Global Aspects Of Classical Integrable Systems

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Global Aspects Of Classical Integrable Systems

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Author : Richard H. Cushman
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Global Aspects Of Classical Integrable Systems written by Richard H. Cushman and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.

This book gives a complete global geometric description of the motion of the two di mensional hannonic oscillator, the Kepler problem, the Euler top, the spherical pendulum and the Lagrange top. These classical integrable Hamiltonian systems one sees treated in almost every physics book on classical mechanics. So why is this book necessary? The answer is that the standard treatments are not complete. For instance in physics books one cannot see the monodromy in the spherical pendulum from its explicit solution in terms of elliptic functions nor can one read off from the explicit solution the fact that a tennis racket makes a near half twist when it is tossed so as to spin nearly about its intermediate axis. Modem mathematics books on mechanics do not use the symplectic geometric tools they develop to treat the qualitative features of these problems either. One reason for this is that their basic tool for removing symmetries of Hamiltonian systems, called regular reduction, is not general enough to handle removal of the symmetries which occur in the spherical pendulum or in the Lagrange top. For these symmetries one needs singular reduction. Another reason is that the obstructions to making local action angle coordinates global such as monodromy were not known when these works were written.

Integrable Systems And Algebraic Geometry

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Author : Ron Donagi
language : en
Publisher: Cambridge University Press
Release Date : 2020-04-02

Integrable Systems And Algebraic Geometry written by Ron Donagi and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-04-02 with Mathematics categories.

A collection of articles discussing integrable systems and algebraic geometry from leading researchers in the field.

Geometry From Dynamics Classical And Quantum

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Author : José F. Cariñena
language : en
Publisher: Springer
Release Date : 2014-09-23

Geometry From Dynamics Classical And Quantum written by José F. Cariñena and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-23 with Science categories.

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

Introduction To Classical Integrable Systems

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Author : Olivier Babelon
language : en
Publisher: Cambridge University Press
Release Date : 2003-04-17

Introduction To Classical Integrable Systems written by Olivier Babelon and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-04-17 with Mathematics categories.

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.

Hamiltonian Systems And Their Integrability

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Author : Mich'le Audin
language : en
Publisher: American Mathematical Soc.
Release Date : 2008

Hamiltonian Systems And Their Integrability written by Mich'le Audin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.

Hamiltonian systems began as a mathematical approach to the study of mechanical systems. As the theory developed, it became clear that the systems that had a sufficient number of conserved quantities enjoyed certain remarkable properties. These are the completely integrable systems. In time, a rich interplay arose between integrable systems and other areas of mathematics, particularly topology, geometry, and group theory.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. Audin has included many examples and exercises. Most of the exercises build on the material in the text. None of the important proofs have been relegated to the exercises. Many of the examples are classical, rather than abstract. This book would be suitable for a graduate course in Hamiltonian systems.

Scaling Limits And Models In Physical Processes

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Author : Carlo Cercignani
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

Scaling Limits And Models In Physical Processes written by Carlo Cercignani and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

This is an introductory text, in two parts, on scaling limits and modelling in equations of mathematical physics. The first part is concerned with basic concepts of the kinetic theory of gases which is not only important in its own right but also as a prototype of a mathematical construct central to the theory of non-equilibrium phenomena in large systems. It also features a very readable historic survey of the field. The second part dwells on the role of integrable systems for modelling weakly nonlinear equations which contain the effects of both dispersion and nonlinearity. Starting with a historical introduction to the subject and a description of numerical techniques, it proceeds to a discussion of the derivation of the Korteweg de Vries and nonlinear Schrödinger equations, followed by a careful treatment of the inverse scattering theory for the Schrödinger operator. The book provides an up-to-date and detailed overview to this very active area of research and is intended as an accessible introduction for non-specialists and graduate students in mathematics, physics and engineering.

Algebraic Integrability Painlev Geometry And Lie Algebras

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Author : Mark Adler
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Algebraic Integrability Painlev Geometry And Lie Algebras written by Mark Adler and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.

Geometric Formulation Of Classical And Quantum Mechanics

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Author : Giovanni Giachetta
language : en
Publisher: World Scientific
Release Date : 2010-10-11

Geometric Formulation Of Classical And Quantum Mechanics written by Giovanni Giachetta and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-10-11 with Science categories.

The geometric formulation of autonomous Hamiltonian mechanics in the terms of symplectic and Poisson manifolds is generally accepted. The literature on this subject is extensive. The present book provides the geometric formulation of non-autonomous mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of mechanics as like as that of classical field theory lies in the framework of general theory of dynamic systems, and Lagrangian and Hamiltonian formalisms on fiber bundles. The reader will find a strict mathematical exposition of non-autonomous dynamic systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics, together with a number of advanced models — superintegrable systems, non-autonomous constrained systems, theory of Jacobi fields, mechanical systems with time-dependent parameters, non-adiabatic Berry phase theory, instantwise quantization, and quantization relative to different reference frames.

