Mathematical Physics III - Integrable Systems of Classical Mechanics PDF Download
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Author: Matteo Petrera
Publisher:
ISBN: 9783832539504
Category : Differential equations, Nonlinear
Languages : en
Pages : 0
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Book Description
These Lecture Notes provide an introduction to the modern theory of classical finite-dimensional integrable systems. The first chapter focuses on some classical topics of differential geometry. This should help the reader to get acquainted with the required language of smooth manifolds, Lie groups and Lie algebras. The second chapter is devoted to Poisson and symplectic geometry with special emphasis on the construction of finite-dimensional Hamiltonian systems. Multi-Hamiltonian systems are also considered. In the third chapter the classical theory of Arnold-Liouville integrability is presented, while chapter four is devoted to a general overview of the modern theory of integrability. Among the topics covered are: Lie-Poisson structures, Lax formalism, double Lie algebras, R-brackets, Adler-Kostant-Symes scheme, Lie bialgebras, r-brackets. Some examples (Toda system, Garnier system, Gaudin system, Lagrange top) are presented in chapter five. They provide a concrete illustration of the theoretical part. Finally, the last chapter is devoted to a short overview of the problem of integrable discretization.
Author: Matteo Petrera
Publisher:
ISBN: 9783832539504
Category : Differential equations, Nonlinear
Languages : en
Pages : 0
Get Book
Book Description
These Lecture Notes provide an introduction to the modern theory of classical finite-dimensional integrable systems. The first chapter focuses on some classical topics of differential geometry. This should help the reader to get acquainted with the required language of smooth manifolds, Lie groups and Lie algebras. The second chapter is devoted to Poisson and symplectic geometry with special emphasis on the construction of finite-dimensional Hamiltonian systems. Multi-Hamiltonian systems are also considered. In the third chapter the classical theory of Arnold-Liouville integrability is presented, while chapter four is devoted to a general overview of the modern theory of integrability. Among the topics covered are: Lie-Poisson structures, Lax formalism, double Lie algebras, R-brackets, Adler-Kostant-Symes scheme, Lie bialgebras, r-brackets. Some examples (Toda system, Garnier system, Gaudin system, Lagrange top) are presented in chapter five. They provide a concrete illustration of the theoretical part. Finally, the last chapter is devoted to a short overview of the problem of integrable discretization.
Author: PERELOMOV
Publisher: Birkhäuser
ISBN: 3034892578
Category : Science
Languages : en
Pages : 312
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Book Description
This book offers a systematic presentation of a variety of methods and results concerning integrable systems of classical mechanics. The investigation of integrable systems was an important line of study in the last century, but up until recently only a small number of examples with two or more degrees of freedom were known. In the last fifteen years however, remarkable progress has been made in this field via the so-called isospectral deformation method which makes extensive use of group-theoretical concepts. The book focuses mainly on the development and applications of this new method, and also gives a fairly complete survey of the older classic results. Chapter 1 contains the necessary background material and outlines the isospectral deformation method in a Lie-algebraic form. Chapter 2 gives an account of numerous previously known integrable systems. Chapter 3 deals with many-body systems of generalized Calogero-Moser type, related to root systems of simple Lie algebras. Chapter 4 is devoted to the Toda lattice and its various modifications seen from the group-theoretic point of view. Chapter 5 investigates some additional topics related to many-body systems. The book will be valuable to students as well as researchers.
Author: Andreas Knauf
Publisher: Springer
ISBN: 3662557746
Category : Science
Languages : en
Pages : 683
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Book Description
As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics. The book addresses the needs of graduate and senior undergraduate students in mathematics and physics, and of researchers interested in approaching classical mechanics from a modern point of view.
