The Inverse Problem Of The Calculus Of Variations For Ordinary Differential Equations

Author : Ian Anderson,Gerard Thompson
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 53,8 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821825334

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The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations by Ian Anderson,Gerard Thompson Pdf

This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centers on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coincides with a given system of ordinary differential equations. The authors rederive the basic necessary and sufficient conditions of Douglas for second order equations and extend them to equations of higher order using methods of the variational bicomplex of Tulcyjew, Vinogradov, and Tsujishita. What emerges is a fundamental dichotomy between second and higher order systems: the most general Lagrangian for any higher order system can depend only upon finitely many constants. The authors present an algorithm, based upon exterior differential systems techniques, for solving the inverse problem for second order equations. A number of new examples illustrate the effectiveness of this approach. The monograph also contains a study of the inverse problem for a pair of geodesic equations arising from a two dimensional symmetric affine connection. The various possible solutions to the inverse problem for these equations are distinguished by geometric properties of the Ricci tensor.

Author : Dmitry V. Zenkov
Publisher : Springer
Page : 296 pages
File Size : 43,6 Mb
Release : 2015-10-15
Category : Mathematics
ISBN : 9789462391093

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The Inverse Problem of the Calculus of Variations by Dmitry V. Zenkov Pdf

The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

Author : M. V. Makarets,V. Yu Reshetnyak
Publisher : World Scientific
Page : 385 pages
File Size : 52,9 Mb
Release : 1995
Category : Mathematics
ISBN : 9789810221911

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Ordinary Differential Equations and Calculus of Variations by M. V. Makarets,V. Yu Reshetnyak Pdf

This problem book contains exercises for courses in differential equations and calculus of variations at universities and technical institutes. It is designed for non-mathematics students and also for scientists and practicing engineers who feel a need to refresh their knowledge. The book contains more than 260 examples and about 1400 problems to be solved by the students ? much of which have been composed by the authors themselves. Numerous references are given at the end of the book to furnish sources for detailed theoretical approaches, and expanded treatment of applications.

Author : David Colton,Heinz W. Engl,Alfred K. Louis,Joyce McLaughlin,William Rundell
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 48,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783709162965

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Surveys on Solution Methods for Inverse Problems by David Colton,Heinz W. Engl,Alfred K. Louis,Joyce McLaughlin,William Rundell Pdf

Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term "regularization methods". This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.

Author : David L. Colton,Richard E. Ewing,William Rundell,Society for Industrial and Applied Mathematics
Publisher : SIAM
Page : 234 pages
File Size : 51,7 Mb
Release : 1990-01-01
Category : Mathematics
ISBN : 0898712521

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Inverse Problems in Partial Differential Equations by David L. Colton,Richard E. Ewing,William Rundell,Society for Industrial and Applied Mathematics Pdf

Author : Angelo Favini,Genni Fragnelli,Rosa Maria Mininni
Publisher : Springer
Page : 472 pages
File Size : 45,8 Mb
Release : 2014-11-27
Category : Mathematics
ISBN : 9783319114064

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New Prospects in Direct, Inverse and Control Problems for Evolution Equations by Angelo Favini,Genni Fragnelli,Rosa Maria Mininni Pdf

This book, based on a selection of talks given at a dedicated meeting in Cortona, Italy, in June 2013, shows the high degree of interaction between a number of fields related to applied sciences. Applied sciences consider situations in which the evolution of a given system over time is observed, and the related models can be formulated in terms of evolution equations (EEs). These equations have been studied intensively in theoretical research and are the source of an enormous number of applications. In this volume, particular attention is given to direct, inverse and control problems for EEs. The book provides an updated overview of the field, revealing its richness and vitality.

Author : Peter Deuflhard,Ernst Hairer
Publisher : Unknown
Page : 384 pages
File Size : 52,8 Mb
Release : 1983
Category : Mathematics
ISBN : UVA:X001413464

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Numerical Treatment of Inverse Problems in Differential and Integral Equations by Peter Deuflhard,Ernst Hairer Pdf

Author : Joseph Grifone,Zoltán Muzsnay
Publisher : World Scientific
Page : 228 pages
File Size : 54,7 Mb
Release : 2000-05-25
Category : Mathematics
ISBN : 9789814495363

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Variational Principles for Second-Order Differential Equations by Joseph Grifone,Zoltán Muzsnay Pdf

The inverse problem of the calculus of variations was first studied by Helmholtz in 1887 and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations. This book looks at second-order differential equations and asks if they can be written as Euler–Lagrangian equations. If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi–Civita for some Riemann metric. To solve the inverse problem, the authors use the formal integrability theory of overdetermined partial differential systems in the Spencer–Quillen–Goldschmidt version. The main theorems of the book furnish a complete illustration of these techniques because all possible situations appear: involutivity, 2-acyclicity, prolongation, computation of Spencer cohomology, computation of the torsion, etc. Contents:An Introduction to Formal Integrability Theory of Partial Differential SystemsFrölicher–Nijenhuis Theory of DerivationsDifferential Algebraic Formalism of ConnectionsNecessary Conditions for Variational SpraysObstructions to the Integrability of the Euler–Lagrange SystemThe Classification of Locally Variational Sprays on Two-Dimensional ManifoldsEuler–Lagrange Systems in the Isotropic Case Readership: Mathematicians. Keywords:Calculus of Variations;Inverse Problem;Euler-Lagrange Equation;Sprays;Formal Integrability;Involution;Janet-Riquier Theory;Spencer TheoryReviews: “Everybody seriously interested in the modern theory of the inverse problem of the calculus of variations should take a look at this book.” Zentralblatt MATH