Geometry Of Nonholonomically Constrained Systems

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Author : Richard H. Cushman
language : en
Publisher: World Scientific
Release Date : 2010

Geometry Of Nonholonomically Constrained Systems written by Richard H. Cushman and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.

1. Nonholonomically constrained motions. 1.1. Newton's equations. 1.2. Constraints. 1.3. Lagrange-d'Alembert equations. 1.4. Lagrange derivative. 1.5. Hamilton-d'Alembert equations. 1.6. Distributional Hamiltonian formulation. 1.7. Almost Poisson brackets. 1.8. Momenta and momentum equation. 1.9. Projection principle. 1.10. Accessible sets. 1.11. Constants of motion. 1.12. Notes -- 2. Group actions and orbit spaces. 2.1. Group actions. 2.2. Orbit spaces. 2.3. Isotropy and orbit types. 2.4. Smooth structure on an orbit space. 2.5. Subcartesian spaces. 2.6. Stratification of the orbit space by orbit types. 2.7. Derivations and vector fields on a differential space. 2.8. Vector fields on a stratified differential space. 2.9. Vector fields on an orbit space. 2.10. Tangent objects to an orbit space. 2.11. Notes -- 3. Symmetry and reduction. 3.1. Dynamical systems with symmetry. 3.2. Nonholonomic singular reduction. 3.3. Nonholonomic regular reduction. 3.4. Chaplygin systems. 3.5. Orbit types and reduction. 3.6. Conservation laws. 3.7. Lifted actions and the momentum equation. 3.8. Notes -- 4. Reconstruction, relative equilibria and relative periodic orbits. 4.1. Reconstruction. 4.2. Relative equilibria. 4.3. Relative periodic orbits. 4.4. Notes -- 5. Carathéodory's sleigh. 5.1. Basic set up. 5.2. Equations of motion. 5.3. Reduction of the E(2) symmetry. 5.4. Motion on the E(2) reduced phase space. 5.5. Reconstruction. 5.6. Notes -- 6. Convex rolling rigid body. 6.1. Basic set up. 6.2. Unconstrained motion. 6.3. Constraint distribution. 6.4. Constrained equations of motion. 6.5. Reduction of the translational [symbol] symmetry. 6.6. Reduction of E(2) symmetry. 6.7. Body of revolution. 6.8. Notes -- 7. The rolling disk. 7.1. General set up. 7.2. Reduction of the E(2) x S[symbol] symmetry. 7.3. Reconstruction. 7.4. Relative equilibria. 7.5. A potential function on an interval. 7.6. Scaling. 7.7. Solutions of the rescaled Chaplygin equations. 7.8. Bifurcations of a vertical disk. 7.9. The global geometry of the degeneracy locus. 7.10. Falling flat. 7.11. Near falling flat. 7.12. The bifurcation diagram. 7.13. The integral map. 7.14. Constant energy slices. 7.15. The spatial rotational shift. 7.16. Notes.

Lectures On Dynamical Systems

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Author : Eduard Zehnder
language : en
Publisher: European Mathematical Society
Release Date : 2010

Lectures On Dynamical Systems written by Eduard Zehnder and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Dynamics categories.

This book originated from an introductory lecture course on dynamical systems given by the author for advanced students in mathematics and physics at ETH Zurich. The first part centers around unstable and chaotic phenomena caused by the occurrence of homoclinic points. The existence of homoclinic points complicates the orbit structure considerably and gives rise to invariant hyperbolic sets nearby. The orbit structure in such sets is analyzed by means of the shadowing lemma, whose proof is based on the contraction principle. This lemma is also used to prove S. Smale's theorem about the embedding of Bernoulli systems near homoclinic orbits. The chaotic behavior is illustrated in the simple mechanical model of a periodically perturbed mathematical pendulum. The second part of the book is devoted to Hamiltonian systems. The Hamiltonian formalism is developed in the elegant language of the exterior calculus. The theorem of V. Arnold and R. Jost shows that the solutions of Hamiltonian systems which possess sufficiently many integrals of motion can be written down explicitly and for all times. The existence proofs of global periodic orbits of Hamiltonian systems on symplectic manifolds are based on a variational principle for the old action functional of classical mechanics. The necessary tools from variational calculus are developed. There is an intimate relation between the periodic orbits of Hamiltonian systems and a class of symplectic invariants called symplectic capacities. From these symplectic invariants one derives surprising symplectic rigidity phenomena. This allows a first glimpse of the fast developing new field of symplectic topology.