Author: Gleb Arutyunov
Publisher: Springer
ISBN: 303024198X
Category : Science
Languages : en
Pages : 414
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Book Description
Integrable models have a fascinating history with many important discoveries that dates back to the famous Kepler problem of planetary motion. Nowadays it is well recognised that integrable systems play a ubiquitous role in many research areas ranging from quantum field theory, string theory, solvable models of statistical mechanics, black hole physics, quantum chaos and the AdS/CFT correspondence, to pure mathematics, such as representation theory, harmonic analysis, random matrix theory and complex geometry. Starting with the Liouville theorem and finite-dimensional integrable models, this book covers the basic concepts of integrability including elements of the modern geometric approach based on Poisson reduction, classical and quantum factorised scattering and various incarnations of the Bethe Ansatz. Applications of integrability methods are illustrated in vast detail on the concrete examples of the Calogero-Moser-Sutherland and Ruijsenaars-Schneider models, the Heisenberg spin chain and the one-dimensional Bose gas interacting via a delta-function potential. This book has intermediate and advanced topics with details to make them clearly comprehensible.
Author: Diarmuid Ó'Mathúna
Publisher: Springer Science & Business Media
ISBN: 0817645950
Category : Science
Languages : en
Pages : 241
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Book Description
Shows that exact solutions to the Kepler (two-body), the Euler (two-fixed center), and the Vinti (earth-satellite) problems can all be put in a form that admits the general representation of the orbits and follows a definite shared pattern Includes a full analysis of the planar Euler problem via a clear generalization of the form of the solution in the Kepler case Original insights that have hithertofore not appeared in book form
Author: Olivier Babelon
Publisher: Cambridge University Press
ISBN: 9780521822671
Category : Mathematics
Languages : en
Pages : 622
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Book Description
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.
Author: Anton Alekseev
Publisher: Springer Nature
ISBN: 3030781488
Category : Mathematics
Languages : en
Pages : 652
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Book Description
Over the course of his distinguished career, Nicolai Reshetikhin has made a number of groundbreaking contributions in several fields, including representation theory, integrable systems, and topology. The chapters in this volume – compiled on the occasion of his 60th birthday – are written by distinguished mathematicians and physicists and pay tribute to his many significant and lasting achievements. Covering the latest developments at the interface of noncommutative algebra, differential and algebraic geometry, and perspectives arising from physics, this volume explores topics such as the development of new and powerful knot invariants, new perspectives on enumerative geometry and string theory, and the introduction of cluster algebra and categorification techniques into a broad range of areas. Chapters will also cover novel applications of representation theory to random matrix theory, exactly solvable models in statistical mechanics, and integrable hierarchies. The recent progress in the mathematical and physicals aspects of deformation quantization and tensor categories is also addressed. Representation Theory, Mathematical Physics, and Integrable Systems will be of interest to a wide audience of mathematicians interested in these areas and the connections between them, ranging from graduate students to junior, mid-career, and senior researchers.
Author: Maciej Błaszak
Publisher: Springer
ISBN: 3030183793
Category : Science
Languages : en
Pages : 460
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Book Description
This accessible monograph introduces physicists to the general relation between classical and quantum mechanics based on the mathematical idea of deformation quantization and describes an original approach to the theory of quantum integrable systems developed by the author.The first goal of the book is to develop of a common, coordinate free formulation of classical and quantum Hamiltonian mechanics, framed in common mathematical language.In particular, a coordinate free model of quantum Hamiltonian systems in Riemannian spaces is formulated, based on the mathematical idea of deformation quantization, as a complete physical theory with an appropriate mathematical accuracy.The second goal is to develop of a theory which allows for a deeper understanding of classical and quantum integrability. For this reason the modern separability theory on both classical and quantum level is presented. In particular, the book presents a modern geometric separability theory, based on bi-Poissonian and bi-presymplectic representations of finite dimensional Liouville integrable systems and their admissible separable quantizations.The book contains also a generalized theory of classical Stäckel transforms and the discussion of the concept of quantum trajectories.In order to make the text consistent and self-contained, the book starts with a compact overview of mathematical tools necessary for understanding the remaining part of the book. However, because the book is dedicated mainly to physicists, despite its mathematical nature, it refrains from highlighting definitions, theorems or lemmas.Nevertheless, all statements presented are either proved or the reader is referred to the literature where the proof is available.
Author: PERELOMOV
Publisher: Birkhäuser
ISBN: 9783034892582
Category : Science
Languages : en
Pages : 308
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Book Description
Author: V.I. Arnol'd
Publisher: Springer Science & Business Media
ISBN: 1475720637
Category : Mathematics
Languages : en
Pages : 530
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Book Description
This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.
