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Exterior Differential Systems And The Calculus Of Variations


Exterior Differential Systems And The Calculus Of Variations
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Exterior Differential Systems And The Calculus Of Variations


Exterior Differential Systems And The Calculus Of Variations
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Author : P.A. Griffiths
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Exterior Differential Systems And The Calculus Of Variations written by P.A. Griffiths 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 2013-06-29 with Mathematics categories.


15 0. PRELIMINARIES a) Notations from Manifold Theory b) The Language of Jet Manifolds c) Frame Manifolds d) Differentia! Ideals e) Exterior Differential Systems EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL SYSTEMS ~liTH ONE I. 32 INDEPENDENT VARIABLE a) Setting up the Problem; Classical Examples b) Variational Equations for Integral Manifolds of Differential Systems c) Differential Systems in Good Form; the Derived Flag, Cauchy Characteristics, and Prolongation of Exterior Differential Systems d) Derivation of the Euler-Lagrange Equations; Examples e) The Euler-Lagrange Differential System; Non-Degenerate Variational Problems; Examples FIRST INTEGRALS OF THE EULER-LAGRANGE SYSTEM; NOETHER'S II. 1D7 THEOREM AND EXAMPLES a) First Integrals and Noether's Theorem; Some Classical Examples; Variational Problems Algebraically Integrable by Quadratures b) Investigation of the Euler-Lagrange System for Some Differential-Geometric Variational Pro~lems: 2 i) ( K ds for Plane Curves; i i) Affine Arclength; 2 iii) f K ds for Space Curves; and iv) Delauney Problem. II I. EULER EQUATIONS FOR VARIATIONAL PROBLEfiJS IN HOMOGENEOUS SPACES 161 a) Derivation of the Equations: i) Motivation; i i) Review of the Classical Case; iii) the Genera 1 Euler Equations 2 K /2 ds b) Examples: i) the Euler Equations Associated to f for lEn; but for Curves in i i) Some Problems as in i) sn; Non- Curves in iii) Euler Equations Associated to degenerate Ruled Surfaces IV.



Exterior Differential Systems


Exterior Differential Systems
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Author : Robert L. Bryant
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29

Exterior Differential Systems written by Robert L. Bryant 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 2013-06-29 with Mathematics categories.


This book gives a treatment of exterior differential systems. It will in clude both the general theory and various applications. An exterior differential system is a system of equations on a manifold defined by equating to zero a number of exterior differential forms. When all the forms are linear, it is called a pfaffian system. Our object is to study its integral manifolds, i. e. , submanifolds satisfying all the equations of the system. A fundamental fact is that every equation implies the one obtained by exterior differentiation, so that the complete set of equations associated to an exterior differential system constitutes a differential ideal in the algebra of all smooth forms. Thus the theory is coordinate-free and computations typically have an algebraic character; however, even when coordinates are used in intermediate steps, the use of exterior algebra helps to efficiently guide the computations, and as a consequence the treatment adapts well to geometrical and physical problems. A system of partial differential equations, with any number of inde pendent and dependent variables and involving partial derivatives of any order, can be written as an exterior differential system. In this case we are interested in integral manifolds on which certain coordinates remain independent. The corresponding notion in exterior differential systems is the independence condition: certain pfaffian forms remain linearly indepen dent. Partial differential equations and exterior differential systems with an independence condition are essentially the same object.



Exterior Differential Systems


Exterior Differential Systems
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Author : Robert L. Bryant
language : en
Publisher:
Release Date : 1991-01

Exterior Differential Systems written by Robert L. Bryant and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-01 with Differential equations, Partial categories.




The Inverse Problem In The Calculus Of Variations Via Exterior Differential Systems


The Inverse Problem In The Calculus Of Variations Via Exterior Differential Systems
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Author : Thi Kim Thoan Do
language : en
Publisher:
Release Date : 2016

The Inverse Problem In The Calculus Of Variations Via Exterior Differential Systems written by Thi Kim Thoan Do and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Calculus of variations categories.




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


The Inverse Problem Of The Calculus Of Variations For Ordinary Differential Equations
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Author : Ian Anderson
language : en
Publisher: American Mathematical Soc.
Release Date : 1992

The Inverse Problem Of The Calculus Of Variations For Ordinary Differential Equations written by Ian Anderson 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 1992 with Mathematics categories.


