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Critical Point Theory For Lagrangian Systems

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Critical Point Theory For Lagrangian Systems

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Author : Marco Mazzucchelli
language : en
Publisher: Birkhäuser
Release Date : 2011-11-23

Critical Point Theory For Lagrangian Systems written by Marco Mazzucchelli and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-11-23 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.

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.

Duality Principles In Nonconvex Systems

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Author : David Yang Gao
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-09

Duality Principles In Nonconvex Systems written by David Yang Gao 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 2013-03-09 with Mathematics categories.

Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics. Topics covered include the classical (minimax) mono-duality of convex static equilibria, the beautiful bi-duality in dynamical systems, the interesting tri-duality in non-convex problems and the complicated multi-duality in general canonical systems. A potentially powerful sequential canonical dual transformation method for solving fully nonlinear problems is developed heuristically and illustrated by use of many interesting examples as well as extensive applications in a wide variety of nonlinear systems, including differential equations, variational problems and inequalities, constrained global optimization, multi-well phase transitions, non-smooth post-bifurcation, large deformation mechanics, structural limit analysis, differential geometry and non-convex dynamical systems. With exceptionally coherent and lucid exposition, the work fills a big gap between the mathematical and engineering sciences. It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science. Much of the book contains material that is new, both in its manner of presentation and in its research development. A self-contained appendix provides some necessary background from elementary functional analysis. Audience: The book will be a valuable resource for students and researchers in applied mathematics, physics, mechanics and engineering. The whole volume or selected chapters can alsobe recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.

Symplectic Twist Maps

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Author : Christophe Golé
language : en
Publisher: World Scientific
Release Date : 2001

Symplectic Twist Maps written by Christophe Golé and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.

0. Introduction. 1. Fall from paradise. 2. Billiards and broken geodesies. 3. An ancestor of symplectic topology -- 1. Twist maps of the annulus. 4. Monotone twist maps of the annulus. 5. Generating functions and variational setting. 6. Examples. 7. The Poincare-Birkhoff theorem -- 2. The Aubry-Mather theorem. 8. Introduction. 9. Cyclically ordered sequences and orbits. 10. Minimizing orbits. 11. CO orbits of all rotation numbers. 12. Aubry-Mather sets -- 3. Ghost circles. 14. Gradient flow of the action. 15. The gradient flow and the Aubry-Mather theorem. 16. Ghost circles. 17. Construction of ghost circles. 18. Construction of disjoint ghost circles. 19. Proof of lemma 18.5. 20. Proof of theorem 18.1. 21. Remarks and applications. 22. Proofs of monotonicity and of the Sturmian lemma -- 4. Symplectic twist maps. 23. Symplectic twist maps of T[symbol] x IR[symbol]. 24. Examples. 25. More on generating functions. 2.6. Symplectic twist maps on general cotangent bundles of compact manifolds -- 5. Periodic orbits for symplectic twist maps of T[symbol] x IR[symbol]. 27. Presentation of the results. 28. Finite dimensional variational setting. 29. Second variation and nondegenerate periodic orbits. 30. The coercive case. 31. Asymptotically linear systems. 32. Ghost tori. 33. Hyperbolicity Vs. action minimizers -- 6. Invariant manifolds. 34. The theory of Kolmogorov-Arnold-Moser. 35. Properties of invariant tori. 36. (Un)stable manifolds and heteroclinic orbits. 37. Instability, transport and diffusion -- 7. Hamiltonian systems vs. twist maps. 38. Case study: The geodesic flow. 39. Decomposition of Hamiltonian maps into twist maps. 40. Return maps in Hamiltonian systems. 41. Suspension of symplectic twist maps by Hamiltonian flows -- 8. Periodic orbits for Hamiltonian systems. 42. Periodic orbits in the cotangent of the n-torus. 43. Periodic orbits in general cotangent spaces. 44. Linking of spheres -- 9. Generalizations of the Aubry-Mather theorem. 45. Theory for functions on lattices and PDE's. 46. Monotone recurrence relationst. 47. Anti-integrable limit. 48. Mather's theory of minimal measures. 49. The case of hyperbolic manifolds. 50. Concluding remarks -- 10. Generating phases and symplectic topology. 51. Chaperon's method and the theorem Of Conley-Zehnder. 52. Generating phases and symplectic geometry.

