Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
Author: Kenneth R. Meyer
Publisher: Springer
Total Pages: 389
Release: 2017-05-04
Genre: Mathematics
ISBN: 3319536915
GET BOOK

Download Introduction to Hamiltonian Dynamical Systems and the N-Body Problem Book in PDF, Epub and Kindle

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)

Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems

Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems
Author: Antonio Giorgilli
Publisher: Cambridge University Press
Total Pages: 474
Release: 2022-05-05
Genre: Science
ISBN: 100917486X
GET BOOK

Download Notes on Hamiltonian Dynamical Systems Notes on Hamiltonian Dynamical Systems Book in PDF, Epub and Kindle

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.

Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces

Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces
Author: Birgit Jacob
Publisher: Springer Science & Business Media
Total Pages: 221
Release: 2012-06-13
Genre: Science
ISBN: 3034803990
GET BOOK

Download Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces Book in PDF, Epub and Kindle

This book provides a self-contained introduction to the theory of infinite-dimensional systems theory and its applications to port-Hamiltonian systems. The textbook starts with elementary known results, then progresses smoothly to advanced topics in current research. Many physical systems can be formulated using a Hamiltonian framework, leading to models described by ordinary or partial differential equations. For the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take into account this interaction with the environment. This book is the first textbook on infinite-dimensional port-Hamiltonian systems. An abstract functional analytical approach is combined with the physical approach to Hamiltonian systems. This combined approach leads to easily verifiable conditions for well-posedness and stability. The book is accessible to graduate engineers and mathematicians with a minimal background in functional analysis. Moreover, the theory is illustrated by many worked-out examples.

Hamiltonian Systems

Hamiltonian Systems
Author: Alfredo M. Ozorio de Almeida
Publisher: Cambridge University Press
Total Pages: 262
Release: 1988
Genre: Mathematics
ISBN: 9780521386708
GET BOOK

Download Hamiltonian Systems Book in PDF, Epub and Kindle

Hamiltonian Systems outlines the main results in the field, and considers the implications for quantum mechanics.

Classical and Quantum Dynamics of Constrained Hamiltonian Systems

Classical and Quantum Dynamics of Constrained Hamiltonian Systems
Author: Heinz J. Rothe
Publisher: World Scientific
Total Pages: 317
Release: 2010
Genre: Science
ISBN: 9814299642
GET BOOK

Download Classical and Quantum Dynamics of Constrained Hamiltonian Systems Book in PDF, Epub and Kindle

This book is an introduction to the field of constrained Hamiltonian systems and their quantization, a topic which is of central interest to theoretical physicists who wish to obtain a deeper understanding of the quantization of gauge theories, such as describing the fundamental interactions in nature. Beginning with the early work of Dirac, the book covers the main developments in the field up to more recent topics, such as the field?antifield formalism of Batalin and Vilkovisky, including a short discussion of how gauge anomalies may be incorporated into this formalism. All topics are well illustrated with examples emphasizing points of central interest. The book should enable graduate students to follow the literature on this subject without much problems, and to perform research in this field.

Metamorphoses of Hamiltonian Systems with Symmetries

Metamorphoses of Hamiltonian Systems with Symmetries
Author: Konstantinos Efstathiou
Publisher: Springer
Total Pages: 155
Release: 2005-01-28
Genre: Science
ISBN: 3540315500
GET BOOK

Download Metamorphoses of Hamiltonian Systems with Symmetries Book in PDF, Epub and Kindle

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.

Hamiltonian Dynamical Systems and Applications

Hamiltonian Dynamical Systems and Applications
Author: Walter Craig
Publisher: Springer Science & Business Media
Total Pages: 450
Release: 2008-02-17
Genre: Mathematics
ISBN: 1402069642
GET BOOK

Download Hamiltonian Dynamical Systems and Applications Book in PDF, Epub and Kindle

This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.

Nearly Integrable Infinite-Dimensional Hamiltonian Systems

Nearly Integrable Infinite-Dimensional Hamiltonian Systems
Author: Sergej B. Kuksin
Publisher: Springer
Total Pages: 128
Release: 2006-11-15
Genre: Mathematics
ISBN: 3540479201
GET BOOK

Download Nearly Integrable Infinite-Dimensional Hamiltonian Systems Book in PDF, Epub and Kindle

The book is devoted to partial differential equations of Hamiltonian form, close to integrable equations. For such equations a KAM-like theorem is proved, stating that solutions of the unperturbed equation that are quasiperiodic in time mostly persist in the perturbed one. The theorem is applied to classical nonlinear PDE's with one-dimensional space variable such as the nonlinear string and nonlinear Schr|dinger equation andshow that the equations have "regular" (=time-quasiperiodic and time-periodic) solutions in rich supply. These results cannot be obtained by other techniques. The book will thus be of interest to mathematicians and physicists working with nonlinear PDE's. An extensivesummary of the results and of related topics is provided in the Introduction. All the nontraditional material used is discussed in the firstpart of the book and in five appendices.

Morse Theory for Hamiltonian Systems

Morse Theory for Hamiltonian Systems
Author: Alberto Abbondandolo
Publisher: CRC Press
Total Pages: 202
Release: 2001-03-15
Genre: Mathematics
ISBN: 1482285746
GET BOOK

Download Morse Theory for Hamiltonian Systems Book in PDF, Epub and Kindle

This Research Note explores existence and multiplicity questions for periodic solutions of first order, non-convex Hamiltonian systems. It introduces a new Morse (index) theory that is easier to use, less technical, and more flexible than existing theories and features techniques and results that, until now, have appeared only in scattered journals

Hamiltonian Mechanical Systems and Geometric Quantization

Hamiltonian Mechanical Systems and Geometric Quantization
Author: Mircea Puta
Publisher: Springer Science & Business Media
Total Pages: 289
Release: 2012-12-06
Genre: Mathematics
ISBN: 9401119929
GET BOOK

Download Hamiltonian Mechanical Systems and Geometric Quantization Book in PDF, Epub and Kindle

This volume presents various aspects of the geometry of symplectic and Poisson manifolds, and applications in Hamiltonian mechanics and geometric quantization are indicated. Chapter 1 presents some general facts about symplectic vector space, symplectic manifolds and symplectic reduction. Chapter 2 deals with the study of Hamiltonian mechanics. Chapter 3 considers some standard facts concerning Lie groups and algebras which lead to the theory of momentum mappings and the Marsden--Weinstein reduction. Chapters 4 and 5 consider the theory and the stability of equilibrium solutions of Hamilton--Poisson mechanical systems. Chapters 6 and 7 are devoted to the theory of geometric quantization. This leads, in Chapter 8, to topics such as foliated cohomology, the theory of the Dolbeault--Kostant complex, and their applications. A discussion of the relation between geometric quantization and the Marsden--Weinstein reduction is presented in Chapter 9. The final chapter considers extending the theory of geometric quantization to Poisson manifolds, via the theory of symplectic groupoids. Each chapter concludes with problems and solutions, many of which present significant applications and, in some cases, major theorems. For graduate students and researchers whose interests and work involve symplectic geometry and Hamiltonian mechanics.