Progress in Mathematics Vol. 25. Edited by J. Coates and S. Helgason. Springer Science+ Business Media, LLC

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2 Progress in Mathematics Vol. 25 Edited by J. Coates and S. Helgason Springer Science+ Business Media, LLC

3 Phillip A. Griffiths Exterior Differential Systems and the Calculus of Variations 1983 Springer Science+ Business Media, LLC

4 Author: Phillip A. Griffiths Department of Mathematics Harvard University Cambridge, MA Library of Congress Cataloging in Publication Data Griffiths, Phillip. Exterior differential systems and the calculus of variations. (Progress in mathematics ; v. 25) Includes index. 1. Calculus of variations. 2. Exterior differential systems. I. Title. II. Series: Progress in mathema ti cs (Cambridge, Mass.) ; v. 25. QA316.G ' CIP-Kurztitelaufnahme der Deutschen Bibliothek Griffiths, Phillip A.: Exterior differential systems and the calculus of variations 1 Phillip A. Griffiths. - Boston Basel ; Stuttgart : Birkhauser, (Progress in mathematics ; Vol. 25) NE: GT ISBN ISBN (ebook) DOI / All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form ar by any means, electronic, mechanical, photocopying, recording or otherwise, wtthout prior permission of the copyright owner Springer Science+Business Media New York Originally published by Birkhauser Boston in 1983

5 To the memory of my mother Jeanette Field Griffiths

6 TABLE OF CONTEHTS INTRODUCTION 0. I. II. 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 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 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: i) ( K 2 ds for Plane Curves; i i) Affine Arclength; iii) f K 2 ds for Space Curves; and iv) Delauney Problem D7 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 b) Examples: i) the Euler Equations Associated to f K 2 /2 ds for Curves in len; i i) Some Problems as in i) but for Curves in sn; iii) Euler Equations Associated to Nondegenerate Ruled Surfaces

7 IV. ENDPOINT CONDITIONS; JACOBI EQUATIONS AND THE 2nd VARIATION; CONJUGATE POINTS; FIELDS AND THE HAMILTON-JACOBI EQUATION; THE LAGRANGE PROBLEt1 199 a) Endpoint Conditions; Well-Posed Variational Problems; Examples b) Jacobi Vector Fields and Conjugate Points; Examples c) Geometry of the Reduced t 1omentum Space; the znd Variation; the Index Form and Sufficient Conditions for a Local 11inimum d) Fields and the Hamilton-Jacobi Equation; Further Sufficient Conditions for a Local Minimum e) t1ixed Endpoint Conditions and the Classical Problem of Lagrange; i) Well-Posed Mixed Variational Problems; ii) The Lagrange Problem; iii) The Classical Approach to the Lagrange Problem; iv) Some Related Examples APPEtlDIX: MISCELLANEOUS REHARKS AND EXAt1PLES 310 a) Problems with Integral Constraints; Examples b) Classical Problems Expressed in Moving Frames INDEX 329 BIBLIOGRAPHY 332 viii

8 LIST OF C0~1t,10NLY USED NOTATIONS (Note: The references for the undefined terms used below may be found in t~ndex.) A''' (X) {1:} (I,w) V( I,w) TN(V(I,w)) (I,w;liJ) Exterior algebra of smooth differential forms on a manifold X Algebraic ideal in set L of forms on generated by a Exterior differential system with independence condition Set of integral manifolds of Tangent space to V(l,w) at N (I,w) Variational Problem (cf. Chapter I, Sec. a) :V(I,w)->-IR Functional on V(l,w) O :TN(I,w) ->-IR V( I,w; (A,B])) Differential of Subset of V(l,w) given by endpoint conditions TN(V(I,w; [A, B))) Tangent space to V( I,w; [A, B]) - mod I Congruence modulo an ideal I ca 1 '(X) Congruence modulo the image of (cf. (ll.b.4)) I A I->- A'''(X) PE Projectivization of a vector space E d8 F (.) G Restriction of 8EA'''(X) to a submanifold NcX Exterior derivative of a differential form; 1 ittle 8 is frequently denoted by capital G Frame manifold Lie derivative of a form l(j along a vector field v y Momentum space associated to (I,w;l(J) Q Reduced momentum space associated to (I,w ;lij)