Hamiltonian Systems And Their Integrability
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Hamiltonian Systems And Their Integrability
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Author : Mich'le Audin
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Hamiltonian Systems And Their Integrability written by Mich'le Audin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
Hamiltonian systems began as a mathematical approach to the study of mechanical systems. As the theory developed, it became clear that the systems that had a sufficient number of conserved quantities enjoyed certain remarkable properties. These are the completely integrable systems. In time, a rich interplay arose between integrable systems and other areas of mathematics, particularly topology, geometry, and group theory.This book presents some modern techniques in the theory of integrable systems viewed as variations on the theme of action-angle coordinates. These techniques include analytical methods coming from the Galois theory of differential equations, as well as more classical algebro-geometric methods related to Lax equations. Audin has included many examples and exercises. Most of the exercises build on the material in the text. None of the important proofs have been relegated to the exercises. Many of the examples are classical, rather than abstract. This book would be suitable for a graduate course in Hamiltonian systems.
Differential Galois Theory And Non Integrability Of Hamiltonian Systems
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Author : Juan J. Morales Ruiz
language : en
Publisher: Springer Science & Business Media
Release Date : 1999-08-01
Differential Galois Theory And Non Integrability Of Hamiltonian Systems written by Juan J. Morales Ruiz 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 1999-08-01 with Mathematics categories.
This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Lyapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)
Mathematical Aspects Of Classical And Celestial Mechanics
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Author : Vladimir I. Arnold
language : en
Publisher: Springer
Release Date : 2010-11-13
Mathematical Aspects Of Classical And Celestial Mechanics written by Vladimir I. Arnold and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-13 with Mathematics categories.
The main purpose of the book is to acquaint mathematicians, physicists and engineers with classical mechanics as a whole, in both its traditional and its contemporary aspects. As such, it describes the fundamental principles, problems, and methods of classical mechanics, with the emphasis firmly laid on the working apparatus, rather than the physical foundations or applications. Chapters cover the n-body problem, symmetry groups of mechanical systems and the corresponding conservation laws, the problem of the integrability of the equations of motion, the theory of oscillations and perturbation theory.
Algebraic Integrability Of Nonlinear Dynamical Systems On Manifolds
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Author : A.K. Prykarpatsky
language : en
Publisher: Springer
Release Date : 1998-06-30
Algebraic Integrability Of Nonlinear Dynamical Systems On Manifolds written by A.K. Prykarpatsky and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-06-30 with Mathematics categories.
Noting that their research is not yet completed, Prykarpatsky (mathematics, U. of Mining and Metallurgy, Cracow, Poland and mechanics and mathematics, NAS, Lviv, Ukraine) and Mykytiuk (mechanics and mathematics, NAS and Lviv Polytechnic State U., Ukraine) describe some of the ideas of Lie algebra that underlie many of the comprehensive integrability theories of nonlinear dynamical systems on manifolds. For each case they analyze, they separate the basic algebraic essence responsible for the complete integrability and explore how it is also in some sense characteristic for all of them. They cover systems with homogeneous configuration spaces, geometric quantization, structures on manifolds, algebraic methods of quantum statistical mechanics and their applications, and algebraic and differential geometric aspects related to infinite-dimensional functional manifolds. They have not indexed their work.
Notes On Hamiltonian Dynamical Systems
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Author : Antonio Giorgilli
language : en
Publisher: Cambridge University Press
Release Date : 2022-05-05
Notes On Hamiltonian Dynamical Systems written by Antonio Giorgilli and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-05 with Science categories.
Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov–Arnold–Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincaré's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincaré and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.
Dynamical Systems Vii
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Author : V.I. Arnol'd
language : en
Publisher: Springer Science & Business Media
Release Date : 1993-12-06
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 1993-12-06 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.
Global Aspects Of Classical Integrable Systems
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Author : Richard H. Cushman
language : en
Publisher: Birkhäuser
Release Date : 2015-06-01
Global Aspects Of Classical Integrable Systems written by Richard H. Cushman and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-01 with Science categories.
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.
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.
Integrable Systems
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Author : N. J. Hitchin
language : en
Publisher: Oxford University Press
Release Date : 1999-03-18
Integrable Systems written by N. J. Hitchin and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-03-18 with Mathematics categories.
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.
Introduction To Classical Integrable Systems
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Author : Olivier Babelon
language : en
Publisher: Cambridge University Press
Release Date : 2003-04-17
Introduction To Classical Integrable Systems written by Olivier Babelon and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-04-17 with Mathematics categories.
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.