Download or read book Hamiltonian Systems with Three or More Degrees of Freedom written by Carles Simó and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 681 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Hamiltonian Dynamical Systems written by R.S MacKay and published by CRC Press. This book was released on 2020-08-17 with total page 797 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classical mechanics is a subject that is teeming with life. However, most of the interesting results are scattered around in the specialist literature, which means that potential readers may be somewhat discouraged by the effort required to obtain them. Addressing this situation, Hamiltonian Dynamical Systems includes some of the most significant papers in Hamiltonian dynamics published during the last 60 years. The book covers bifurcation of periodic orbits, the break-up of invariant tori, chaotic behavior in hyperbolic systems, and the intricacies of real systems that contain coexisting order and chaos. It begins with an introductory survey of the subjects to help readers appreciate the underlying themes that unite an apparently diverse collection of articles. The book concludes with a selection of papers on applications, including in celestial mechanics, plasma physics, chemistry, accelerator physics, fluid mechanics, and solid state mechanics, and contains an extensive bibliography. The book provides a worthy introduction to the subject for anyone with an undergraduate background in physics or mathematics, and an indispensable reference work for researchers and graduate students interested in any aspect of classical mechanics.

Download or read book Chaotic Worlds from Order to Disorder in Gravitational N Body Dynamical Systems written by B.A. Steves and published by Springer Science & Business Media. This book was released on 2006-09-22 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on the recent NATO Advanced Study Institute "Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems", this state of the art textbook, written by internationally renowned experts, provides an invaluable reference volume for all students and researchers in gravitational n-body systems. The contributions are especially designed to give a systematic development from the fundamental mathematics which underpin modern studies of ordered and chaotic behaviour in n-body dynamics to their application to real motion in planetary systems. This volume presents an up-to-date synoptic view of the subject.

Download or read book Local and Semi Local Bifurcations in Hamiltonian Dynamical Systems written by Heinz Hanßmann and published by Springer. This book was released on 2006-10-18 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book demonstrates that while elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Therefore, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system, absent untypical conditions or external parameters. The text moves logically from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations must be replaced by Cantor sets.

Download or read book Handbook of Dynamical Systems written by H. Broer and published by Elsevier. This book was released on 2010-11-10 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this volume, the authors present a collection of surveys on various aspects of the theory of bifurcations of differentiable dynamical systems and related topics. By selecting these subjects, they focus on those developments from which research will be active in the coming years. The surveys are intended to educate the reader on the recent literature on the following subjects: transversality and generic properties like the various forms of the so-called Kupka-Smale theorem, the Closing Lemma and generic local bifurcations of functions (so-called catastrophe theory) and generic local bifurcations in 1-parameter families of dynamical systems, and notions of structural stability and moduli. Covers recent literature on various topics related to the theory of bifurcations of differentiable dynamical systems Highlights developments that are the foundation for future research in this field Provides material in the form of surveys, which are important tools for introducing the bifurcations of differentiable dynamical systems

Download or read book Multiple Time Scale Dynamical Systems written by Christopher K.R.T. Jones and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

Download or read book Quasi Periodic Motions in Families of Dynamical Systems written by Hendrik W. Broer and published by Springer. This book was released on 2009-01-25 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems - or contexts - are considered.

Download or read book Progress and Challenges in Dynamical Systems written by Santiago Ibáñez and published by Springer Science & Business Media. This book was released on 2013-09-20 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains papers based on talks given at the International Conference Dynamical Systems: 100 years after Poincaré held at the University of Oviedo, Gijón in Spain, September 2012. It provides an overview of the state of the art in the study of dynamical systems. This book covers a broad range of topics, focusing on discrete and continuous dynamical systems, bifurcation theory, celestial mechanics, delay difference and differential equations, Hamiltonian systems and also the classic challenges in planar vector fields. It also details recent advances and new trends in the field, including applications to a wide range of disciplines such as biology, chemistry, physics and economics. The memory of Henri Poincaré, who laid the foundations of the subject, inspired this exploration of dynamical systems. In honor of this remarkable mathematician, theoretical physicist, engineer and philosopher, the authors have made a special effort to place the reader at the frontiers of current knowledge in the discipline.

Download or read book An Investigation of the Stability of Three Degree of Freedom Hamiltonian Systems in Presence of a Third Order Resonance written by David Louis Richardson and published by . This book was released on 1972 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Perturbation Theory written by Giuseppe Gaeta and published by Springer Nature. This book was released on 2022-12-16 with total page 601 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, is devoted to the fundamentals of Perturbation Theory (PT) as well as key applications areas such as Classical and Quantum Mechanics, Celestial Mechanics, and Molecular Dynamics. Less traditional fields of application, such as Biological Evolution, are also discussed. Leading scientists in each area of the field provide a comprehensive picture of the landscape and the state of the art, with the specific goal of combining mathematical rigor, explicit computational methods, and relevance to concrete applications. New to this edition are chapters on Water Waves, Rogue Waves, Multiple Scales methods, legged locomotion, Condensed Matter among others, while all other contributions have been revised and updated. Coverage includes the theory of (Poincare’-Birkhoff) Normal Forms, aspects of PT in specific mathematical settings (Hamiltonian, KAM theory, Nekhoroshev theory, and symmetric systems), technical problems arising in PT with solutions, convergence of series expansions, diagrammatic methods, parametric resonance, systems with nilpotent real part, PT for non-smooth systems, and on PT for PDEs [write out this acronym partial differential equations]. Another group of papers is focused specifically on applications to Celestial Mechanics, Quantum Mechanics and the related semiclassical PT, Quantum Bifurcations, Molecular Dynamics, the so-called choreographies in the N-body problem, as well as Evolutionary Theory. Overall, this unique volume serves to demonstrate the wide utility of PT, while creating a foundation for innovations from a new generation of graduate students and professionals in Physics, Mathematics, Mechanics, Engineering and the Biological Sciences.

