Exact Symplectic Twist Maps In Higher Dimensions

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Symplectic Twist Maps

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Author : Christophe Golé
language : en
Publisher: World Scientific
Release Date : 2001

Symplectic Twist Maps written by Christophe Golé and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.

0. Introduction. 1. Fall from paradise. 2. Billiards and broken geodesies. 3. An ancestor of symplectic topology -- 1. Twist maps of the annulus. 4. Monotone twist maps of the annulus. 5. Generating functions and variational setting. 6. Examples. 7. The Poincare-Birkhoff theorem -- 2. The Aubry-Mather theorem. 8. Introduction. 9. Cyclically ordered sequences and orbits. 10. Minimizing orbits. 11. CO orbits of all rotation numbers. 12. Aubry-Mather sets -- 3. Ghost circles. 14. Gradient flow of the action. 15. The gradient flow and the Aubry-Mather theorem. 16. Ghost circles. 17. Construction of ghost circles. 18. Construction of disjoint ghost circles. 19. Proof of lemma 18.5. 20. Proof of theorem 18.1. 21. Remarks and applications. 22. Proofs of monotonicity and of the Sturmian lemma -- 4. Symplectic twist maps. 23. Symplectic twist maps of T[symbol] x IR[symbol]. 24. Examples. 25. More on generating functions. 2.6. Symplectic twist maps on general cotangent bundles of compact manifolds -- 5. Periodic orbits for symplectic twist maps of T[symbol] x IR[symbol]. 27. Presentation of the results. 28. Finite dimensional variational setting. 29. Second variation and nondegenerate periodic orbits. 30. The coercive case. 31. Asymptotically linear systems. 32. Ghost tori. 33. Hyperbolicity Vs. action minimizers -- 6. Invariant manifolds. 34. The theory of Kolmogorov-Arnold-Moser. 35. Properties of invariant tori. 36. (Un)stable manifolds and heteroclinic orbits. 37. Instability, transport and diffusion -- 7. Hamiltonian systems vs. twist maps. 38. Case study: The geodesic flow. 39. Decomposition of Hamiltonian maps into twist maps. 40. Return maps in Hamiltonian systems. 41. Suspension of symplectic twist maps by Hamiltonian flows -- 8. Periodic orbits for Hamiltonian systems. 42. Periodic orbits in the cotangent of the n-torus. 43. Periodic orbits in general cotangent spaces. 44. Linking of spheres -- 9. Generalizations of the Aubry-Mather theorem. 45. Theory for functions on lattices and PDE's. 46. Monotone recurrence relationst. 47. Anti-integrable limit. 48. Mather's theory of minimal measures. 49. The case of hyperbolic manifolds. 50. Concluding remarks -- 10. Generating phases and symplectic topology. 51. Chaperon's method and the theorem Of Conley-Zehnder. 52. Generating phases and symplectic geometry.

Exact Symplectic Twist Maps In Higher Dimensions

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Author : Héctor Ernesto Lomelí Ortega
language : en
Publisher:
Release Date : 1995

Exact Symplectic Twist Maps In Higher Dimensions written by Héctor Ernesto Lomelí Ortega and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with categories.

Hamiltonian Systems With Three Or More Degrees Of Freedom

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Author : Carles Simó
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Hamiltonian Systems With Three Or More Degrees Of Freedom written by Carles Simó 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 2012-12-06 with Mathematics categories.

