Geometry And Dynamics Of Integrable Systems

Author : Alexey Bolsinov
Publisher : Birkhäuser
ISBN 13 : 3319335030
Total Pages : 140 pages
Book Rating : 4.32/5 ( download)

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Download or read book Geometry and Dynamics of Integrable Systems written by Alexey Bolsinov and published by Birkhäuser. This book was released on 2016-10-27 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : V.I. Arnol'd
Publisher : Springer Science & Business Media
ISBN 13 : 366206796X
Total Pages : 346 pages
Book Rating : 4.63/5 ( download)

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Download or read book Dynamical Systems VII written by V.I. Arnol'd and published by Springer Science & Business Media. This book was released on 2013-12-14 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : A.V. Bolsinov
Publisher : CRC Press
ISBN 13 : 0203643429
Total Pages : 752 pages
Book Rating : 4.26/5 ( download)

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Download or read book Integrable Hamiltonian Systems written by A.V. Bolsinov and published by CRC Press. This book was released on 2004-02-25 with total page 752 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Alexander I. Bobenko
Publisher : Clarendon Press
ISBN 13 : 9780198501602
Total Pages : 466 pages
Book Rating : 4.09/5 ( download)

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Download or read book Discrete Integrable Geometry and Physics written by Alexander I. Bobenko and published by Clarendon Press. This book was released on 1999 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Alain Goriely
Publisher : World Scientific
ISBN 13 : 981023533X
Total Pages : 435 pages
Book Rating : 4.38/5 ( download)

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Download or read book Integrability and Nonintegrability of Dynamical Systems written by Alain Goriely and published by World Scientific. This book was released on 2001 with total page 435 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Xiang Zhang
Publisher : Springer
ISBN 13 : 9811042268
Total Pages : 380 pages
Book Rating : 4.63/5 ( download)

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Download or read book Integrability of Dynamical Systems: Algebra and Analysis written by Xiang Zhang and published by Springer. This book was released on 2017-03-30 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to systematically state the fundamental theory of integrability and its development of ordinary differential equations with emphasis on the Darboux theory of integrability and local integrability together with their applications. It summarizes the classical results of Darboux integrability and its modern development together with their related Darboux polynomials and their applications in the reduction of Liouville and elementary integrabilty and in the center—focus problem, the weakened Hilbert 16th problem on algebraic limit cycles and the global dynamical analysis of some realistic models in fields such as physics, mechanics and biology. Although it can be used as a textbook for graduate students in dynamical systems, it is intended as supplementary reading for graduate students from mathematics, physics, mechanics and engineering in courses related to the qualitative theory, bifurcation theory and the theory of integrability of dynamical systems.

Author : Velimir Jurdjevic
Publisher : Cambridge University Press
ISBN 13 : 1316586332
Total Pages : 437 pages
Book Rating : 4.34/5 ( download)

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Download or read book Optimal Control and Geometry: Integrable Systems written by Velimir Jurdjevic and published by Cambridge University Press. This book was released on 2016-07-04 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.

Author : José F. Cariñena
Publisher : Springer
ISBN 13 : 9401792208
Total Pages : 739 pages
Book Rating : 4.02/5 ( download)

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Download or read book Geometry from Dynamics, Classical and Quantum written by José F. Cariñena and published by Springer. This book was released on 2014-09-23 with total page 739 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : V.I. Arnol'd
Publisher : Springer
ISBN 13 : 9783662067970
Total Pages : 344 pages
Book Rating : 4.78/5 ( download)

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Download or read book Dynamical Systems VII written by V.I. Arnol'd and published by Springer. This book was released on 2014-10-05 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Alain Goriely
Publisher : World Scientific
ISBN 13 : 9789812811943
Total Pages : 438 pages
Book Rating : 4.4X/5 ( download)

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Download or read book Integrability and Nonintegrability of Dynamical Systems written by Alain Goriely and published by World Scientific. This book was released on 2001 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.