Nonlinear Poisson Brackets

Author : Mikhail Vladimirovich Karasev,V. P. Maslov
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 53,5 Mb
Release : 1993
Category : Mathematics
ISBN : 0821845969

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Nonlinear Poisson Brackets by Mikhail Vladimirovich Karasev,V. P. Maslov Pdf

This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to describe in detail the new objects that appear in the solution of these problems. Many ideas of algebra, modern differential geometry, algebraic topology, and operator theory are synthesized here. The authors prove all statements in detail, thus making the book accessible to graduate students.

Author : Mihail Vladimirovi_ Karasev,Viktor Pavlovi_ Maslov,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 366 pages
File Size : 48,6 Mb
Release : 2012-06-06
Category : Electronic
ISBN : 9780821887967

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Nonlinear Poisson Brackets by Mihail Vladimirovi_ Karasev,Viktor Pavlovi_ Maslov,American Mathematical Society Pdf

This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to describe in detail the new objects that appear in the solution of these problems. Many ideas of algebra, modern differential geometry, algebraic topology, and operator theory are synthesized here. The authors prove all statements in detail, thus making the book accessible to graduate students.

Author : Anonim
Publisher : American Mathematical Soc.
Page : 382 pages
File Size : 46,6 Mb
Release : 1989
Category : Electronic
ISBN : 9780821897461

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Nonlinear Poisson Brackets by Anonim Pdf

A hierarchy of vector fields (master symmetries) and homogeneous nonlinear Poisson structures associated with the Toda lattice are constructed and the various connections between them are investigated. Among their properties: new brackets are generated from old ones by using Lie-derivatives in the direction of certain vector fields; the infinite sequences obtained consist of compatible Poisson brackets in which the constants of motion for the Toda lattice are in involution. The vector fields in the construction are unique up to addition of a Hamiltonian vector field. Similarly the Poisson brackets are unique up to addition of a trivial Poisson bracket. These are Poisson tensors generated by wedge products of Hamiltonian vector fields. The non-trivial brackets may also be obtained by the use of r-matrices; we give formulas and prove this for the quadratic and cubic Toda brackets. We also indicate how these results can be generalized to other (semisimple) Toda flows and we give explicit formulas for the rank 2 Lie algebra of type B2. The main tool in this calculation is Dirac's constraint bracket formula. Finally we study nonlinear Poisson brackets associated with orbits through nilpotent conjugacy classes in gl(n, R) and formulate some conjectures. We determine the degree of the transverse Poisson structure through such nilpotent elements in gl(n, R) for n ≤ 7. This is accomplished also by the use of Dirac's bracket formula.

Author : Mikhail Vladimirovich Karasev,M. V. KARASEV;V. P. MASLOV.
Publisher : Unknown
Page : 382 pages
File Size : 51,5 Mb
Release : 2024-05-12
Category : Hamiltonian systems
ISBN : 1470416190

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Nonlinear Poisson Brackets by Mikhail Vladimirovich Karasev,M. V. KARASEV;V. P. MASLOV. Pdf

This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to descri.

Author : Lev V. Prokhorov,Sergei V. Shabanov
Publisher : Cambridge University Press
Page : 485 pages
File Size : 44,5 Mb
Release : 2011-09-22
Category : Science
ISBN : 9781139500906

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Hamiltonian Mechanics of Gauge Systems by Lev V. Prokhorov,Sergei V. Shabanov Pdf

The principles of gauge symmetry and quantization are fundamental to modern understanding of the laws of electromagnetism, weak and strong subatomic forces and the theory of general relativity. Ideal for graduate students and researchers in theoretical and mathematical physics, this unique book provides a systematic introduction to Hamiltonian mechanics of systems with gauge symmetry. The book reveals how gauge symmetry may lead to a non-trivial geometry of the physical phase space and studies its effect on quantum dynamics by path integral methods. It also covers aspects of Hamiltonian path integral formalism in detail, along with a number of related topics such as the theory of canonical transformations on phase space supermanifolds, non-commutativity of canonical quantization and elimination of non-physical variables. The discussion is accompanied by numerous detailed examples of dynamical models with gauge symmetries, clearly illustrating the key concepts.

