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A Variational Method For Finding Periodic Solutions Of Differential Equations
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Book Synopsis A Variational Method for Finding Periodic Solutions of Differential Equations by : Paul H. Rabinowitz
Download or read book A Variational Method for Finding Periodic Solutions of Differential Equations written by Paul H. Rabinowitz and published by . This book was released on 1978 with total page 31 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper concerns the use of minimax and approximation techniques from the calculus of variations to prove the existence of periodic solutions of Hamiltonian systems of ordinary differential equations. Most of the results are for equations where the period is prescribed and assumptions are made about the growth of the Hamiltonian near infinity. However it is also shown how such results can give information about solutions having prescribed energy. (Author).
Download or read book Variational Methods written by BERESTYCKI and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the framework of the "Annee non lineaire" (the special nonlinear year) sponsored by the C.N.R.S. (the French National Center for Scien tific Research), a meeting was held in Paris in June 1988. It took place in the Conference Hall of the Ministere de la Recherche and had as an organizing theme the topic of "Variational Problems." Nonlinear analysis has been one of the leading themes in mathemat ical research for the past decade. The use of direct variational methods has been particularly successful in understanding problems arising from physics and geometry. The growth of nonlinear analysis is largely due to the wealth of ap plications from various domains of sciences and industrial applica tions. Most of the papers gathered in this volume have their origin in applications: from mechanics, the study of Hamiltonian systems, from physics, from the recent mathematical theory of liquid crystals, from geometry, relativity, etc. Clearly, no single volume could pretend to cover the whole scope of nonlinear variational problems. We have chosen to concentrate on three main aspects of these problems, organizing them roughly around the following topics: 1. Variational methods in partial differential equations in mathemat ical physics 2. Variational problems in geometry 3. Hamiltonian systems and related topics.
Book Synopsis Variational Methods by : Michael Struwe
Download or read book Variational Methods written by Michael Struwe and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hilberts talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateaus problem by Douglas and Rad. This third edition gives a concise introduction to variational methods and presents an overview of areas of current research in the field, plus a survey on new developments.
Book Synopsis Progress in Variational Methods by : Chungen Liu
Download or read book Progress in Variational Methods written by Chungen Liu and published by World Scientific. This book was released on 2010-09-07 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last forty years, nonlinear analysis has been broadly and rapidly developed. Lectures presented in the International Conference on Variational Methods at the Chern Institute of Mathematics in Tianjin of May 2009 reflect this development from different angles. This volume contains articles based on lectures in the following areas of nonlinear analysis: critical point theory, Hamiltonian dynamics, partial differential equations and systems, KAM theory, bifurcation theory, symplectic geometry, geometrical analysis, and celestial mechanics. Combinations of topological, analytical (especially variational), geometrical, and algebraic methods in these researches play important roles. In this proceedings, introductory materials on new theories and surveys on traditional topics are also given. Further perspectives and open problems on hopeful research topics in related areas are described and proposed. Researchers, graduate and postgraduate students from a wide range of areas in mathematics and physics will find contents in this proceedings are helpful.
Book Synopsis A Variational Approach to Heteroclinic Orbits for a Class of Hamiltonian Systems by : Paul H. Rabinowitz
Download or read book A Variational Approach to Heteroclinic Orbits for a Class of Hamiltonian Systems written by Paul H. Rabinowitz and published by . This book was released on 1990 with total page 17 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large literature has developed in the last decade in which methods from the calculus of variations have been used to prove the periodic solutions of Hamiltonian systems of ordinary differential equations. The recent monograph of Mawhin and Willem provides a sizable bibliography of such works. Aside from equilibria, periodic solutions are the simplest global in time solutions of differential equations. It is only within the past one - two years that attempts have begun to extend the variational approach to such systems to find other kinds of global solutions of Hamiltonian systems. Thus far mainly homoclinic orbits have been treated. However heteroclinic orbits were studied in an earlier work by the author, entitled Periodic and Heteroclinic orbits for a Periodic Hamiltonian system, for the class of second order Hamiltonian systems. The author's goal in this paper is to extend one of the main results in his earlier work mentioned above. (KR).