Download or read book Introduction to Applied Nonlinear Dynamical Systems and Chaos written by Stephen Wiggins and published by Springer Science & Business Media. This book was released on 2003-10-01 with total page 860 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to applied nonlinear dynamics and chaos places emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about their behavior. The new edition has been updated and extended throughout, and contains a detailed glossary of terms. From the reviews: "Will serve as one of the most eminent introductions to the geometric theory of dynamical systems." --Monatshefte für Mathematik

Download or read book Regular and Stochastic Motion written by A. J. Lichtenberg and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book treats stochastic motion in nonlinear oscillator systems. It describes a rapidly growing field of nonlinear mechanics with applications to a number of areas in science and engineering, including astronomy, plasma physics, statistical mechanics and hydrodynamics. The main em phasis is on intrinsic stochasticity in Hamiltonian systems, where the stochastic motion is generated by the dynamics itself and not by external noise. However, the effects of noise in modifying the intrinsic motion are also considered. A thorough introduction to chaotic motion in dissipative systems is given in the final chapter. Although the roots of the field are old, dating back to the last century when Poincare and others attempted to formulate a theory for nonlinear perturbations of planetary orbits, it was new mathematical results obtained in the 1960's, together with computational results obtained using high speed computers, that facilitated our new treatment of the subject. Since the new methods partly originated in mathematical advances, there have been two or three mathematical monographs exposing these developments. However, these monographs employ methods and language that are not readily accessible to scientists and engineers, and also do not give explicit tech niques for making practical calculations. In our treatment of the material, we emphasize physical insight rather than mathematical rigor. We present practical methods for describing the motion, for determining the transition from regular to stochastic behavior, and for characterizing the stochasticity. We rely heavily on numerical computations to illustrate the methods and to validate them.

Download or read book Mathematical Reviews written by and published by . This book was released on 2005 with total page 1084 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Rigid Body Dynamics written by Hamad M. Yehia and published by Springer Nature. This book was released on 2022-06-27 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a complete and up-to-date examination of rigid body dynamics using a Lagrangian approach. All known integrable cases, which were previously scattered throughout the literature, are collected here for convenient reference. Also contained are particular solutions to diverse problems treated within rigid body dynamics. The first seven chapters introduce the elementary dynamics of the rigid body and its main problems. A full historical account of the discovery and development of each of the integrable cases is included as well. Instructors will find this portion of the book well-suited for an undergraduate course, having been formulated by the author in the classroom over many years. The second part includes more advanced topics and some of the author’s original research, highlighting several unique methods he developed that have led to significant results. Some of the specific topics covered include the twelve known solutions of the equations of motion in the classical problem, which has not previously appeared in English before; a collection of completely new integrable cases; and the motion of a rigid body around a fixed point under the action of an asymmetric combination of potential and gyroscopic forces. Rigid Body Dynamics will appeal to researchers in the area as well as those studying dynamical and integrable systems theory.

Download or read book Dynamics of Small Solar System Bodies and Exoplanets written by Jean J. Souchay and published by Springer. This book was released on 2010-03-10 with total page 523 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book on recent investigations of the dynamics of celestial bodies in the solar and extra-Solar System is based on the elaborated lecture notes of a thematic school on the topic, held as a result of cooperation between the SYRTE Department of Paris Observatory and the section of astronomy of the Vienna University. Each chapter corresponds to a lecture of several hours given by its author(s). The book therefore represents a necessary and very precious document for teachers, students, and researchers in the ?eld. The ?rst two chapters by A. Lemaˆ ?tre and H. Skokos deal with standard topics of celestial mechanics: the ?rst one explains the basic principles of resonances in mechanics and their studies in the case of the Solar System. The differences between the various cases of resonance (mean motion, secular, etc. ) are emphasized together with resonant effects on celestial bodies moving around the Sun. The second one deals with approximative methods of describing chaos. These methods, some of them being classical, as the Lyapounov exponents, other ones being developed in the very recent past, are explained in full detail. The second one explains the basic principles of resonances in mechanics and their studies in the case of the Solar System. The differences between the various cases of resonance (mean motion, s- ular, etc. ) are emphasized together with resonant effects on celestial bodies moving around the Sun. The following three chapters by A. Cellino, by P. Robutel and J.

Download or read book Differential Galois Theory and Non Integrability of Hamiltonian Systems written by Juan J. Morales Ruiz and published by Birkhäuser. This book was released on 2012-12-06 with total page 177 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Lyapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)

Download or read book Order and Chaos in Dynamical Astronomy written by George Contopoulos and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 633 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is one of the first to provide a general overview of order and chaos in dynamical astronomy. The progress of the theory of chaos has a profound impact on galactic dynamics. It has even invaded celestial mechanics, since chaos was found in the solar system which in the past was considered as a prototype of order. The book provides a unifying approach to these topics from an author who has spent more than 50 years of research in the field. The first part treats order and chaos in general. The other two parts deal with order and chaos in galaxies and with other applications in dynamical astronomy, ranging from celestial mechanics to general relativity and cosmology.