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Author: A.K. Prykarpatsky Publisher: Springer Science & Business Media ISBN: 9401149941 Category : Science Languages : en Pages : 555
Book Description
In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).
Author: A.K. Prykarpatsky Publisher: Springer Science & Business Media ISBN: 9401149941 Category : Science Languages : en Pages : 555
Book Description
In recent times it has been stated that many dynamical systems of classical mathematical physics and mechanics are endowed with symplectic structures, given in the majority of cases by Poisson brackets. Very often such Poisson structures on corresponding manifolds are canonical, which gives rise to the possibility of producing their hidden group theoretical essence for many completely integrable dynamical systems. It is a well understood fact that great part of comprehensive integrability theories of nonlinear dynamical systems on manifolds is based on Lie-algebraic ideas, by means of which, in particular, the classification of such compatibly bi Hamiltonian and isospectrally Lax type integrable systems has been carried out. Many chapters of this book are devoted to their description, but to our regret so far the work has not been completed. Hereby our main goal in each analysed case consists in separating the basic algebraic essence responsible for the complete integrability, and which is, at the same time, in some sense universal, i. e. , characteristic for all of them. Integrability analysis in the framework of a gradient-holonomic algorithm, devised in this book, is fulfilled through three stages: 1) finding a symplectic structure (Poisson bracket) transforming an original dynamical system into a Hamiltonian form; 2) finding first integrals (action variables or conservation laws); 3) defining an additional set of variables and some functional operator quantities with completely controlled evolutions (for instance, as Lax type representation).
Author: Andrei N. Leznov Publisher: Birkhäuser ISBN: 3034886381 Category : Mathematics Languages : en Pages : 308
Book Description
The book reviews a large number of 1- and 2-dimensional equations that describe nonlinear phenomena in various areas of modern theoretical and mathematical physics. It is meant, above all, for physicists who specialize in the field theory and physics of elementary particles and plasma, for mathe maticians dealing with nonlinear differential equations, differential geometry, and algebra, and the theory of Lie algebras and groups and their representa tions, and for students and post-graduates in these fields. We hope that the book will be useful also for experts in hydrodynamics, solid-state physics, nonlinear optics electrophysics, biophysics and physics of the Earth. The first two chapters of the book present some results from the repre sentation theory of Lie groups and Lie algebras and their counterpart on supermanifolds in a form convenient in what follows. They are addressed to those who are interested in integrable systems but have a scanty vocabulary in the language of representation theory. The experts may refer to the first two chapters only occasionally. As we wanted to give the reader an opportunity not only to come to grips with the problem on the ideological level but also to integrate her or his own concrete nonlinear equations without reference to the literature, we had to expose in a self-contained way the appropriate parts of the representation theory from a particular point of view.
Author: Denis Blackmore Publisher: World Scientific ISBN: 9814462713 Category : Mathematics Languages : en Pages : 563
Book Description
This distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field — including some innovations by the authors themselves — that have not appeared in any other book.The exposition begins with an introduction to modern integrable dynamical systems theory, treating such topics as Liouville-Arnold and Mischenko-Fomenko integrability. This sets the stage for such topics as new formulations of the gradient-holonomic algorithm for Lax integrability, novel treatments of classical integration by quadratures, Lie-algebraic characterizations of integrability, and recent results on tensor Poisson structures. Of particular note is the development via spectral reduction of a generalized de Rham-Hodge theory, related to Delsarte-Lions operators, leading to new Chern type classes useful for integrability analysis. Also included are elements of quantum mathematics along with applications to Whitham systems, gauge theories, hadronic string models, and a supplement on fundamental differential-geometric concepts making this volume essentially self-contained.This book is ideal as a reference and guide to new directions in research for advanced students and researchers interested in the modern theory and applications of integrable (especially infinite-dimensional) dynamical systems.
