Geometric Dynamics

Author : Constantin Udriște
Publisher : Springer Science & Business Media
Page : 416 pages
File Size : 46,7 Mb
Release : 2000
Category : Mathematics
ISBN : 0792364015

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Geometric Dynamics by Constantin Udriște Pdf

The theme of this text is the philosophy that any particle flow generates a particle dynamics, in a suitable geometrical framework. It covers topics that include: geometrical and physical vector fields; field lines; flows; stability of equilibrium points; potential systems and catastrophe geometry; field hypersurfaces; bifurcations; distribution orthogonal to a vector field; extrema with nonholonomic constraints; thermodynamic systems; energies; geometric dynamics induced by a vector field; magnetic fields around piecewise rectilinear electric circuits; geometric magnetic dynamics; and granular materials and their mechanical behaviour. The text should be useful for first-year graduate students in mathematics, mechanics, physics, engineering, biology, chemistry, and economics. It can also be addressed to professors and researchers whose work involves mathematics, mechanics, physics, engineering, biology, chemistry, and economics.

Author : C. Udriste
Publisher : Springer Science & Business Media
Page : 406 pages
File Size : 41,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401141871

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Geometric Dynamics by C. Udriste Pdf

Geometric dynamics is a tool for developing a mathematical representation of real world phenomena, based on the notion of a field line described in two ways: -as the solution of any Cauchy problem associated to a first-order autonomous differential system; -as the solution of a certain Cauchy problem associated to a second-order conservative prolongation of the initial system. The basic novelty of our book is the discovery that a field line is a geodesic of a suitable geometrical structure on a given space (Lorentz-Udri~te world-force law). In other words, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, ensuring that all trajectories of a given vector field are geodesics. This is our contribution to an old open problem studied by H. Poincare, S. Sasaki and others. From the kinematic viewpoint of corpuscular intuition, a field line shows the trajectory followed by a particle at a point of the definition domain of a vector field, if the particle is sensitive to the related type of field. Therefore, field lines appear in a natural way in problems of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.

Author : J.Jr. Palis
Publisher : Springer
Page : 835 pages
File Size : 51,7 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540409694

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Geometric Dynamics by J.Jr. Palis Pdf

Author : Holm Darryl D
Publisher : World Scientific Publishing Company
Page : 468 pages
File Size : 47,9 Mb
Release : 2011-07-13
Category : Mathematics
ISBN : 9781911298656

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Geometric Mechanics - Part I: Dynamics And Symmetry (2nd Edition) by Holm Darryl D Pdf

See also GEOMETRIC MECHANICS — Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level.The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie-Poisson Hamiltonian formulations and momentum maps in physical applications.The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly.The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications./a

Author : Christian Bonatti,Lorenzo J. Díaz,Marcelo Viana
Publisher : Springer Science & Business Media
Page : 390 pages
File Size : 45,6 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9783540268444

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Dynamics Beyond Uniform Hyperbolicity by Christian Bonatti,Lorenzo J. Díaz,Marcelo Viana Pdf

What is Dynamics about? In broad terms, the goal of Dynamics is to describe the long term evolution of systems for which an "infinitesimal" evolution rule is known. Examples and applications arise from all branches of science and technology, like physics, chemistry, economics, ecology, communications, biology, computer science, or meteorology, to mention just a few. These systems have in common the fact that each possible state may be described by a finite (or infinite) number of observable quantities, like position, velocity, temperature, concentration, population density, and the like. Thus, m the space of states (phase space) is a subset M of an Euclidean space M . Usually, there are some constraints between these quantities: for instance, for ideal gases pressure times volume must be proportional to temperature. Then the space M is often a manifold, an n-dimensional surface for some n

Author : José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi
Publisher : Springer
Page : 739 pages
File Size : 49,8 Mb
Release : 2014-09-23
Category : Science
ISBN : 9789401792202

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Geometry from Dynamics, Classical and Quantum by José F. Cariñena,Alberto Ibort,Giuseppe Marmo,Giuseppe Morandi Pdf

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

Author : J. Jr. Palis,W. de Melo
Publisher : Springer Science & Business Media
Page : 208 pages
File Size : 54,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461257035

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Geometric Theory of Dynamical Systems by J. Jr. Palis,W. de Melo Pdf

... cette etude qualitative (des equations difj'erentielles) aura par elle-m me un inter t du premier ordre ... HENRI POINCARE, 1881. We present in this book a view of the Geometric Theory of Dynamical Systems, which is introductory and yet gives the reader an understanding of some of the basic ideas involved in two important topics: structural stability and genericity. This theory has been considered by many mathematicians starting with Poincare, Liapunov and Birkhoff. In recent years some of its general aims were established and it experienced considerable development. More than two decades passed between two important events: the work of Andronov and Pontryagin (1937) introducing the basic concept of structural stability and the articles of Peixoto (1958-1962) proving the density of stable vector fields on surfaces. It was then that Smale enriched the theory substantially by defining as a main objective the search for generic and stable properties and by obtaining results and proposing problems of great relevance in this context. In this same period Hartman and Grobman showed that local stability is a generic property. Soon after this Kupka and Smale successfully attacked the problem for periodic orbits. We intend to give the reader the flavour of this theory by means of many examples and by the systematic proof of the Hartman-Grobman and the Stable Manifold Theorems (Chapter 2), the Kupka-Smale Theorem (Chapter 3) and Peixoto's Theorem (Chapter 4). Several ofthe proofs we give vii Introduction Vlll are simpler than the original ones and are open to important generalizations.