Author : S. de Filippo,M. Marinaro,G. Marmo,G. Vilasi
Publisher : Elsevier
Page : 258 pages
File Size : 46,5 Mb
Release : 2012-12-02
Category : Science
ISBN : 9780444598714

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Geometrical and Algebraic Aspects of Nonlinear Field Theory by S. de Filippo,M. Marinaro,G. Marmo,G. Vilasi Pdf

Experts in general relativity, particle physics and mathematical physics discuss aspects of their recent research. The main emphasis is on the geometrical and algebraic methods used in solving a wide range of problems.

Author : Library of Congress,Library of Congress. Office for Subject Cataloging Policy
Publisher : Unknown
Page : 1668 pages
File Size : 44,9 Mb
Release : 2011
Category : Subject headings, Library of Congress
ISBN : MINN:30000009886320

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Library of Congress Subject Headings by Library of Congress,Library of Congress. Office for Subject Cataloging Policy Pdf

Author : Library of Congress. Cataloging Policy and Support Office
Publisher : Unknown
Page : 1596 pages
File Size : 46,7 Mb
Release : 2009
Category : Subject headings, Library of Congress
ISBN : UOM:39015079817063

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Library of Congress Subject Headings by Library of Congress. Cataloging Policy and Support Office Pdf

Author : E.I. Kats,V.V. Lebedev
Publisher : Springer Science & Business Media
Page : 191 pages
File Size : 43,8 Mb
Release : 2012-12-06
Category : Science
ISBN : 9781461243328

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Fluctuational Effects in the Dynamics of Liquid Crystals by E.I. Kats,V.V. Lebedev Pdf

Liquid crystals, widely used in displays for electronic equipment and other applications, have highly unusual properties arising from the anisotropy of their molecules. It appears that some aspects of the fluid dynamics of liquid crystals, such as their viscosity, can be understood only by considering the role played by thermal fluctuations. In order to provide a theoretical framework for understanding the experimental results, the authors devote a large part of the book to a derivation of the nonlinear dynamic equations and to a discussion of linearized equations for the various types of liquid crystals. The diagrammatic and other techniques they use are of general use in condensed matter physics, and this exposition should thus be of interest to all condensed-matter theorists.

Author : Albert W Stetz
Publisher : World Scientific Publishing Company
Page : 140 pages
File Size : 45,6 Mb
Release : 2016-06-17
Category : Science
ISBN : 9789813141377

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Lectures in Nonlinear Mechanics and Chaos Theory by Albert W Stetz Pdf

This elegant book presents a rigorous introduction to the theory of nonlinear mechanics and chaos. It turns out that many simple mechanical systems suffer from a peculiar malady. They are deterministic in the sense that their motion can be described with partial differential equations, but these equations have no proper solutions and the behavior they describe can be wildly unpredictable. This is implicit in Newtonian physics, and although it was analyzed in the pioneering work of Poincaré in the 19th century, its full significance has only been realized since the advent of modern computing. This book follows this development in the context of classical mechanics as it is usually taught in most graduate programs in physics. It starts with the seminal work of Laplace, Hamilton, and Liouville in the early 19th century and shows how their formulation of mechanics inevitably leads to systems that cannot be "solved" in the usual sense of the word. It then discusses perturbation theory which, rather than providing approximate solutions, fails catastrophically due to the problem of small denominators. It then goes on to describe chaotic motion using the tools of discrete maps and Poincaré sections. This leads to the two great landmarks of chaos theory, the Poincaré–Birkhoff theorem and the so-called KAM theorem, one of the signal results in modern mathematics. The book concludes with an appendix discussing the relevance of the KAM theorem to the ergodic hypothesis and the second law of thermodynamics. Lectures on Nonlinear Mechanics and Chaos Theory is written in the easy conversational style of a great teacher. It features numerous computer-drawn figures illustrating the behavior of nonlinear systems. It also contains homework exercises and a selection of more detailed computational projects. The book will be valuable to students and faculty in physics, mathematics, and engineering. See Press Release: Problems in mechanics open the door to the orderly world of chaos

Author : Jerrold E. Marsden, George W. Patrick, and William F. Shadwick
Publisher : American Mathematical Soc.
Page : 260 pages
File Size : 55,9 Mb
Release : 2024-05-12
Category : Differentiable dynamical systems
ISBN : 0821871188

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Integration Algorithms and Classical Mechanics by Jerrold E. Marsden, George W. Patrick, and William F. Shadwick Pdf

Dedicated to the late Juan Carlos Simo, this volume contains the proceedings of a workshop held at the Fields Institute in October 1993. The articles focus on current algorithms for the integration of mechanical systems, from systems in celestial mechanics to coupled rigid bodies to fluid mechanics. The scope of the articles ranges from symplectic integration methods to energy-momentum methods and related themes.

