The Inverse Problem in the Calculus of Variations Via Exterior Differential Systems
Title | The Inverse Problem in the Calculus of Variations Via Exterior Differential Systems PDF eBook |
Author | Thi Kim Thoan Do |
Publisher | |
Pages | 1224 |
Release | 2016 |
Genre | Calculus of variations |
ISBN |
The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations
Title | The Inverse Problem of the Calculus of Variations for Ordinary Differential Equations PDF eBook |
Author | Ian Anderson |
Publisher | American Mathematical Soc. |
Pages | 122 |
Release | 1992 |
Genre | Mathematics |
ISBN | 082182533X |
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.
The Inverse Problem of the Calculus of Variations
Title | The Inverse Problem of the Calculus of Variations PDF eBook |
Author | Dmitry V. Zenkov |
Publisher | Springer |
Pages | 296 |
Release | 2015-10-15 |
Genre | Mathematics |
ISBN | 9462391092 |
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).
Variational Principles For Second-order Differential Equations, Application Of The Spencer Theory Of
Title | Variational Principles For Second-order Differential Equations, Application Of The Spencer Theory Of PDF eBook |
Author | Joseph Grifone |
Publisher | World Scientific |
Pages | 229 |
Release | 2000-05-25 |
Genre | Mathematics |
ISBN | 9814495360 |
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.
Variational Principles for Second-order Differential Equations
Title | Variational Principles for Second-order Differential Equations PDF eBook |
Author | J. Grifone |
Publisher | World Scientific |
Pages | 236 |
Release | 2000 |
Genre | Mathematics |
ISBN | 9789810237349 |
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.
Exterior Differential Systems and the Calculus of Variations
Title | Exterior Differential Systems and the Calculus of Variations PDF eBook |
Author | P.A. Griffiths |
Publisher | Springer Science & Business Media |
Pages | 348 |
Release | 2013-06-29 |
Genre | Mathematics |
ISBN | 1461581664 |
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.
Inverse Problems in Partial Differential Equations
Title | Inverse Problems in Partial Differential Equations PDF eBook |
Author | David L. Colton |
Publisher | SIAM |
Pages | 234 |
Release | 1990-01-01 |
Genre | Mathematics |
ISBN | 9780898712520 |