Author : J. Grifone,Zolt n Muzsnay
Publisher : World Scientific
Page : 236 pages
File Size : 53,7 Mb
Release : 2000
Category : Mathematics
ISBN : 9810237340

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Variational Principles for Second-order Differential Equations by J. Grifone,Zolt n Muzsnay Pdf

The inverse problem of the calculus of variations was first studied by Helmholtz in 1887 and it is entirely solved for the differential operators, but only a few results are known in the more general case of differential equations. This book looks at second-order differential equations and asks if they can be written as Euler-Lagrangian equations. If the equations are quadratic, the problem reduces to the characterization of the connections which are Levi-Civita for some Riemann metric.To solve the inverse problem, the authors use the formal integrability theory of overdetermined partial differential systems in the Spencer-Quillen-Goldschmidt version. The main theorems of the book furnish a complete illustration of these techniques because all possible situations appear: involutivity, 2-acyclicity, prolongation, computation of Spencer cohomology, computation of the torsion, etc.

Author : Demeter Krupka,David Saunders
Publisher : Elsevier
Page : 1243 pages
File Size : 51,7 Mb
Release : 2011-08-11
Category : Mathematics
ISBN : 9780080556734

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Handbook of Global Analysis by Demeter Krupka,David Saunders Pdf

This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis , dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics.This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry. - Comprehensive coverage of modern global analysis and geometrical mathematical physics- Written by world-experts in the field- Up-to-date contents

Author : Olga Krupkova
Publisher : Springer
Page : 261 pages
File Size : 40,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540696575

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The Geometry of Ordinary Variational Equations by Olga Krupkova Pdf

The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea - to build up these theories as related with the class of equivalent Lagrangians - distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.

Author : P.L. Antonelli
Publisher : Springer Science & Business Media
Page : 305 pages
File Size : 48,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401142359

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Finslerian Geometries by P.L. Antonelli Pdf

The International Conference on Finsler and Lagrange Geometry and its Applications: A Meeting of Minds, took place August 13-20, 1998 at the University of Alberta in Edmonton, Canada. The main objective of this meeting was to help acquaint North American geometers with the extensive modern literature on Finsler geometry and Lagrange geometry of the Japanese and European schools, each with its own venerable history, on the one hand, and to communicate recent advances in stochastic theory and Hodge theory for Finsler manifolds by the younger North American school, on the other. The intent was to bring together practitioners of these schools of thought in a Canadian venue where there would be ample opportunity to exchange information and have cordial personal interactions. The present set of refereed papers begins ·with the Pedagogical Sec tion I, where introductory and brief survey articles are presented, one from the Japanese School and two from the European School (Romania and Hungary). These have been prepared for non-experts with the intent of explaining basic points of view. The Section III is the main body of work. It is arranged in alphabetical order, by author. Section II gives a brief account of each of these contribu tions with a short reference list at the end. More extensive references are given in the individual articles.

Author : George M. Rassias,Themistocles M. Rassias
Publisher : CRC Press
Page : 544 pages
File Size : 53,8 Mb
Release : 2023-05-31
Category : Mathematics
ISBN : 9781000943948

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Differential Geometry, Calculus of Variations, and Their Applications by George M. Rassias,Themistocles M. Rassias Pdf

This book contains a series of papers on some of the longstanding research problems of geometry, calculus of variations, and their applications. It is suitable for advanced graduate students, teachers, research mathematicians, and other professionals in mathematics.

Author : G. Marmo,David Martín de Diego,Miguel Muñoz Lecanda
Publisher : Springer Nature
Page : 388 pages
File Size : 46,6 Mb
Release : 2019-10-26
Category : Science
ISBN : 9783030247485

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Classical and Quantum Physics by G. Marmo,David Martín de Diego,Miguel Muñoz Lecanda Pdf

This proceedings is based on the interdisciplinary workshop held in Madrid, 5-9 March 2018, dedicated to Alberto Ibort on his 60th birthday. Alberto has great and significantly contributed to many fields of mathematics and physics, always with highly original and innovative ideas.Most of Albertos’s scientific activity has been motivated by geometric ideas, concepts and tools that are deeply related to the framework of classical dynamics and quantum mechanics.Let us mention some of the fields of expertise of Alberto Ibort:Geometric Mechanics; Constrained Systems; Variational Principles; Multisymplectic structures for field theories; Super manifolds; Inverse problem for Bosonic and Fermionic systems; Quantum Groups, Integrable systems, BRST Symmetries; Implicit differential equations; Yang-Mills Theories; BiHamiltonian Systems; Topology Change and Quantum Boundary Conditions; Classical and Quantum Control; Orthogonal Polynomials; Quantum Field Theory and Noncommutative Spaces; Classical and Quantum Tomography; Quantum Mechanics on phase space; Wigner-Weyl formalism; Lie-Jordan Algebras, Classical and Quantum; Quantum-to-Classical transition; Contraction of Associative Algebras; contact geometry, among many others.In each contribution, one may find not only technical novelties but also completely new way of looking at the considered problems. Even an experienced reader, reading Alberto's contributions on his field of expertise, will find new perspectives on the considered topic.His enthusiasm is happily contagious, for this reason he has had, and still has, very bright students wishing to elaborate their PhD thesis under his guidance.What is more impressive, is the broad list of rather different topics on which he has contributed.

Author : G. Anger
Publisher : Walter de Gruyter GmbH & Co KG
Page : 256 pages
File Size : 54,5 Mb
Release : 1990-12-31
Category : Mathematics
ISBN : 9783112707173

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Inverse Problems in Differential Equations by G. Anger Pdf

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