Download or read book Stability of the equilibrium points in Hamiltonian systems with two degrees of freedom written by Julij S. Ilʹjašenko and published by . This book was released on 1990 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Stability of the Equilibrium Points in Hamiltonian Systems written by I︠U︡. S. Ilʹi︠a︡shenko and published by . This book was released on 1990 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Stability of the Equilibrium Points in Hamiltonian Systems with Two Degrees of Freedom written by Ju. S. Il'yashenko and published by . This book was released on 1990 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Normal Forms and Stability of Hamiltonian Systems written by Hildeberto E. Cabral and published by Springer Nature. This book was released on 2023-09-12 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the reader to the study of Hamiltonian systems, focusing on the stability of autonomous and periodic systems and expanding to topics that are usually not covered by the canonical literature in the field. It emerged from lectures and seminars given at the Federal University of Pernambuco, Brazil, known as one of the leading research centers in the theory of Hamiltonian dynamics. This book starts with a brief review of some results of linear algebra and advanced calculus, followed by the basic theory of Hamiltonian systems. The study of normal forms of Hamiltonian systems is covered by Ch.3, while Chapters 4 and 5 treat the normalization of Hamiltonian matrices. Stability in non-linear and linear systems are topics in Chapters 6 and 7. This work finishes with a study of parametric resonance in Ch. 8. All the background needed is presented, from the Hamiltonian formulation of the laws of motion to the application of the Krein-Gelfand-Lidskii theory of strongly stable systems. With a clear, self-contained exposition, this work is a valuable help to advanced undergraduate and graduate students, and to mathematicians and physicists doing research on this topic.
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 Introduction to Hamiltonian Dynamical Systems and the N Body Problem written by Kenneth R. Meyer and published by Springer. This book was released on 2017-05-04 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This third edition text provides expanded material on the restricted three body problem and celestial mechanics. With each chapter containing new content, readers are provided with new material on reduction, orbifolds, and the regularization of the Kepler problem, all of which are provided with applications. The previous editions grew out of graduate level courses in mathematics, engineering, and physics given at several different universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. This text provides a mathematical structure of celestial mechanics ideal for beginners, and will be useful to graduate students and researchers alike. Reviews of the second edition: "The primary subject here is the basic theory of Hamiltonian differential equations studied from the perspective of differential dynamical systems. The N-body problem is used as the primary example of a Hamiltonian system, a touchstone for the theory as the authors develop it. This book is intended to support a first course at the graduate level for mathematics and engineering students. ... It is a well-organized and accessible introduction to the subject ... . This is an attractive book ... ." (William J. Satzer, The Mathematical Association of America, March, 2009) “The second edition of this text infuses new mathematical substance and relevance into an already modern classic ... and is sure to excite future generations of readers. ... This outstanding book can be used not only as an introductory course at the graduate level in mathematics, but also as course material for engineering graduate students. ... it is an elegant and invaluable reference for mathematicians and scientists with an interest in classical and celestial mechanics, astrodynamics, physics, biology, and related fields.” (Marian Gidea, Mathematical Reviews, Issue 2010 d)
Download or read book Formal Stability of Hamiltonian Systems with Two Degrees of Freedom written by Stephen P. Diliberto and published by . This book was released on 1966 with total page 42 pages. Available in PDF, EPUB and Kindle. Book excerpt: Motion near periodic solutions is characterized by the eigen-values of the linear terms of the differential equation in local coordinates. When these local coordinates have purely imaginary characteristic roots the possibility of stability exists. When these roots are commensurable with the frequency of the periodic solution the system is in general unstable. It was believed that there were an infinite set of algebraic conditions necessary for formal stability. These are herein to reduce to two for a Hamiltonian system with two degrees of freedom. (Author).
Download or read book Seeking Stability in Hamiltonian Systems with Two Degrees of Freedom written by Mark Francis Demers and published by . This book was released on 1994 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Introduction to Hamiltonian Dynamical Systems and the N Body Problem written by Kenneth Meyer and published by Springer Science & Business Media. This book was released on 2008-12-05 with total page 404 pages. Available in PDF, EPUB and Kindle. Book excerpt: Arising from a graduate course taught to math and engineering students, this text provides a systematic grounding in the theory of Hamiltonian systems, as well as introducing the theory of integrals and reduction. A number of other topics are covered too.
