Geometry And Dynamics Of Integrable Systems

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Geometry and Dynamics of Integrable Systems

Alexey Bolsinov,Juan J. Morales-Ruiz,Nguyen Tien Zung
Geometry and Dynamics of Integrable Systems Book PDF

Author : Alexey Bolsinov,Juan J. Morales-Ruiz,Nguyen Tien Zung
Publisher : Birkhäuser
Page : 140 pages
File Size : 42,9 Mb
Release : 2016-10-27
Category : Mathematics
ISBN : 9783319335032

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Geometry and Dynamics of Integrable Systems by Alexey Bolsinov,Juan J. Morales-Ruiz,Nguyen Tien Zung Pdf

Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields. Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mirror symmetry). As such, the book will appeal to experts with a wide range of backgrounds.

Integrability, Quantization, and Geometry: I. Integrable Systems

Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Integrability Quantization and Geometry I Integrable Systems Book PDF

Author : Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman
Publisher : American Mathematical Soc.
Page : 516 pages
File Size : 55,9 Mb
Release : 2021-04-12
Category : Education
ISBN : 9781470455910

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Integrability, Quantization, and Geometry: I. Integrable Systems by Sergey Novikov,Igor Krichever,Oleg Ogievetsky,Senya Shlosman Pdf

This book is a collection of articles written in memory of Boris Dubrovin (1950–2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions to this collection of papers are split into two parts: “Integrable Systems” and “Quantum Theories and Algebraic Geometry”, reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Integrable Hamiltonian Systems

A.V. Bolsinov,A.T. Fomenko
Integrable Hamiltonian Systems Book PDF

Author : A.V. Bolsinov,A.T. Fomenko
Publisher : CRC Press
Page : 752 pages
File Size : 40,6 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,

Discrete Integrable Geometry and Physics

Alexander I. Bobenko,Ruedi Seiler
Discrete Integrable Geometry and Physics Book PDF

Author : Alexander I. Bobenko,Ruedi Seiler
Publisher : Clarendon Press
Page : 466 pages
File Size : 48,5 Mb
Release : 1999
Category : Mathematics
ISBN : 0198501609

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Discrete Integrable Geometry and Physics by Alexander I. Bobenko,Ruedi Seiler Pdf

Recent interactions between the fields of geometry, classical and quantum dynamical systems, and visualization of geometric objects such as curves and surfaces have led to the observation that most concepts of surface theory and of the theory of integrable systems have natural discreteanalogues. These are characterized by the property that the corresponding difference equations are integrable, and has led in turn to some important applications in areas of condensed matter physics and quantum field theory, amongst others. The book combines the efforts of a distinguished team ofauthors from various fields in mathematics and physics in an effort to provide an overview of the subject. The mathematical concepts of discrete geometry and discrete integrable systems are firstly presented as fundamental and valuable theories in themselves. In the following part these concepts areput into the context of classical and quantum dynamics.

Dynamical Systems VII

V.I. Arnol'd,S.P. Novikov
Dynamical Systems VII Book PDF

Author : V.I. Arnol'd,S.P. Novikov
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 51,5 Mb
Release : 2013-12-14
Category : Mathematics
ISBN : 9783662067963

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Dynamical Systems VII by V.I. Arnol'd,S.P. Novikov Pdf

A collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.

Integrability and Nonintegrability of Dynamical Systems

Alain Goriely
Integrability and Nonintegrability of Dynamical Systems Book PDF

Author : Alain Goriely
Publisher : World Scientific
Page : 438 pages
File Size : 48,8 Mb
Release : 2001
Category : Science
ISBN : 981281194X

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Integrability and Nonintegrability of Dynamical Systems by Alain Goriely Pdf

This invaluable book examines qualitative and quantitative methods for nonlinear differential equations, as well as integrability and nonintegrability theory. Starting from the idea of a constant of motion for simple systems of differential equations, it investigates the essence of integrability, its geometrical relevance and dynamical consequences. Integrability theory is approached from different perspectives, first in terms of differential algebra, then in terms of complex time singularities and finally from the viewpoint of phase geometry (for both Hamiltonian and non-Hamiltonian systems). As generic systems of differential equations cannot be exactly solved, the book reviews the different notions of nonintegrability and shows how to prove the nonexistence of exact solutions and/or a constant of motion. Finally, nonintegrability theory is linked to dynamical systems theory by showing how the property of complete integrability, partial integrability or nonintegrability can be related to regular and irregular dynamics in phase space. Contents: Integrability: An Algebraic Approach; Integrability: An Analytic Approach; Polynomial and Quasi-Polynomial Vector Fields; Nonintegrability; Hamiltonian Systems; Nearly Integrable Dynamical Systems; Open Problems. Readership: Mathematical and theoretical physicists and astronomers and engineers interested in dynamical systems.

