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Discrete Hamiltonian Systems
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Book Synopsis Discrete Hamiltonian Systems by : Calvin Ahlbrandt
Download or read book Discrete Hamiltonian Systems written by Calvin Ahlbrandt and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book should be accessible to students who have had a first course in matrix theory. The existence and uniqueness theorem of Chapter 4 requires the implicit function theorem, but we give a self-contained constructive proof ofthat theorem. The reader willing to accept the implicit function theorem can read the book without an advanced calculus background. Chapter 8 uses the Moore-Penrose pseudo-inverse, but is accessible to students who have facility with matrices. Exercises are placed at those points in the text where they are relevant. For U. S. universities, we intend for the book to be used at the senior undergraduate level or beginning graduate level. Chapter 2, which is on continued fractions, is not essential to the material of the remaining chapters, but is intimately related to the remaining material. Continued fractions provide closed form representations of the extreme solutions of some discrete matrix Riccati equations. Continued fractions solution methods for Riccati difference equations provide an approach analogous to series solution methods for linear differential equations. The book develops several topics which have not been available at this level. In particular, the material of the chapters on continued fractions (Chapter 2), symplectic systems (Chapter 3), and discrete variational theory (Chapter 4) summarize recent literature. Similarly, the material on transforming Riccati equations presented in Chapter 3 gives a self-contained unification of various forms of Riccati equations. Motivation for our approach to difference equations came from the work of Harris, Vaughan, Hartman, Reid, Patula, Hooker, Erbe & Van, and Bohner.
Book Synopsis Discrete Hamiltonian Systems by : Calvin Ahlbrandt
Download or read book Discrete Hamiltonian Systems written by Calvin Ahlbrandt and published by Springer. This book was released on 1996-10-31 with total page 376 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book should be accessible to students who have had a first course in matrix theory. The existence and uniqueness theorem of Chapter 4 requires the implicit function theorem, but we give a self-contained constructive proof ofthat theorem. The reader willing to accept the implicit function theorem can read the book without an advanced calculus background. Chapter 8 uses the Moore-Penrose pseudo-inverse, but is accessible to students who have facility with matrices. Exercises are placed at those points in the text where they are relevant. For U. S. universities, we intend for the book to be used at the senior undergraduate level or beginning graduate level. Chapter 2, which is on continued fractions, is not essential to the material of the remaining chapters, but is intimately related to the remaining material. Continued fractions provide closed form representations of the extreme solutions of some discrete matrix Riccati equations. Continued fractions solution methods for Riccati difference equations provide an approach analogous to series solution methods for linear differential equations. The book develops several topics which have not been available at this level. In particular, the material of the chapters on continued fractions (Chapter 2), symplectic systems (Chapter 3), and discrete variational theory (Chapter 4) summarize recent literature. Similarly, the material on transforming Riccati equations presented in Chapter 3 gives a self-contained unification of various forms of Riccati equations. Motivation for our approach to difference equations came from the work of Harris, Vaughan, Hartman, Reid, Patula, Hooker, Erbe & Van, and Bohner.
Book Synopsis Construction of Mappings for Hamiltonian Systems and Their Applications by : Sadrilla S. Abdullaev
Download or read book Construction of Mappings for Hamiltonian Systems and Their Applications written by Sadrilla S. Abdullaev and published by Springer. This book was released on 2006-08-02 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the method of canonical transformation of variables and the classical perturbation theory, this innovative book treats the systematic theory of symplectic mappings for Hamiltonian systems and its application to the study of the dynamics and chaos of various physical problems described by Hamiltonian systems. It develops a new, mathematically-rigorous method to construct symplectic mappings which replaces the dynamics of continuous Hamiltonian systems by the discrete ones. Applications of the mapping methods encompass the chaos theory in non-twist and non-smooth dynamical systems, the structure and chaotic transport in the stochastic layer, the magnetic field lines in magnetically confinement devices of plasmas, ray dynamics in waveguides, etc. The book is intended for postgraduate students and researches, physicists and astronomers working in the areas of plasma physics, hydrodynamics, celestial mechanics, dynamical astronomy, and accelerator physics. It should also be useful for applied mathematicians involved in analytical and numerical studies of dynamical systems.