Author : M. Daniel,K. M. Ramizhmani,R. Sahadevan
Publisher : Alpha Science Int'l Ltd.
Page : 490 pages
File Size : 45,7 Mb
Release : 2000
Category : Mathematics
ISBN : 8173193266

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Nonlinear Dynamics by M. Daniel,K. M. Ramizhmani,R. Sahadevan Pdf

Contributed articles presented at the International Conference on Nonlinear Dynamics: Integrability and Chaos held at Bharathidasan University during 12-16 Feb., 1998. In honor of Prof. M. Lakshmanan.

Author : Alexey Borisov,Ivan S. Mamaev
Publisher : Walter de Gruyter GmbH & Co KG
Page : 533 pages
File Size : 53,7 Mb
Release : 2018-12-03
Category : Science
ISBN : 9783110544442

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Rigid Body Dynamics by Alexey Borisov,Ivan S. Mamaev Pdf

This book provides an up-to-date overview of results in rigid body dynamics, including material concerned with the analysis of nonintegrability and chaotic behavior in various related problems. The wealth of topics covered makes it a practical reference for researchers and graduate students in mathematics, physics and mechanics. Contents Rigid Body Equations of Motion and Their Integration The Euler – Poisson Equations and Their Generalizations The Kirchhoff Equations and Related Problems of Rigid Body Dynamics Linear Integrals and Reduction Generalizations of Integrability Cases. Explicit Integration Periodic Solutions, Nonintegrability, and Transition to Chaos Appendix A : Derivation of the Kirchhoff, Poincaré – Zhukovskii, and Four-Dimensional Top Equations Appendix B: The Lie Algebra e(4) and Its Orbits Appendix C: Quaternion Equations and L-A Pair for the Generalized Goryachev – Chaplygin Top Appendix D: The Hess Case and Quantization of the Rotation Number Appendix E: Ferromagnetic Dynamics in a Magnetic Field Appendix F: The Landau – Lifshitz Equation, Discrete Systems, and the Neumann Problem Appendix G: Dynamics of Tops and Material Points on Spheres and Ellipsoids Appendix H: On the Motion of a Heavy Rigid Body in an Ideal Fluid with Circulation Appendix I: The Hamiltonian Dynamics of Self-gravitating Fluid and Gas Ellipsoids

Author : S.S. Chern
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 51,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461211105

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Seminar on Nonlinear Partial Differential Equations by S.S. Chern Pdf

When the Mathematical Sciences Research Institute was started in the Fall of 1982, one of the programs was "non-linear partial differential equations". A seminar was organized whose audience consisted of graduate students of the University and mature mathematicians who are not experts in the field. This volume contains 18 of these lectures. An effort is made to have an adequate Bibliography for further information. The Editor wishes to take this opportunity to thank all the speakers and the authors of the articles presented in this volume for their cooperation. S. S. Chern, Editor Table of Contents Geometrical and Analytical Questions Stuart S. Antman 1 in Nonlinear Elasticity An Introduction to Euler's Equations Alexandre J. Chorin 31 for an Incompressible Fluid Linearizing Flows and a Cohomology Phillip Griffiths 37 Interpretation of Lax Equations The Ricci Curvature Equation Richard Hamilton 47 A Walk Through Partial Differential Fritz John 73 Equations Remarks on Zero Viscosity Limit for Tosio Kato 85 Nonstationary Navier-Stokes Flows with Boundary Free Boundary Problems in Mechanics Joseph B. Keller 99 The Method of Partial Regularity as Robert V.

Author : Jerrold E. Marsden,Tudor S. Ratiu
Publisher : Springer Science & Business Media
Page : 593 pages
File Size : 53,6 Mb
Release : 2013-03-19
Category : Science
ISBN : 9780387217925

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Introduction to Mechanics and Symmetry by Jerrold E. Marsden,Tudor S. Ratiu Pdf

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.