Global Aspects of Classical Integrable Systems

Global Aspects of Classical Integrable Systems
Author: Richard H. Cushman
Publisher: Birkhäuser
Total Pages: 493
Release: 2015-06-01
Genre: Science
ISBN: 3034809182

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.

Introduction to Classical Integrable Systems

Introduction to Classical Integrable Systems
Author: Olivier Babelon
Publisher: Cambridge University Press
Total Pages: 622
Release: 2003-04-17
Genre: Mathematics
ISBN: 9780521822671

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.

Integrable Hamiltonian Systems

Integrable Hamiltonian Systems
Author: A.V. Bolsinov
Publisher: CRC Press
Total Pages: 752
Release: 2004-02-25
Genre: Mathematics
ISBN: 0203643429

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,

Methods of Qualitative Theory of Differential Equations and Related Topics

Methods of Qualitative Theory of Differential Equations and Related Topics
Author: Lev M. Lerman
Publisher: American Mathematical Soc.
Total Pages: 58
Release: 2000
Genre: Mathematics
ISBN: 9780821826638

Dedicated to the memory of Professor E. A. Leontovich-Andronova, this book was composed by former students and colleagues who wished to mark her contributions to the theory of dynamical systems. A detailed introduction by Leontovich-Andronova's close colleague, L. Shilnikov, presents biographical data and describes her main contribution to the theory of bifurcations and dynamical systems. The main part of the volume is composed of research papers presenting the interests of Leontovich-Andronova, her students and her colleagues. Included are articles on traveling waves in coupled circle maps, bifurcations near a homoclinic orbit, polynomial quadratic systems on the plane, foliations on surfaces, homoclinic bifurcations in concrete systems, topology of plane controllability regions, separatrix cycle with two saddle-foci, dynamics of 4-dimensional symplectic maps, torus maps from strong resonances, structure of 3 degree-of-freedom integrable Hamiltonian systems, splitting separatrices in complex differential equations, Shilnikov's bifurcation for C1-smooth systems and "blue sky catastrophe" for periodic orbits.

The Problem of Integrable Discretization

The Problem of Integrable Discretization
Author: Yuri B. Suris
Publisher: Birkhäuser
Total Pages: 1078
Release: 2012-12-06
Genre: Mathematics
ISBN: 3034880162

An exploration of the theory of discrete integrable systems, with an emphasis on the following general problem: how to discretize one or several of independent variables in a given integrable system of differential equations, maintaining the integrability property? This question (related in spirit to such a modern branch of numerical analysis as geometric integration) is treated in the book as an immanent part of the theory of integrable systems, also commonly termed as the theory of solitons. Most of the results are only available from recent journal publications, many of them are new. Thus, the book is a kind of encyclopedia on discrete integrable systems. It unifies the features of a research monograph and a handbook. It is supplied with an extensive bibliography and detailed bibliographic remarks at the end of each chapter. Largely self-contained, it will be accessible to graduate and post-graduate students as well as to researchers in the area of integrable dynamical systems.

Accelerating Expansion

Accelerating Expansion
Author: Gordon Belot
Publisher: Oxford University Press
Total Pages: 241
Release: 2023-08-24
Genre: Mathematics
ISBN: 019286646X

Accelerating Expansion explores some of the philosophical implications of modern cosmology, focused on the significance that the discovery of the accelerating expansion of the Universe has for our understanding of time, geometry, and physics. The appearance of the cosmological constant in the equations of general relativity allows one to model universes in which space has an inherent tendency towards expansion. This constant, introduced by Einstein but subsequently abandoned by him, returned to centre stage with the discovery of the accelerating expansion. This pedagogically-oriented essay begins with a study of the most basic and elegant relativistic world that involves a positive cosmological constant, de Sitter spacetime. It then turns to the relatives of de Sitter spacetime that dominate modern relativistic cosmology. Some of the topics considered include: the nature of time and simultaneity in de Sitter worlds; the sense in which de Sitter spacetime is a powerful dynamical attractor; the limited extent to which observation can give us information about the topology of space in a world undergoing accelerated expansion; and cosmologists' favourite sceptical worry about the reliability of evidence and the possibility of knowledge, the problem of Boltzmann brains.

Dynamical Systems and Chaos

Dynamical Systems and Chaos
Author: Henk Broer
Publisher: Springer Science & Business Media
Total Pages: 313
Release: 2010-10-20
Genre: Mathematics
ISBN: 1441968709

Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.

General Theory of Lie Groupoids and Lie Algebroids

General Theory of Lie Groupoids and Lie Algebroids
Author: Kirill C. H. Mackenzie
Publisher: Cambridge University Press
Total Pages: 540
Release: 2005-06-09
Genre: Mathematics
ISBN: 0521499283

This a comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry andgeneral connection theory. It covers much work done since the mid 1980s including the first treatment in book form of Poisson groupoids, Lie bialgebroids and double vector bundles. As such, this book will be of great interest to all those working in or wishing to learn the modern theory of Lie groupoids and Lie algebroids.

The Oxford Handbook of Philosophy of Physics

The Oxford Handbook of Philosophy of Physics
Author: Robert Batterman
Publisher: Oxford University Press
Total Pages: 701
Release: 2013-03-14
Genre: Philosophy
ISBN: 0195392043

This Oxford Handbook provides an overview of many of the topics that currently engage philosophers of physics. It surveys new issues and the problems that have become a focus of attention in recent years. It also provides up-to-date discussions of the still very important problems that dominated the field in the past. In the late 20th Century, the philosophy of physics was largely focused on orthodox Quantum Mechanics and Relativity Theory. The measurement problem, the question of the possibility of hidden variables, and the nature of quantum locality dominated the literature on the quantum mechanics, whereas questions about relationalism vs. substantivalism, and issues about underdetermination of theories dominated the literature on spacetime. These issues still receive considerable attention from philosophers, but many have shifted their attentions to other questions related to quantum mechanics and to spacetime theories. Quantum field theory has become a major focus, particularly from the point of view of algebraic foundations. Concurrent with these trends, there has been a focus on understanding gauge invariance and symmetries. The philosophy of physics has evolved even further in recent years with attention being paid to theories that, for the most part, were largely ignored in the past. For example, the relationship between thermodynamics and statistical mechanics—-once thought to be a paradigm instance of unproblematic theory reduction—-is now a hotly debated topic. The implicit, and sometimes explicit, reductionist methodology of both philosophers and physicists has been severely criticized and attention has now turned to the explanatory and descriptive roles of "non-fundamental,'' phenomenological theories. This shift of attention includes "old'' theories such as classical mechanics, once deemed to be of little philosophical interest. Furthermore, some philosophers have become more interested in "less fundamental'' contemporary physics such as condensed matter theory. Questions abound with implications for the nature of models, idealizations, and explanation in physics. This Handbook showcases all these aspects of this complex and dynamic discipline.

Topology in Condensed Matter

Topology in Condensed Matter
Author: Michael I. Monastyrsky
Publisher: Springer Science & Business Media
Total Pages: 263
Release: 2006-02-04
Genre: Science
ISBN: 3540312641

This book reports new results in condensed matter physics for which topological methods and ideas are important. It considers, on the one hand, recently discovered systems such as carbon nanocrystals and, on the other hand, new topological methods used to describe more traditional systems such as the Fermi surfaces of normal metals, liquid crystals and quasicrystals. The authors of the book are renowned specialists in their fields and present the results of ongoing research, some of it obtained only very recently and not yet published in monograph form.