Book Synopsis Introduction to the Perturbation Theory of Hamiltonian Systems by : Dmitry Treschev
Download or read book Introduction to the Perturbation Theory of Hamiltonian Systems written by Dmitry Treschev and published by Springer Science & Business Media. This book was released on 2009-10-08 with total page 221 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an extended version of lectures given by the ?rst author in 1995–1996 at the Department of Mechanics and Mathematics of Moscow State University. We believe that a major part of the book can be regarded as an additional material to the standard course of Hamiltonian mechanics. In comparison with the original Russian 1 version we have included new material, simpli?ed some proofs and corrected m- prints. Hamiltonian equations ?rst appeared in connection with problems of geometric optics and celestial mechanics. Later it became clear that these equations describe a large classof systemsin classical mechanics,physics,chemistry,and otherdomains. Hamiltonian systems and their discrete analogs play a basic role in such problems as rigid body dynamics, geodesics on Riemann surfaces, quasi-classic approximation in quantum mechanics, cosmological models, dynamics of particles in an accel- ator, billiards and other systems with elastic re?ections, many in?nite-dimensional models in mathematical physics, etc. In this book we study Hamiltonian systems assuming that they depend on some parameter (usually?), where for?= 0 the dynamics is in a sense simple (as a rule, integrable). Frequently such a parameter appears naturally. For example, in celestial mechanics it is accepted to take? equal to the ratio: the mass of Jupiter over the mass of the Sun. In other cases it is possible to introduce the small parameter ar- ?cially.
Book Synopsis Metamorphoses of Hamiltonian Systems with Symmetries by : Konstantinos Efstathiou
Download or read book Metamorphoses of Hamiltonian Systems with Symmetries written by Konstantinos Efstathiou and published by Springer. This book was released on 2005-01-28 with total page 155 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern notions and important tools of classical mechanics are used in the study of concrete examples that model physically significant molecular and atomic systems. The parametric nature of these examples leads naturally to the study of the major qualitative changes of such systems (metamorphoses) as the parameters are varied. The symmetries of these systems, discrete or continuous, exact or approximate, are used to simplify the problem through a number of mathematical tools and techniques like normalization and reduction. The book moves gradually from finding relative equilibria using symmetry, to the Hamiltonian Hopf bifurcation and its relation to monodromy and, finally, to generalizations of monodromy.
Book Synopsis Port-Hamiltonian Systems Theory by : Schaft Van Der
Download or read book Port-Hamiltonian Systems Theory written by Schaft Van Der and published by . This book was released on 2014-06-13 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: Port-Hamiltonian Systems Theory: An Introductory Overview provides a concise and easily accessible description of the foundations underpinning the subject and emphasizes novel developments in the field, which will be of interest to a broad range of researchers.
Book Synopsis The Problem of Integrable Discretization by : Yuri B. Suris
Download or read book The Problem of Integrable Discretization written by Yuri B. Suris and published by Birkhäuser. This book was released on 2012-12-06 with total page 1078 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Hamiltonian Approach to Dynamic Economics by : David Cass
Download or read book The Hamiltonian Approach to Dynamic Economics written by David Cass and published by Academic Press. This book was released on 2014-05-10 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Hamiltonian Approach to Dynamic Economics focuses on the application of the Hamiltonian approach to dynamic economics and attempts to provide some unification of the theory of heterogeneous capital. Emphasis is placed on the stability of long-run steady-state equilibrium in models of heterogeneous capital accumulation. Generalizations of the Samuelson-Scheinkman approach are also given. Moreover, conditions are sought on the geometry of the Hamiltonian function (that is, on static technology) that suffice to preserve under (not necessarily small) perturbation the basic properties of the Hamiltonian dynamical system. Comprised of eight essays, this book begins with an introduction to Hamiltonian dynamics in economics, followed by a discussion on optimal steady states of n-sector growth models when utility is discounted. Optimal growth and decentralized or descriptive growth models in both continuous and discrete time are treated as applications of Hamiltonian dynamics. Theproblem of optimal growth with zero discounting is considered, with emphasis on a steepness condition on the Hamiltonian function. The general problem of decentralized growth with instantaneously adjusted expectations about price changes is also analyzed, along with the global asymptotic stability of optimal control systems with applications to the theory of economic growth. This monograph will be of value to mathematicians and economists.
Book Synopsis Time-Varying Discrete Linear Systems by : Aristide Halanay
Download or read book Time-Varying Discrete Linear Systems written by Aristide Halanay and published by Birkhäuser. This book was released on 2012-12-06 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete-time systems arise as a matter of course in modelling biological or economic processes. For systems and control theory they are of major importance, particularly in connection with digital control applications. If sampling is performed in order to control periodic processes, almost periodic systems are obtained. This is a strong motivation to investigate the discrete-time systems with time-varying coefficients. This research monograph contains a study of discrete-time nodes, the discrete counterpart of the theory elaborated by Bart, Gohberg and Kaashoek for the continuous case, discrete-time Lyapunov and Riccati equations, discrete-time Hamiltonian systems in connection with input-output operators and associated Hankel and Toeplitz operators. All these tools aim to solve the problems of stabilization and attenuation of disturbances in the framework of H2- and H-control theory. The book is the first of its kind to be devoted to these topics and consists mainly of original, recently obtained results.