Author: Yvette Kosmann-Schwarzbach Publisher: Springer ISBN: Category : Mathematics Languages : en Pages : 396
Book Description
The theory of nonlinear systems and, in particular, of integrable systems is related to several very active fields of research in theoretical physics. Many mathematical aspects of nonlinear systems, both continuous and discrete, are analyzed here with particular emphasis on the domains of inverse-scattering techniques, singularity analysis, the bilinear formalism, chaos in nonlinear oscillators, Lie-algebraic and group-theoretical methods, classical and quantum integrability, bihamiltonian structures. The book will be of considerable interest to those who wish to study integrable systems, and to follow the future developments, both in mathematics and in theoretical physics, of the theory of integrability.
Author: Piotr Kielanowski Publisher: Birkhäuser ISBN: 3319635948 Category : Mathematics Languages : en Pages : 280
Book Description
This book features a selection of articles based on the XXXV Białowieża Workshop on Geometric Methods in Physics, 2016. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, and with applications to classical and quantum physics. In 2016 the special session "Integrability and Geometry" in particular attracted pioneers and leading specialists in the field. Traditionally, the Białowieża Workshop is followed by a School on Geometry and Physics, for advanced graduate students and early-career researchers, and the book also includes extended abstracts of the lecture series.
Author: Vitaliy Zhurbenko Publisher: BoD – Books on Demand ISBN: 9533073047 Category : Science Languages : en Pages : 526
Book Description
This book is dedicated to various aspects of electromagnetic wave theory and its applications in science and technology. The covered topics include the fundamental physics of electromagnetic waves, theory of electromagnetic wave propagation and scattering, methods of computational analysis, material characterization, electromagnetic properties of plasma, analysis and applications of periodic structures and waveguide components, and finally, the biological effects and medical applications of electromagnetic fields.
Author: Claire David Publisher: Bentham Science Publishers ISBN: 1608051404 Category : Science Languages : en Pages : 267
Book Description
Since the first description by John Scott Russel in 1834, the solitary wave phenomenon has attracted considerable interests from scientists. The most interesting discovery since then has been the ability to integrate most of the nonlinear wave equations which govern solitary waves, from the Korteweg-de Vries equation to the nonlinear Schrodinger equation, in the 1960's. From that moment, a huge amount of theoretical works can be found on solitary waves. Due to the fact that many physical phenomena can be described by a soliton model, applications have followed each other, in telecommunications
Author: Piotr Kielanowski Publisher: Springer Nature ISBN: 3030533050 Category : Mathematics Languages : en Pages : 373
Book Description
The book consists of articles based on the XXXVIII Białowieża Workshop on Geometric Methods in Physics, 2019. The series of Białowieża workshops, attended by a community of experts at the crossroads of mathematics and physics, is a major annual event in the field. The works in this book, based on presentations given at the workshop, are previously unpublished, at the cutting edge of current research, typically grounded in geometry and analysis, with applications to classical and quantum physics. For the past eight years, the Białowieża Workshops have been complemented by a School on Geometry and Physics, comprising series of advanced lectures for graduate students and early-career researchers. The extended abstracts of the five lecture series that were given in the eighth school are included. The unique character of the Workshop-and-School series draws on the venue, a famous historical, cultural and environmental site in the Białowieża forest, a UNESCO World Heritage Centre in the east of Poland: lectures are given in the Nature and Forest Museum and local traditions are interwoven with the scientific activities. The chapter “Toeplitz Extensions in Noncommutative Topology and Mathematical Physics” is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
Author: Lawrence Markus Publisher: American Mathematical Soc. ISBN: 0821818449 Category : Differential equations Languages : en Pages : 58
Book Description
This memoir gives an introduction to Hamiltonian dynamical systems on symplectic manifolds, including definitions of Hamiltonian vector fields, Poisson brackets, integrals of motion, complete integrability, and ergodicity. A particularly complete treatment of action-angle coordinates is given. Historical background into the question of ergodicity and integrability in Hamiltonian systems is also given.
Author: A.K. Prykarpatsky
Author: A.K. Prykarpatsky
Author: Andrei N. Leznov
Author: Denis Blackmore
Author: P. G. L. Leach
Author: Yvette Kosmann-Schwarzbach
Author: Piotr Kielanowski
Author: Vitaliy Zhurbenko
Author: Claire David
Author: Piotr Kielanowski
Author: Lawrence Markus