Book Synopsis Critical Point Theory and Hamiltonian Systems by : Jean Mawhin
Download or read book Critical Point Theory and Hamiltonian Systems written by Jean Mawhin and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: FACHGEB The last decade has seen a tremendous development in critical point theory in infinite dimensional spaces and its application to nonlinear boundary value problems. In particular, striking results were obtained in the classical problem of periodic solutions of Hamiltonian systems. This book provides a systematic presentation of the most basic tools of critical point theory: minimization, convex functions and Fenchel transform, dual least action principle, Ekeland variational principle, minimax methods, Lusternik- Schirelmann theory for Z2 and S1 symmetries, Morse theory for possibly degenerate critical points and non-degenerate critical manifolds. Each technique is illustrated by applications to the discussion of the existence, multiplicity, and bifurcation of the periodic solutions of Hamiltonian systems. Among the treated questions are the periodic solutions with fixed period or fixed energy of autonomous systems, the existence of subharmonics in the non-autonomous case, the asymptotically linear Hamiltonian systems, free and forced superlinear problems. Application of those results to the equations of mechanical pendulum, to Josephson systems of solid state physics and to questions from celestial mechanics are given. The aim of the book is to introduce a reader familiar to more classical techniques of ordinary differential equations to the powerful approach of modern critical point theory. The style of the exposition has been adapted to this goal. The new topological tools are introduced in a progressive but detailed way and immediately applied to differential equation problems. The abstract tools can also be applied to partial differential equations and the reader will also find the basic references in this direction in the bibliography of more than 500 items which concludes the book. ERSCHEIN
Book Synopsis Nonlinear Evolution Equations by : Michael G. Crandall
Download or read book Nonlinear Evolution Equations written by Michael G. Crandall and published by Elsevier. This book was released on 2014-05-10 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Evolution Equation covers the proceedings of the Symposium by the same title, conducted by the Mathematics Research Center at the University of Wisconsin, Madison on October 17-19, 1977. This book is divided into 13 chapters and begins with reviews of the uniqueness of solution to systems of conservation laws and the computational aspects of Glimm's method. The next chapters examine the theoretical and practical aspects of Boltzmann, Navier-Stokes, and evolution equations. These topics are followed by discussions of the practical applications of Trotter's product formula for some nonlinear semigroups and the finite time blow-up in nonlinear problems. The closing chapters deal with a Hamiltonian approach to the K-dV and other equations, along with a variational method for finding periodic solutions of differential equations. This book will prove useful to mathematicians and engineers.
Book Synopsis Almost Periodic Differential Equations by : A.M. Fink
Download or read book Almost Periodic Differential Equations written by A.M. Fink and published by Springer. This book was released on 2006-11-15 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Progress In Variational Methods - Proceedings Of The International Conference On Variational Methods by : Chungen Liu
Download or read book Progress In Variational Methods - Proceedings Of The International Conference On Variational Methods written by Chungen Liu and published by World Scientific. This book was released on 2010-09-07 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last forty years, nonlinear analysis has been broadly and rapidly developed. Lectures presented in the International Conference on Variational Methods at the Chern Institute of Mathematics in Tianjin of May 2009 reflect this development from different angles. This volume contains articles based on lectures in the following areas of nonlinear analysis: critical point theory, Hamiltonian dynamics, partial differential equations and systems, KAM theory, bifurcation theory, symplectic geometry, geometrical analysis, and celestial mechanics. Combinations of topological, analytical (especially variational), geometrical, and algebraic methods in these researches play important roles. In this proceedings, introductory materials on new theories and surveys on traditional topics are also given. Further perspectives and open problems on hopeful research topics in related areas are described and proposed. Researchers, graduate and postgraduate students from a wide range of areas in mathematics and physics will find contents in this proceedings are helpful.
Book Synopsis Nonlinear Oscillations of Hamiltonian PDEs by : Massimiliano Berti
Download or read book Nonlinear Oscillations of Hamiltonian PDEs written by Massimiliano Berti and published by Springer Science & Business Media. This book was released on 2007-10-01 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many partial differential equations (PDEs) that arise in physics can be viewed as infinite-dimensional Hamiltonian systems. This monograph presents recent existence results of nonlinear oscillations of Hamiltonian PDEs, particularly of periodic solutions for completely resonant nonlinear wave equations. The text serves as an introduction to research in this fascinating and rapidly growing field. Graduate students and researchers interested in variational techniques and nonlinear analysis applied to Hamiltonian PDEs will find inspiration in the book.
