rigid bodies provided by classical mechanics. A modern view of the role played by algebraic geometry has been established iby many mathematicians. This book presents some of these techniques, which fall within the orbit of finite dimensional integrable systems. The main body of the text presents a rich assortment of methods and ideas from algebraic geometry prompted by classical mechanics, whilst in appendices the general, abstract theory is described. The methods are given a topological application to the study of Liouville tori and their bifurcations. The book is based on courses for graduate students given by the author at Strasbourg University but the wealth of original ideas will make it also appeal to researchers.

Author : Ladislav Šamaj,Zoltán Bajnok
Publisher : Cambridge University Press
Page : 525 pages
File Size : 54,7 Mb
Release : 2013-05-16
Category : Science
ISBN : 9781107067660

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Introduction to the Statistical Physics of Integrable Many-body Systems by Ladislav Šamaj,Zoltán Bajnok Pdf

Including topics not traditionally covered in literature, such as (1+1)-dimensional QFT and classical 2D Coulomb gases, this book considers a wide range of models and demonstrates a number of situations to which they can be applied. Beginning with a treatise of nonrelativistic 1D continuum Fermi and Bose quantum gases of identical spinless particles, the book describes the quantum inverse scattering method and the analysis of the related Yang–Baxter equation and integrable quantum Heisenberg models. It also discusses systems within condensed matter physics, the complete solution of the sine-Gordon model and modern trends in the thermodynamic Bethe ansatz. Each chapter concludes with problems and solutions to help consolidate the reader's understanding of the theory and its applications. Basic knowledge of quantum mechanics and equilibrium statistical physics is assumed, making this book suitable for graduate students and researchers in statistical physics, quantum mechanics and mathematical and theoretical physics.