Book Synopsis Soliton Equations and Hamiltonian Systems by : L.A. Dickey
Download or read book Soliton Equations and Hamiltonian Systems written by L.A. Dickey and published by World Scientific. This book was released on 1991 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of soliton equations and integrable systems has developed rapidly during the last 20 years with numerous applications in mechanics and physics. For a long time books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this followed one single work by Gardner, Greene, Kruskal, and Miura about the Korteweg-de Vries equation (KdV) which, had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water.
Book Synopsis Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems by : Antonio Giorgilli
Download or read book Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems written by Antonio Giorgilli and published by Cambridge University Press. This book was released on 2022-05-05 with total page 474 pages. Available in PDF, EPUB and Kindle. Book excerpt: Starting with the basics of Hamiltonian dynamics and canonical transformations, this text follows the historical development of the theory culminating in recent results: the Kolmogorov–Arnold–Moser theorem, Nekhoroshev's theorem and superexponential stability. Its analytic approach allows students to learn about perturbation methods leading to advanced results. Key topics covered include Liouville's theorem, the proof of Poincaré's non-integrability theorem and the nonlinear dynamics in the neighbourhood of equilibria. The theorem of Kolmogorov on persistence of invariant tori and the theory of exponential stability of Nekhoroshev are proved via constructive algorithms based on the Lie series method. A final chapter is devoted to the discovery of chaos by Poincaré and its relations with integrability, also including recent results on superexponential stability. Written in an accessible, self-contained way with few prerequisites, this book can serve as an introductory text for senior undergraduate and graduate students.
Download or read book Galileo Unbound written by David D. Nolte and published by Oxford University Press. This book was released on 2018-07-12 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: Galileo Unbound traces the journey that brought us from Galileo's law of free fall to today's geneticists measuring evolutionary drift, entangled quantum particles moving among many worlds, and our lives as trajectories traversing a health space with thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets or the collapse of stars into black holes. This book tells the history of spaces of expanding dimension and increasing abstraction and how they continue today to give new insight into the physics of complex systems. Galileo published the first modern law of motion, the Law of Fall, that was ideal and simple, laying the foundation upon which Newton built the first theory of dynamics. Early in the twentieth century, geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, forcing light rays to bend past the Sun. Possibly more radical was Feynman's dilemma of quantum particles taking all paths at once — setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant, the need to track ever more complex changes and to capture their essence, to find patterns in the chaos as we try to predict and control our world.
Book Synopsis Difference Equations, Discrete Dynamical Systems and Applications by : Sorin Olaru
Download or read book Difference Equations, Discrete Dynamical Systems and Applications written by Sorin Olaru and published by Springer Nature. This book was released on with total page 423 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Differential Dynamical Systems, Revised Edition by : James D. Meiss
Download or read book Differential Dynamical Systems, Revised Edition written by James D. Meiss and published by SIAM. This book was released on 2017-01-24 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics. Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts?flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. This new edition contains several important updates and revisions throughout the book. Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple, Mathematica, and MATLAB software to give students practice with computation applied to dynamical systems problems.
Book Synopsis Discrete Systems and Integrability by : J. Hietarinta
Download or read book Discrete Systems and Integrability written by J. Hietarinta and published by Cambridge University Press. This book was released on 2016-09 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: A first introduction to the theory of discrete integrable systems at a level suitable for students and non-experts.
Book Synopsis The Physics of Chaos in Hamiltonian Systems by : George M. Zaslavsky
Download or read book The Physics of Chaos in Hamiltonian Systems written by George M. Zaslavsky and published by Imperial College Press. This book was released on 2007 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to familiarize the reader with the essential properties of the chaotic dynamics of Hamiltonian systems by avoiding specialized mathematical tools, thus making it easily accessible to a broader audience of researchers and students. Unique material on the most intriguing and fascinating topics of unsolved and current problems in contemporary chaos theory is presented. The coverage includes: separatrix chaos; properties and a description of systems with non-ergodic dynamics; the distribution of Poincar(r) recurrences and their role in transport theory; dynamical models of the MaxwellOCOs Demon, the occurrence of persistent fluctuations, and a detailed discussion of their role in the problem underlying the foundation of statistical physics; the emergence of stochastic webs in phase space and their link to space tiling with periodic (crystal type) and aperiodic (quasi-crystal type) symmetries. This second edition expands on pseudochaotic dynamics with weak mixing and the new phenomenon of fractional kinetics, which is crucial to the transport properties of chaotic motion. The book is ideally suited to all those who are actively working on the problems of dynamical chaos as well as to those looking for new inspiration in this area. It introduces the physicist to the world of Hamiltonian chaos and the mathematician to actual physical problems.The material can also be used by graduate students."
Book Synopsis Ordinary Differential Equations and Dynamical Systems by : Gerald Teschl
Download or read book Ordinary Differential Equations and Dynamical Systems written by Gerald Teschl and published by American Mathematical Society. This book was released on 2024-01-12 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm–Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincaré–Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.
