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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,
Nearly Integrable Infinite-Dimensional Hamiltonian Systems by Sergej B. Kuksin Pdf
The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.
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.
Lectures: J. Guckenheimer: Bifurcations of dynamical systems.- J. Moser: Various aspects of integrable.- S. Newhouse: Lectures on dynamical systems.- Seminars: A. Chenciner: Hopf bifurcation for invariant tori.- M. Misiurewicz: Horseshoes for continuous mappings of an interval.
Integrable Hamiltonian systems and spectral theory by Jürgen Moser Pdf
These notes are based on six Fermi Lectures held at the Scuola Normale Superiore in Pisa in March and April 1981. The topics treated depend on basic concepts of classical mechanics, elementary geometry, complex analysis as well as spectral theory and are meant for mathematicians and theoretical physicists alike. These lectures weave together a number of threads from various fields of mathematics impinging on the subject of inverse spectral theory. I did not try to give an overview over this fast moving subject but rather tie various aspects together by one guiding theme: the construction of all potentials for the one-dimensional Schrödinger equation which gives rise to finite band potentials, which is done by reducing it to solving a system of differential equations. In fact, we will see that the problem of finding all almost periodic potentials having finitely many intervals as its spectrum is equivalent to the study of the geodesics on an ellipsoid. To make this connection clear we have carried together several facts from classical mechanics and from spectral theory and we give a self-contained exposition of the construction of these finite band potentials.
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.
Hamiltonian Systems and Their Integrability by Mich'le Audin Pdf
"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. This book would be suitable for a graduate course in Hamiltonian systems."--BOOK JACKET.
Mainly drawing on explicit examples, the author introduces the reader to themost recent techniques to study finite and infinite dynamical systems. Without any knowledge of differential geometry or lie groups theory the student can follow in a series of case studies the most recent developments. r-matrices for Calogero-Moser systems and Toda lattices are derived. Lax pairs for nontrivial infinite dimensionalsystems are constructed as limits of classical matrix algebras. The reader will find explanations of the approach to integrable field theories, to spectral transform methods and to solitons. New methods are proposed, thus helping students not only to understand established techniques but also to interest them in modern research on dynamical systems.
Integrable Hamiltonian Hierarchies by Vladimir Gerdjikov,Gaetano Vilasi,Alexandar Borisov Yanovski Pdf
This book presents a detailed derivation of the spectral properties of the Recursion Operators allowing one to derive all the fundamental properties of the soliton equations and to study their hierarchies.
What Is Integrability? by Vladimir E. Zakharov Pdf
The idea of devoting a complete book to this topic was born at one of the Workshops on Nonlinear and Turbulent Processes in Physics taking place reg ularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors of this volume were participants at the third of these workshops. All of them were acquainted with each other and with each other's work. Yet it seemed to be somewhat of a discovery that all of them were and are trying to understand the same problem - the problem of integrability of dynamical systems, primarily Hamiltonian ones with an infinite number of degrees of freedom. No doubt that they (or to be more exact, we) were led to this by the logical process of scientific evolution which often leads to independent, almost simultaneous discoveries. Integrable, or, more accurately, exactly solvable equations are essential to theoretical and mathematical physics. One could say that they constitute the "mathematical nucleus" of theoretical physics whose goal is to describe real clas sical or quantum systems. For example, the kinetic gas theory may be considered to be a theory of a system which is trivially integrable: the system of classical noninteracting particles. One of the main tasks of quantum electrodynamics is the development of a theory of an integrable perturbed quantum system, namely, noninteracting electromagnetic and electron-positron fields.
Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt
Author : Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt Publisher : Springer Science & Business Media Page : 505 pages File Size : 49,5 Mb Release : 2007-07-05 Category : Mathematics ISBN : 9783540489269
Mathematical Aspects of Classical and Celestial Mechanics by Vladimir I. Arnold,Valery V. Kozlov,Anatoly I. Neishtadt Pdf
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.
Integrable and Superintegrable Systems by Boris A. Kupershmidt Pdf
Some of the most active practitioners in the field of integrable systems have been asked to describe what they think of as the problems and results which seem to be most interesting and important now and are likely to influence future directions. The papers in this collection, representing their authors' responses, offer a broad panorama of the subject as it enters the 1990's.
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.
Multi-Hamiltonian Theory of Dynamical Systems by Maciej Blaszak Pdf
This book offers a modern introduction to the Hamiltonian theory of dynamical systems, presenting a unified treatment of all types of dynamical systems, i.e., finite, lattice, and field. Particular attention is paid to nonlinear systems that have more than one Hamiltonian formulation in a single coordinate system. As this property is closely related to integrability, this book presents an algebraic theory of integrable.