Book Synopsis Periodic Solutions of Scalar Second Order Differential Equations with a Singularity by : Alessandro Fonda
Download or read book Periodic Solutions of Scalar Second Order Differential Equations with a Singularity written by Alessandro Fonda and published by . This book was released on 1993 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Multiplicity Results of Periodic Solutions for Two Classes of Nonlinear Problems by : Kazuya Hata
Download or read book Multiplicity Results of Periodic Solutions for Two Classes of Nonlinear Problems written by Kazuya Hata and published by . This book was released on 2014 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations: I) (Special) Relativistic Pendulum Equations (RPEs); II) (2-coupled) Gross-Pitaevskii Equations (GPEs). The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will ow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation is called RPE. Our result answers some open questions about the existence of multiple periodic solutions for RPEs. GPEs are sometimes called coupled nonlinear schrodinger equations. the Schrodinger equation is the fundamental equation of Quantum Mechanics which is the \exotic" probabilistic fundamental physics law of the \micro" world { the world of atoms and molecules. A well-known physicist and Nobel laureate, R. Feynman, said \I think I can safely say that nobody understands quantum mechanics." which indicates the physical/ philosophical difficulty of interpretations. It raises paradoxical problems such the well-known Schrodinger's Cat. Setting aside these difficult, if we combine Special Relativity and Quantum Mechanics as a many-body system, then we have Quantum Field Theory (QFT) which is more deterministic, and governs even elementary particle physics. GPEs are also related to QFT. For example, superconductivity and Bose Einstein Condensates (BEC). These phenomena in condensed matter physics can be thought of as the emergence of the mysterious micro world physics at \macro" level. We study these equations from the viewpoint of mathematical interest. It is generally difficult to solve nonlinear differential equations. It is also generally difficult even to prove the existence of solutions. Although we show there exist solutions, we still do not know how to solve the differential equations analytically. Variational Methods (or Calculus of Variations) are useful tools to show there exist solutions of differential equations. The idea is to convert the problem of solving equations into the problem of finding critical points (i.e. minimum/maximum points or saddle points) of a functional, and each critical point can generally correspond to a weak solution. However, it is also generally difficult to find out such critical points because we look for critical points in an infinite-dimensional functions space. Thus many advanced mathematical theories or tools have been developed and used for decades in nonlinear analysis. We use some topological theories. From information of the functional's shape, these theories deduce if there exists a critical point, or how many critical points exist. The key of these theories is to use the symmetry of the equations. We also investigate bifurcation structures for II), i.e. the connection structures between the solutions. By linearizations which look at the equations \locally," we reduce the problem in the infinite dimension to one in a finite dimension. Furthermore, it allows us to apply Morse Theory, which connects between local and global aspects of the functional's information. In several cases, we show that there are infinitely many bifurcation points that give rise to global bifurcation branches.
Book Synopsis Ergodic Theory and Dynamical Systems II by : Katok
Download or read book Ergodic Theory and Dynamical Systems II written by Katok and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Metrical Almost Periodicity and Applications to Integro-Differential Equations by : Marko Kostić
Download or read book Metrical Almost Periodicity and Applications to Integro-Differential Equations written by Marko Kostić and published by Walter de Gruyter GmbH & Co KG. This book was released on 2023-06-06 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Handbook of Differential Equations: Ordinary Differential Equations by : A. Canada
Download or read book Handbook of Differential Equations: Ordinary Differential Equations written by A. Canada and published by Elsevier. This book was released on 2005-09-02 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: This handbook is the second volume in a series devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, writtenby leading researchers in the area. All contributors have made an additional effort to achieve readability for mathematicians and scientists from other related fields, in order to make the chapters of the volume accessible to a wide audience. . Six chapters covering a variety of problems in ordinary differential equations. . Both, pure mathematical research and real word applications are reflected. Written by leading researchers in the area.
Book Synopsis Methods of Bifurcation Theory by : S.-N. Chow
Download or read book Methods of Bifurcation Theory written by S.-N. Chow and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider we have presented in detail much of the relevant background audience, material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the mate rial, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable.
Book Synopsis Minimax Methods in Critical Point Theory with Applications to Differential Equations by : Paul H. Rabinowitz
Download or read book Minimax Methods in Critical Point Theory with Applications to Differential Equations written by Paul H. Rabinowitz and published by American Mathematical Soc.. This book was released on 1986-07-01 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book provides an introduction to minimax methods in critical point theory and shows their use in existence questions for nonlinear differential equations. An expanded version of the author's 1984 CBMS lectures, this volume is the first monograph devoted solely to these topics. Among the abstract questions considered are the following: the mountain pass and saddle point theorems, multiple critical points for functionals invariant under a group of symmetries, perturbations from symmetry, and variational methods in bifurcation theory. The book requires some background in functional analysis and differential equations, especially elliptic partial differential equations. It is addressed to mathematicians interested in differential equations and/or nonlinear functional analysis, particularly critical point theory.