Billiards A Genetic Introduction To The Dynamics Of Systems With Impacts

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Author : Valeriĭ Viktorovich Kozlov
language : en
Publisher: American Mathematical Soc.
Release Date : 1991-08-05

Billiards A Genetic Introduction To The Dynamics Of Systems With Impacts written by Valeriĭ Viktorovich Kozlov 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 1991-08-05 with Mathematics categories.

Starting with the work of G D Birkhoff, billiards have been a popular research topic drawing on such areas as ergodic theory, Morse theory, and KAM theory. Billiard systems are also remarkable in that they arise naturally in a number of important problems of mechanics and physics. This book is devoted to mathematical aspects of the theory of dynamical systems of billiard type. Focusing on the genetic approach, the authors strive to clarify the genesis of the basic ideas and concepts of the theory of dynamical systems with impact intereactions and also to demonstrate that these methods are natural and effective. Recent limit theorems, which justify various mathematical models of impact theory, are key features. Questions of existence and stability of periodic trajectories of elastic billiards occupy a special place in the book, and considerable attention is devoted to integrable billiards. A brief survey is given of work on billiards with ergodic behaviour. Each chapter ends with a list of problems.

Morse Theory For Hamiltonian Systems

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Author : Alberto Abbondandolo
language : en
Publisher: CRC Press
Release Date : 2001-03-15

Morse Theory For Hamiltonian Systems written by Alberto Abbondandolo and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-03-15 with Mathematics categories.

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

Reformulation Nonsmooth Piecewise Smooth Semismooth And Smoothing Methods

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Author : Masao Fukushima
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Reformulation Nonsmooth Piecewise Smooth Semismooth And Smoothing Methods written by Masao Fukushima 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 2013-04-17 with Mathematics categories.

The concept of "reformulation" has long been playing an important role in mathematical programming. A classical example is the penalization technique in constrained optimization that transforms the constraints into the objective function via a penalty function thereby reformulating a constrained problem as an equivalent or approximately equivalent unconstrained problem. More recent trends consist of the reformulation of various mathematical programming prob lems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable. Because of the recent advent of various tools in nonsmooth analysis, the reformulation approach has become increasingly profound and diversified. In view of growing interests in this active field, we planned to organize a cluster of sessions entitled "Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods" in the 16th International Symposium on Mathematical Programming (ismp97) held at Lausanne EPFL, Switzerland on August 24-29, 1997. Responding to our invitation, thirty-eight people agreed to give a talk within the cluster, which enabled us to organize thirteen sessions in total. We think that it was one of the largest and most exciting clusters in the symposium. Thanks to the earnest support by the speakers and the chairpersons, the sessions attracted much attention of the participants and were filled with great enthusiasm of the audience.

Geometrical Methods In Variational Problems

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Author : N.A. Bobylov
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Geometrical Methods In Variational Problems written by N.A. Bobylov 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.

This self-contained monograph presents methods for the investigation of nonlinear variational problems. These methods are based on geometric and topological ideas such as topological index, degree of a mapping, Morse-Conley index, Euler characteristics, deformation invariant, homotopic invariant, and the Lusternik-Shnirelman category. Attention is also given to applications in optimisation, mathematical physics, control, and numerical methods. Audience: This volume will be of interest to specialists in functional analysis and its applications, and can also be recommended as a text for graduate and postgraduate-level courses in these fields.

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.

Basic Theory Of Fractional Differential Equations Third Edition

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Author : Yong Zhou
language : en
Publisher: World Scientific
Release Date : 2023-10-06

Basic Theory Of Fractional Differential Equations Third Edition written by Yong Zhou and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-10-06 with Mathematics categories.

This accessible monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary differential equations and evolution equations. It is self-contained and unified in presentation, and provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, Picard operators technique, critical point theory and semigroups theory. This book is based on the research work done so far by the author and other experts, and contains comprehensive up-to-date materials on the topic.In this third edition, four new topics have been added: Hilfer fractional evolution equations and infinite interval problems, oscillations and nonoscillations, fractional Hamiltonian systems, fractional Rayleigh-Stokes equations, and wave equations. The bibliography has also been updated and expanded.This book is useful to researchers, graduate or PhD students dealing with fractional calculus and applied analysis, differential equations, and related areas of research.

Critical Point Theory For Lagrangian Systems

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