This monograph explores various aspects of the inverse problem of the calculus of variations for systems of ordinary differential equations. The main problem centres on determining the existence and degree of generality of Lagrangians whose system of Euler-Lagrange equations coicides 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. 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.



Exterior Differential Systems And Equivalence Problems


Exterior Differential Systems And Equivalence Problems
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Author : Kichoon Yang
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Exterior Differential Systems And Equivalence Problems written by Kichoon Yang 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 2013-03-14 with Mathematics categories.


This monograph presents a concise yet elementary account of exterior differential system theory so that it can be quickly applied to problems. The first part of the monograph, Chapters 1-5, deals with the general theory: the Cartan-Kaehler theorem is proved, the notions of involution and prolongation are carefully laid out, quasi-linear differential systems are examined in detail, and explicit examples of the Spencer cohomology groups and the characteristic variety are given. The second part of the monograph, Chapters 6 and 7, deals with applications to problems in differential geometry: the isometric embedding theorem of Cartan-Janet and its various geometric ramifications are discussed, a proof of the Andreotti-Hill theorem on the O-R embedding problem is given, and embeddings of abstract projective structures are discussed. For researchers and graduate students who would like a good introduction to exterior differential systems. This volume will also be particularly useful to those whose work involves differential geometry and partial differential equations.



Applied Exterior Calculus


Applied Exterior Calculus
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Author : Dominic G. B. Edelen
language : en
Publisher: Courier Corporation
Release Date : 2005-01-01

Applied Exterior Calculus written by Dominic G. B. Edelen and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-01-01 with Mathematics categories.


This text begins with the essentials, advancing to applications and studies of physical disciplines, including classical and irreversible thermodynamics, electrodynamics, and the theory of gauge fields. Geared toward advanced undergraduates and graduate students, it develops most of the theory and requires only a familiarity with upper-division algebra and mathematical analysis. "Essential." — SciTech Book News. 1985 edition.



Differential Systems And Isometric Embeddings


Differential Systems And Isometric Embeddings
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Author : Phillip A. Griffiths
language : en
Publisher: Princeton University Press
Release Date : 1987-05-21

Differential Systems And Isometric Embeddings written by Phillip A. Griffiths and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1987-05-21 with Mathematics categories.


The theory of exterior differential systems provides a framework for systematically addressing the typically non-linear, and frequently overdetermined, partial differential equations that arise in differential geometry. Adaptation of the techniques of microlocalization to differential systems have led to recent activity on the foundations of the theory; in particular, the fundamental role of the characteristic variety in geometric problems is now clearly established. In this book the general theory is explained in a relatively quick and concrete manner, and then this general theory is applied to the recent developments in the classical problem of isometric embeddings of Riemannian manifolds.



The Geometry Of Ordinary Variational Equations


The Geometry Of Ordinary Variational Equations
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Author : Olga Krupkova
language : en
Publisher: Springer
Release Date : 2006-11-14

The Geometry Of Ordinary Variational Equations written by Olga Krupkova and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-14 with Mathematics categories.


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.



Exterior Differential Systems And Euler Lagrange Partial Differential Equations


Exterior Differential Systems And Euler Lagrange Partial Differential Equations
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Author : Robert Bryant
language : en
Publisher: University of Chicago Press
Release Date : 2003-07

Exterior Differential Systems And Euler Lagrange Partial Differential Equations written by Robert Bryant and has been published by University of Chicago Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07 with Mathematics categories.


In Exterior Differential Systems, the authors present the results of their ongoing development of a theory of the geometry of differential equations, focusing especially on Lagrangians and Poincaré-Cartan forms. They also cover certain aspects of the theory of exterior differential systems, which provides the language and techniques for the entire study. Because it plays a central role in uncovering geometric properties of differential equations, the method of equivalence is particularly emphasized. In addition, the authors discuss conformally invariant systems at length, including results on the classification and application of symmetries and conservation laws. The book also covers the Second Variation, Euler-Lagrange PDE systems, and higher-order conservation laws. This timely synthesis of partial differential equations and differential geometry will be of fundamental importance to both students and experienced researchers working in geometric analysis.