Geometry from Dynamics, Classical and Quantum

José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi
Geometry from Dynamics Classical and Quantum Book PDF

Author : José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi
Publisher : Springer
Page : 719 pages
File Size : 48,7 Mb
Release : 2014-09-23
Category : Science
ISBN : 9789401792202

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Geometry from Dynamics, Classical and Quantum by José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi Pdf

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

Integrable Systems in the realm of Algebraic Geometry

Pol Vanhaecke
Integrable Systems in the realm of Algebraic Geometry Book PDF

Author : Pol Vanhaecke
Publisher : Springer
Page : 226 pages
File Size : 40,5 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662215357

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Integrable Systems in the realm of Algebraic Geometry by Pol Vanhaecke Pdf

Integrable systems are related to algebraic geometry in many different ways. This book deals with some aspects of this relation, the main focus being on the algebraic geometry of the level manifolds of integrable systems and the construction of integrable systems, starting from algebraic geometric data. For a rigorous account of these matters, integrable systems are defined on affine algebraic varieties rather than on smooth manifolds. The exposition is self-contained and is accessible at the graduate level; in particular, prior knowledge of integrable systems is not assumed.

Integrable Systems, Geometry, and Topology

Chuu-lian Terng
Integrable Systems Geometry and Topology Book PDF

Author : Chuu-lian Terng
Publisher : American Mathematical Soc.
Page : 270 pages
File Size : 45,9 Mb
Release : 2006
Category : Geometry
ISBN : 9780821840481

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Integrable Systems, Geometry, and Topology by Chuu-lian Terng Pdf

The articles in this volume are based on lectures from a program on integrable systems and differential geometry held at Taiwan's National Center for Theoretical Sciences. As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems. The article by Burstall gives a beautiful exposition on isothermic surfaces and theirrelations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental's quantum differential equations. The article by Heintze, Liu,and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of Yang-Mills-Higgs equations on Riemann surfaces. The article by Terng and Uhlenbeck explains the gauge equivalence of the matrix non-linear Schrödinger equation, the Schrödinger flow on Grassmanian, and the Heisenberg Feromagnetic model. The bookprovides an introduction to integrable systems and their relation to differential geometry. It is suitable for advanced graduate students and research mathematicians. Information for our distributors: Titles in this series are copublished with International Press, Cambridge, MA.

Integrable Systems and Algebraic Geometry

Ron Donagi,Tony Shaska
Integrable Systems and Algebraic Geometry Book PDF

Author : Ron Donagi,Tony Shaska
Publisher : Cambridge University Press
Page : 421 pages
File Size : 41,9 Mb
Release : 2020-04-02
Category : Mathematics
ISBN : 9781108715744

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Integrable Systems and Algebraic Geometry by Ron Donagi,Tony Shaska Pdf

A collection of articles discussing integrable systems and algebraic geometry from leading researchers in the field.

A Memoir on Integrable Systems

Yuri Fedorov,Valerij Vasilievich Kozlov
A Memoir on Integrable Systems Book PDF

Author : Yuri Fedorov,Valerij Vasilievich Kozlov
Publisher : Springer
Page : 0 pages
File Size : 47,5 Mb
Release : 2017-03-14
Category : Mathematics
ISBN : 3540590005

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A Memoir on Integrable Systems by Yuri Fedorov,Valerij Vasilievich Kozlov Pdf

This book considers the larger class of systems which are not (at least a priori) Hamiltonian but possess tensor invariants, in particular, an invariant measure. Several integrability theorems related to the existence of tensor invariants are formulated, and the authors illustrate the geometrical background of some classical and new hierarchies of integrable systems and give their explicit solution in terms of theta-functions. Most of the results discussed have not been published before, making this book immensely useful both to specialists in analytical dynamics who are interested in integrable problems and those in algebraic geometry who are looking for applications.

The Geometry of Infinite-Dimensional Groups

Boris Khesin,Robert Wendt
The Geometry of Infinite Dimensional Groups Book PDF

Author : Boris Khesin,Robert Wendt
Publisher : Springer Science & Business Media
Page : 304 pages
File Size : 55,7 Mb
Release : 2008-09-28
Category : Mathematics
ISBN : 9783540772637

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The Geometry of Infinite-Dimensional Groups by Boris Khesin,Robert Wendt Pdf

This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.

Symplectic Geometry of Integrable Hamiltonian Systems

Michèle Audin,Ana Cannas da Silva,Eugene Lerman
Symplectic Geometry of Integrable Hamiltonian Systems Book PDF

Author : Michèle Audin,Ana Cannas da Silva,Eugene Lerman
Publisher : Springer Science & Business Media
Page : 240 pages
File Size : 44,5 Mb
Release : 2003-04-24
Category : Mathematics
ISBN : 3764321679

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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.

Integrable Systems

N. J. Hitchin,G. B. Segal,R. S. Ward
Integrable Systems Book PDF

Author : N. J. Hitchin,G. B. Segal,R. S. Ward
Publisher : Oxford University Press
Page : 150 pages
File Size : 42,9 Mb
Release : 1999-03-18
Category : Mathematics
ISBN : 0198504217

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

This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The authors are internationally renowned both as researchers and expositors, and the book is written in an informal and accessible style.