Author : Dong Eui Chang,Darryl D. Holm,George Patrick,Tudor Ratiu
Publisher : Springer
Page : 506 pages
File Size : 48,7 Mb
Release : 2015-04-16
Category : Mathematics
ISBN : 9781493924417

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Geometry, Mechanics, and Dynamics by Dong Eui Chang,Darryl D. Holm,George Patrick,Tudor Ratiu Pdf

This book illustrates the broad range of Jerry Marsden’s mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy. The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well.

Author : Instituto de Matemática Pura e Aplicada
Publisher : Lecture Notes in Mathematics
Page : 850 pages
File Size : 49,9 Mb
Release : 1983-09
Category : Mathematics
ISBN : UOM:39015015609426

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Geometric Dynamics by Instituto de Matemática Pura e Aplicada Pdf

Author : Constantin Udriste
Publisher : Springer
Page : 395 pages
File Size : 49,5 Mb
Release : 2011-11-10
Category : Mathematics
ISBN : 9401141886

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Geometric Dynamics by Constantin Udriste Pdf

Geometric dynamics is a tool for developing a mathematical representation of real world phenomena, based on the notion of a field line described in two ways: -as the solution of any Cauchy problem associated to a first-order autonomous differential system; -as the solution of a certain Cauchy problem associated to a second-order conservative prolongation of the initial system. The basic novelty of our book is the discovery that a field line is a geodesic of a suitable geometrical structure on a given space (Lorentz-Udri~te world-force law). In other words, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, ensuring that all trajectories of a given vector field are geodesics. This is our contribution to an old open problem studied by H. Poincare, S. Sasaki and others. From the kinematic viewpoint of corpuscular intuition, a field line shows the trajectory followed by a particle at a point of the definition domain of a vector field, if the particle is sensitive to the related type of field. Therefore, field lines appear in a natural way in problems of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.

Author : Jared Maruskin
Publisher : Walter de Gruyter GmbH & Co KG
Page : 348 pages
File Size : 53,8 Mb
Release : 2018-08-21
Category : Science
ISBN : 9783110597806

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Dynamical Systems and Geometric Mechanics by Jared Maruskin Pdf

Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explore similar systems that instead evolve on differentiable manifolds. The first part discusses the linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincaré maps, Floquet theory, the Poincaré-Bendixson theorem, bifurcations, and chaos. The second part of the book begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms.

Author : Darryl D. Holm
Publisher : Imperial College Press
Page : 375 pages
File Size : 40,7 Mb
Release : 2008-01-01
Category : Mathematics
ISBN : 9781848161955

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Geometric Mechanics: Dynamics and symmetry by Darryl D. Holm Pdf

Advanced undergraduate and graduate students in mathematics, physics and engineering.

Author : Workshop on Dynamical Systems and Related Topics
Publisher : American Mathematical Soc.
Page : 358 pages
File Size : 55,7 Mb
Release : 2008
Category : Differentiable dynamical systems
ISBN : 9780821842867

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Geometric and Probabilistic Structures in Dynamics by Workshop on Dynamical Systems and Related Topics Pdf

"This book presents a collection of articles that cover areas of mathematics related to dynamical systems. The authors are well-known experts who use geometric and probabilistic methods to study interesting problems in the theory of dynamical systems and its applications. Some of the articles are surveys while others are original contributions. The topics covered include: Riemannian geometry, models in mathematical physics and mathematical biology, symbolic dynamics, random and stochastic dynamics. This book can be used by graduate students and researchers in dynamical systems and its applications."--BOOK JACKET.

Author : Taeyoung Lee,Melvin Leok,N. Harris McClamroch
Publisher : Springer
Page : 539 pages
File Size : 45,5 Mb
Release : 2017-08-14
Category : Mathematics
ISBN : 9783319569536

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Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds by Taeyoung Lee,Melvin Leok,N. Harris McClamroch Pdf

This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities. The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems. This book is written for a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics.

Author : N. Pytheas Fogg
Publisher : Springer
Page : 404 pages
File Size : 41,5 Mb
Release : 2003-10-24
Category : Mathematics
ISBN : 9783540457145

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Substitutions in Dynamics, Arithmetics and Combinatorics by N. Pytheas Fogg Pdf

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems.