Geometry And Dynamics Of Integrable Systems
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Geometry And Dynamics Of Integrable Systems
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Author : Alexey Bolsinov
language : en
Publisher: Birkhäuser
Release Date : 2016-10-27
Geometry And Dynamics Of Integrable Systems written by Alexey Bolsinov and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-10-27 with Mathematics categories.
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.
Integrable Hamiltonian Systems
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Author : A.V. Bolsinov
language : en
Publisher: CRC Press
Release Date : 2004-02-25
Integrable Hamiltonian Systems written by A.V. Bolsinov and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-02-25 with Mathematics categories.
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
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Author : Alexander I. Bobenko
language : en
Publisher: Clarendon Press
Release Date : 1999
Discrete Integrable Geometry And Physics written by Alexander I. Bobenko and has been published by Clarendon Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999 with Mathematics categories.
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.
Integrability And Nonintegrability Of Dynamical Systems
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Author : Alain Goriely
language : en
Publisher: World Scientific
Release Date : 2001
Integrability And Nonintegrability Of Dynamical Systems written by Alain Goriely and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Science categories.
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.
Dynamical Systems Vii
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Author : V.I. Arnol'd
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-12-14
Dynamical Systems Vii written by V.I. Arnol'd and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-12-14 with Mathematics categories.
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.
Integrable Systems In The Realm Of Algebraic Geometry
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Author : Pol Vanhaecke
language : en
Publisher: Springer
Release Date : 2013-11-11
Integrable Systems In The Realm Of Algebraic Geometry written by Pol Vanhaecke and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
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.
The Geometry Of Infinite Dimensional Groups
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Author : Boris Khesin
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-09-28
The Geometry Of Infinite Dimensional Groups written by Boris Khesin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-28 with Mathematics categories.
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.
Vii Dynamical Systems
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Author : S.P. Novikov
language : en
Publisher:
Release Date : 1994
Vii Dynamical Systems written by S.P. Novikov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Chaotic behavior in systems categories.
中国科学院科学出版基金资助出版
Geometry From Dynamics Classical And Quantum
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Author : José F. Carinena
language : en
Publisher: Springer
Release Date : 2014-09-28
Geometry From Dynamics Classical And Quantum written by José F. Carinena and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-28 with Science categories.
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.
A Memoir On Integrable Systems
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Author : Yuri Fedorov
language : en
Publisher: Springer
Release Date : 2017-03-14
A Memoir On Integrable Systems written by Yuri Fedorov and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-14 with Mathematics categories.
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.