Download or read book Long Term Dynamical Behaviour of Natural and Artificial N Body Systems written by Archie E. Roy and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: The reader will find in this volume the Proceedings of the NATO Advanced Study Institute held in Cortina d'Ampezzo, Italy between August 3 and August 13, 1987 under the title "Long Term Dynamical Behaviour of Natural and Artificial N-body Systems". The Institute was the latest in a series held in 1972, 1975, 1978, 1981, 1984 in dynamical astronomy, theoretical mechanics and celestial mechanics under the Directorship of Professor Victor Szebehely. These previous institutes, held in high esteem by the international community of research workers, have resulted in a series of well-received and valuable Proceedings. In correspondence with Professor Szebehely and in long discussions with him in Colorado in August 1985, I agreed to his request that I undertake the preparation of a new ASI. I was happy to do so knowing I could call upon his vast experience in overseeing such ASI's. The last quarter century has been a period in which increasingly rapid progress has been made in celestial mechanics and related subjects not only because of the appearance of new problems urgently requiring solution but also because of the advent of new analytical techniques and powerful computer hardware and software.
Download or read book Transient Chaos written by Ying-Cheng Lai and published by Springer Science & Business Media. This book was released on 2011-02-26 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this Book is to give an overview, based on the results of nearly three decades of intensive research, of transient chaos. One belief that motivates us to write this book is that, transient chaos may not have been appreciated even within the nonlinear-science community, let alone other scientific disciplines.
Download or read book Capture Dynamics and Chaotic Motions in Celestial Mechanics written by Edward Belbruno and published by Princeton University Press. This book was released on 2004 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book describes a revolutionary new approach to determining low energy routes for spacecraft and comets by exploiting regions in space where motion is very sensitive (or chaotic). It also represents an ideal introductory text to celestial mechanics, dynamical systems, and dynamical astronomy. Bringing together wide-ranging research by others with his own original work, much of it new or previously unpublished, Edward Belbruno argues that regions supporting chaotic motions, termed weak stability boundaries, can be estimated. Although controversial until quite recently, this method was in fact first applied in 1991, when Belbruno used a new route developed from this theory to get a stray Japanese satellite back on course to the moon. This application provided a major verification of his theory, representing the first application of chaos to space travel. Since that time, the theory has been used in other space missions, and NASA is implementing new applications under Belbruno's direction. The use of invariant manifolds to find low energy orbits is another method here addressed. Recent work on estimating weak stability boundaries and related regions has also given mathematical insight into chaotic motion in the three-body problem. Belbruno further considers different capture and escape mechanisms, and resonance transitions. Providing a rigorous theoretical framework that incorporates both recent developments such as Aubrey-Mather theory and established fundamentals like Kolmogorov-Arnold-Moser theory, this book represents an indispensable resource for graduate students and researchers in the disciplines concerned as well as practitioners in fields such as aerospace engineering.
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 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 Global Dynamics Phase Space Transport Orbits Homoclinic to Resonances and Applications written by Stephen Wiggins and published by American Mathematical Soc.. This book was released on with total page 538 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph, which grew out of a series of lectures delivered by Stephen Wiggins at the Fields Institute in early 1993, is concerned with the geometrical viewpoint of the global dynamics of nonlinear dynamical systems. With appropriate examples and concise explanations, Wiggins unites many different topics into one volume and makes a unique contribution to the field. Engineers, physicists, chemists, and mathematicians who work on issues related to the global dynamics of nonlinear dynamical systems will find these lectures very useful.
Download or read book Global Transversality Resonance and Chaotic Dynamics written by Albert C. J. Luo and published by World Scientific. This book was released on 2008 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique book presents a different point of view on the fundamental theory of global transversality, resonance and chaotic dynamics in n -dimensional nonlinear dynamic systems. The methodology and techniques presented in this book are applicable to nonlinear dynamical systems in general. This book provides useful tools for analytical and numerical predictions of chaos in nonlinear Hamiltonian and dissipative systems. All theoretical results are strictly proved. However, the ideas presented in this book are less formal and rigorous in an informal and lively manner. The author hopes the initial ideas may give some inspirations in the field of nonlinear dynamics. With physical concepts, the author also used the simple, mathematical language to write this book. Therefore, this book is very readable, which can be either a textbook for senior undergraduate and graduate students or a reference book for researches in nonlinear dynamics. Sample Chapter(s). Chapter 1: Introduction (1,196 KB). Contents: Differential Geometry of Flows; Global Transversality in Continuous Dynamical Systems; Chaotic Layer Dynamics; Two-Dimensional Stochastic Layers; Stochasticity in Resonant Separatrix Layers; Nonlinear Dynamics on an Equi-energy Surface; Stability and Grazing in Dissipative Systems; Global Dynamics in Two-Dimensional Dynamical Systems; Flow Switchability in Discontinuous Dynamical Systems. Readership: Mathematicians, physicists, researchers and engineers in mechanical engineering and electrical engineering as well as university professors and students.
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)
