Convexity Methods In Hamiltonian Mechanics
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Convexity Methods In Hamiltonian Mechanics
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Author : Ivar Ekeland
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Convexity Methods In Hamiltonian Mechanics written by Ivar Ekeland 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.
In the case of completely integrable systems, periodic solutions are found by inspection. For nonintegrable systems, such as the three-body problem in celestial mechanics, they are found by perturbation theory: there is a small parameter € in the problem, the mass of the perturbing body for instance, and for € = 0 the system becomes completely integrable. One then tries to show that its periodic solutions will subsist for € -# 0 small enough. Poincare also introduced global methods, relying on the topological properties of the flow, and the fact that it preserves the 2-form L~=l dPi 1\ dqi' The most celebrated result he obtained in this direction is his last geometric theorem, which states that an area-preserving map of the annulus which rotates the inner circle and the outer circle in opposite directions must have two fixed points. And now another ancient theme appear: the least action principle. It states that the periodic solutions of a Hamiltonian system are extremals of a suitable integral over closed curves. In other words, the problem is variational. This fact was known to Fermat, and Maupertuis put it in the Hamiltonian formalism. In spite of its great aesthetic appeal, the least action principle has had little impact in Hamiltonian mechanics. There is, of course, one exception, Emmy Noether's theorem, which relates integrals ofthe motion to symmetries of the equations. But until recently, no periodic solution had ever been found by variational methods.
Symplectic Invariants And Hamiltonian Dynamics
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Author : Helmut Hofer
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Symplectic Invariants And Hamiltonian Dynamics written by Helmut Hofer and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
The discoveries of the past decade have opened new perspectives for the old field of Hamiltonian systems and led to the creation of a new field: symplectic topology. Surprising rigidity phenomena demonstrate that the nature of symplectic map pings is very different from that of volume preserving mappings which raised new questions, many of them still unanswered. On the other hand, due to the analysis of an old variational principle in classical mechanics, global periodic phenomena in Hamiltonian systems have been established. As it turns out, these seemingly differ ent phenomena are mysteriously related. One of the links is a class of symplectic invariants, called symplectic capacities. These invariants are the main theme of this book which grew out of lectures given by the authors at Rutgers University, the RUB Bochum and at the ETH Zurich (1991) and also at the Borel Seminar in Bern 1992. Since the lectures did not require any previous knowledge, only a few and rather elementary topics were selected and proved in detail. Moreover, our se lection has been prompted by a single principle: the action principle of mechanics. The action functional for loops in the phase space, given by 1 Fh) = J pdq -J H(t, 'Y(t)) dt , 'Y 0 differs from the old Hamiltonian principle in the configuration space defined by a Lagrangian. The critical points of F are those loops 'Y which solve the Hamiltonian equations associated with the Hamiltonian H and hence are the periodic orbits.
Critical Point Theory For Lagrangian Systems
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Author : Marco Mazzucchelli
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-11-16
Critical Point Theory For Lagrangian Systems written by Marco Mazzucchelli 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 2011-11-16 with Science categories.
Lagrangian systems constitute a very important and old class in dynamics. Their origin dates back to the end of the eighteenth century, with Joseph-Louis Lagrange’s reformulation of classical mechanics. The main feature of Lagrangian dynamics is its variational flavor: orbits are extremal points of an action functional. The development of critical point theory in the twentieth century provided a powerful machinery to investigate existence and multiplicity questions for orbits of Lagrangian systems. This monograph gives a modern account of the application of critical point theory, and more specifically Morse theory, to Lagrangian dynamics, with particular emphasis toward existence and multiplicity of periodic orbits of non-autonomous and time-periodic systems.
Dynamics Of Nonlinear Waves In Dissipative Systems Reduction Bifurcation And Stability
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Author : G Dangelmayr
language : en
Publisher: CRC Press
Release Date : 1996-08-01
Dynamics Of Nonlinear Waves In Dissipative Systems Reduction Bifurcation And Stability written by G Dangelmayr and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996-08-01 with Mathematics categories.
The mathematical description of complex spatiotemporal behaviour observed in dissipative continuous systems is a major challenge for modern research in applied mathematics. While the behaviour of low-dimensional systems, governed by the dynamics of a finite number of modes is well understood, systems with large or unbounded spatial domains show intrinsic infinite-dimensional behaviour --not a priori accessible to the methods of finite dimensionaldynamical systems. The purpose of the four contributions in this book is to present some recent and active lines of research in evolution equations posed in large or unbounded domains. One of the most prominent features of these systems is the propagation of various types of patterns in the form of waves, such as travelling and standing waves and pulses and fronts. Different approaches to studying these kinds of phenomena are discussed in the book. A major theme is the reduction of an original evolution equation in the form of a partial differential equation system to a simpler system of equations, either a system of ordinary differential equation or a canonical system of PDEs. The study of the reduced equations provides insight into the bifurcations from simple to more complicated solutions and their stabilities. .
Symplectic Geometry And Topology
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Author : Yakov Eliashberg
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Symplectic Geometry And Topology written by Yakov Eliashberg 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 2004 with Mathematics categories.
Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introductionto Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristicsand Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. Information for our distributors: Titles in this series are copublished with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
Canonical Duality Theory
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Author : David Yang Gao
language : en
Publisher: Springer
Release Date : 2017-10-09
Canonical Duality Theory written by David Yang Gao and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-09 with Mathematics categories.
This book on canonical duality theory provides a comprehensive review of its philosophical origin, physics foundation, and mathematical statements in both finite- and infinite-dimensional spaces. A ground-breaking methodological theory, canonical duality theory can be used for modeling complex systems within a unified framework and for solving a large class of challenging problems in multidisciplinary fields in engineering, mathematics, and the sciences. This volume places a particular emphasis on canonical duality theory’s role in bridging the gap between non-convex analysis/mechanics and global optimization. With 18 total chapters written by experts in their fields, this volume provides a nonconventional theory for unified understanding of the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization. Additionally, readers will find a unified methodology and powerful algorithms for solving challenging problems in complex systems with real-world applications in non-convex analysis, non-monotone variational inequalities, integer programming, topology optimization, post-buckling of large deformed structures, etc. Researchers and graduate students will find explanation and potential applications in multidisciplinary fields.
Index Theory For Symplectic Paths With Applications
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Author : Yiming Long
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Index Theory For Symplectic Paths With Applications written by Yiming Long and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This book is based upon my monograph Index Theory for Hamiltonian Systems with Applications published in 1993 in Chinese, and my notes for lectures and courses given at Nankai University, Brigham Young University, ICTP-Trieste, and the Institute of Mathematics of Academia Sinica during the last ten years. The aim of this book is twofold: (1) to give an introduction to the index theory for symplectic matrix paths and its iteration theory, which form a basis for the Morse theoretical study on Hamilto nian systems, and to give applications of this theory to periodic boundary value problems of nonlinear Hamiltonian systems. Here the iteration theory means the index theory of iterations of periodic solutions and symplectic matrix paths. (2) to serve as a reference book on these topics. There are many different ways to introduce the index theory for symplectic paths in order to establish Morse type index theory of Hamiltonian systems. In this book, I have chosen a relatively elementary way, i.e., the homotopy classification method of symplectic matrix paths. It depends only on linear algebra, point set topology, and certain basic parts of linear functional analysis. I have tried to make this part of the book self-contained and at the same time include all of the major results on these topics so that researchers and students interested in them can read it without substantial difficulties, and can learn the main results in this area for their possible applications.
Topics In Nonlinear Functional Analysis
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Author : L. Nirenberg
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Topics In Nonlinear Functional Analysis written by L. Nirenberg 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.
Since its first appearance as a set of lecture notes published by the Courant Institute in 1974, this book served as an introduction to various subjects in nonlinear functional analysis. The current edition is a reprint of these notes, with added bibliographic references. Topological and analytic methods are developed for treating nonlinear ordinary and partial differential equations. The first two chapters of the book introduce the notion of topological degree and develop its basic properties. These properties are used in later chapters in the discussion of bifurcation theory (the possible branching of solutions as parameters vary), including the proof of Rabinowitz global bifurcation theorem. Stability of the branches is also studied. The book concludes with a presentation of some generalized implicit function theorems of Nash-Moser type with applications to Kolmogorov-Arnold-Moser theory and to conjugacy problems. For more than 20 years, this book continues to be an excellent graduate level textbook and a useful supplementary course text. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.
Stability And Bifurcation Theory For Non Autonomous Differential Equations
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Author : Anna Capietto
language : en
Publisher: Springer
Release Date : 2012-12-14
Stability And Bifurcation Theory For Non Autonomous Differential Equations written by Anna Capietto and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-14 with Mathematics categories.
This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session "Stability and Bifurcation Problems for Non-Autonomous Differential Equations," held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.
Advances In Automatic Control
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Author : Mihail Voicu
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Advances In Automatic Control written by Mihail Voicu 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 Technology & Engineering categories.
During the academic year 2002-2003, the Faculty of Automatic Control and Computer Engineering of Ia~i (Romania), and its Departments of Automatic Control and Industrial Informatics and of Computer Engineering respectively, celebrated 25 years from the establishment of the specialization named Automatic Control and Computer Engineering within the framework of the former Faculty of Electrical Engineering of Ia~i, and, at the same time, 40 years since the first courses on Automatic Control and Computers respectively, were introduced in the curricula of the former specializations of Electromechanical Engineering and Electrical Power Engineering at the already mentioned Faculty of Electrical Engineering. The reader interested to know some important moments ofour evolution during the last five decades is invited to see the Addendum ofthis volume, where a short history is presented. And, to highlight once more the nice coincidences, it must be noted here that in 2003 our Technical University "Gheorghe Asachi" of Ia~i celebrated 190 years from the emergence of the first cadastral engineering degree course in Ia~i (thanks to the endeavor ofGheorghe Asachi), which is today considered to be the beginningofthe engineering higher education in Romania. Generally speaking, an anniversary is a celebration meant to mark special events ofthe past, with festivities to be performed solemnly and publicly according to a specific ritual.