A survey of current knowledge about Hamiltonian systems with three or more degrees of freedom and related topics. The Hamiltonian systems appearing in most of the applications are non-integrable. Hence methods to prove non-integrability results are presented and the different meaning attributed to non-integrability are discussed. For systems near an integrable one, it can be shown that, under suitable conditions, some parts of the integrable structure, most of the invariant tori, survive. Many of the papers discuss near-integrable systems. From a topological point of view, some singularities must appear in different problems, either caustics, geodesics, moving wavefronts, etc. This is also related to singularities in the projections of invariant objects, and can be used as a signature of these objects. Hyperbolic dynamics appear as a source on unpredictable behaviour and several mechanisms of hyperbolicity are presented. The destruction of tori leads to Aubrey-Mather objects, and this is touched on for a related class of systems. Examples without periodic orbits are constructed, against a classical conjecture. Other topics concern higher dimensional systems, either finite (networks and localised vibrations on them) or infinite, like the quasiperiodic Schrödinger operator or nonlinear hyperbolic PDE displaying quasiperiodic solutions. Most of the applications presented concern celestial mechanics problems, like the asteroid problem, the design of spacecraft orbits, and methods to compute periodic solutions.

The Geometry Of Hamiltonian Systems

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Author : Tudor Ratiu
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Geometry Of Hamiltonian Systems written by Tudor Ratiu 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 2012-12-06 with Mathematics categories.

The papers in this volume are an outgrowth of the lectures and informal discussions that took place during the workshop on "The Geometry of Hamiltonian Systems" which was held at MSRl from June 5 to 16, 1989. It was, in some sense, the last major event of the year-long program on Symplectic Geometry and Mechanics. The emphasis of all the talks was on Hamiltonian dynamics and its relationship to several aspects of symplectic geometry and topology, mechanics, and dynamical systems in general. The organizers of the conference were R. Devaney (co-chairman), H. Flaschka (co-chairman), K. Meyer, and T. Ratiu. The entire meeting was built around two mini-courses of five lectures each and a series of two expository lectures. The first of the mini-courses was given by A. T. Fomenko, who presented the work of his group at Moscow University on the classification of integrable systems. The second mini course was given by J. Marsden of UC Berkeley, who spoke about several applications of symplectic and Poisson reduction to problems in stability, normal forms, and symmetric Hamiltonian bifurcation theory. Finally, the two expository talks were given by A. Fathi of the University of Florida who concentrated on the links between symplectic geometry, dynamical systems, and Teichmiiller theory.

Introduction To Hamiltonian Dynamical Systems And The N Body Problem

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Author : Kenneth Meyer
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Introduction To Hamiltonian Dynamical Systems And The N Body Problem written by Kenneth Meyer 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 2013-04-17 with Mathematics categories.

The theory of Hamiltonian systems is a vast subject which can be studied from many different viewpoints. This book develops the basic theory of Hamiltonian differential equations from a dynamical systems point of view. That is, the solutions of the differential equations are thought of as curves in a phase space and it is the geometry of these curves that is the important object of study. The analytic underpinnings of the subject are developed in detail. The last chapter on twist maps has a more geometric flavor. It was written by Glen R. Hall. The main example developed in the text is the classical N-body problem, i.e., the Hamiltonian system of differential equations which describe the motion of N point masses moving under the influence of their mutual gravitational attraction. Many of the general concepts are applied to this example. But this is not a book about the N-body problem for its own sake. The N-body problem is a subject in its own right which would require a sizable volume of its own. Very few of the special results which only apply to the N-body problem are given.

Lectures On Symplectic Geometry

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Author : Ana Cannas da Silva
language : en
Publisher: Springer
Release Date : 2004-10-27

Lectures On Symplectic Geometry written by Ana Cannas da Silva and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-10-27 with Mathematics categories.

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Hamiltonian Dynamical Systems

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Author : Kenneth Ray Meyer
language : en
Publisher:
Release Date : 1988

Hamiltonian Dynamical Systems written by Kenneth Ray Meyer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.

This volume contains contributions by participants in the AMS-IMS-SIAM Summer Research Conference on Hamiltonian Dynamical Systems, held at the University of Colorado in June 1984. The conference brought together researchers from a wide spectrum of areas in Hamiltonian dynamics. The papers vary from expository descriptions of recent developments to fairly technical presentations with new results. Collectively, they provide an excellent survey of contemporary work in this area. The field of Hamiltonian dynamics has its roots in Newton's application of the science of dynamics to the emerging problems of orbital mechanics and in the development of celestial mechanics. Indeed, many of the talks at the conference emphasized topics directly concerned with such questions as the Newtonian $n$-body problem, the three-body problem, and the artificial earth satellite. Some speakers focused on those dynamical issues--such as integrability, KAM, and extensions of the Poincare-Birkhoff results--that emerged from celestial mechanics and extend to wider classes of dynamical systems. Other topics covered include periodic orbits with variation methods, twist and annulus maps, stable mainfold theory, almost periodic motion, and heteroclinic and homoclinic orbits. By bringing together papers from such a diverse range of topics, this book may serve to stimulate further development in this area.

Symplectic Geometric Algorithms For Hamiltonian Systems

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Author : Kang Feng
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-10-18

Symplectic Geometric Algorithms For Hamiltonian Systems written by Kang Feng 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 2010-10-18 with Mathematics categories.

"Symplectic Geometric Algorithms for Hamiltonian Systems" will be useful not only for numerical analysts, but also for those in theoretical physics, computational chemistry, celestial mechanics, etc. The book generalizes and develops the generating function and Hamilton-Jacobi equation theory from the perspective of the symplectic geometry and symplectic algebra. It will be a useful resource for engineers and scientists in the fields of quantum theory, astrophysics, atomic and molecular dynamics, climate prediction, oil exploration, etc. Therefore a systematic research and development of numerical methodology for Hamiltonian systems is well motivated. Were it successful, it would imply wide-ranging applications.

Asymptotics Beyond All Orders

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Author : Harvey Segur
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Asymptotics Beyond All Orders written by Harvey Segur 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 2012-12-06 with Science categories.

An asymptotic expansion is a series that provides a sequence of increasingly accurate approximations to a function in a particular limit. The formal definition, given by Poincare (1886, Acta Math. 8:295), is as follows. Given a function,

Smooth Ergodic Theory And Its Applications

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Author : A. B. Katok
language : en
Publisher: American Mathematical Soc.
Release Date : 2001

Smooth Ergodic Theory And Its Applications written by A. B. Katok 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 2001 with Mathematics categories.

During the past decade, there have been several major new developments in smooth ergodic theory, which have attracted substantial interest to the field from mathematicians as well as scientists using dynamics in their work. In spite of the impressive literature, it has been extremely difficult for a student-or even an established mathematician who is not an expert in the area-to acquire a working knowledge of smooth ergodic theory and to learn how to use its tools. Accordingly, the AMS Summer Research Institute on Smooth Ergodic Theory and Its Applications (Seattle, WA) had a strong educational component, including ten mini-courses on various aspects of the topic that were presented by leading experts in the field. This volume presents the proceedings of that conference. Smooth ergodic theory studies the statistical properties of differentiable dynamical systems, whose origin traces back to the seminal works of Poincare and later, many great mathematicians who made contributions to the development of the theory. The main topic of this volume, smooth ergodic theory, especially the theory of nonuniformly hyperbolic systems, provides the principle paradigm for the rigorous study of complicated or chaotic behavior in deterministic systems. This paradigm asserts that if a non-linear dynamical system exhibits sufficiently pronounced exponential behavior, then global properties of the system can be deduced from studying the linearized system. One can then obtain detailed information on topological properties (such as the growth of periodic orbits, topological entropy, and dimension of invariant sets including attractors), as well as statistical properties (such as the existence of invariant measures, asymptotic behavior of typical orbits, ergodicity, mixing, decay of corre This volume serves a two-fold purpose: first, it gives a useful gateway to smooth ergodic theory for students and nonspecialists, and second, it provides a state-of-the-art report on important current aspects of the subject. The book is divided into three parts: lecture notes consisting of three long expositions with proofs aimed to serve as a comprehensive and self-contained introduction to a particular area of smooth ergodic theory; thematic sections based on mini-courses or surveys held at the conference; and original contributions presented at the meeting or closely related to the topics that were discussed there.