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Birkhäuser

Exterior Differential Systems and the Calculus of Variations

  • Book
  • © 1983

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Authors:
  1. Phillip A. Griffiths

    1. Department of Mathematics, Harvard University, Cambridge, USA

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Part of the book series: Progress in Mathematics (PM, volume 25)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction

    • Phillip A. Griffiths
    Pages 1-14

  3. Preliminaries

    • Phillip A. Griffiths
    Pages 15-31

  4. Euler-Lagrange Equations for Differential Systems with One Independent Variable

    • Phillip A. Griffiths
    Pages 32-106

  5. First Integrals of the Euler-Lagrange System; Noether’s Theorem and Examples

    • Phillip A. Griffiths
    Pages 107-160

  6. Euler Equations for Variational Problems in Homogeneous Spaces

    • Phillip A. Griffiths
    Pages 161-198

  7. Endpoint Conditions; Jacobi Equations and the 2nd Variation; the Hamilton-Jacobi Equation

    • Phillip A. Griffiths
    Pages 199-309

  8. Back Matter

    Pages 310-339
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Harvard University, Cambridge, USA

    Phillip A. Griffiths

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