Exterior Differential Systems and the Calculus of Variations
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) Set
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Springer Science+ Business Media, LLC
Phillip A. Griffiths
Exte rior Diffe renti al Syst ems and the Calc ulus of Varia tions
1983
Springer Science+ Business Media, LLC
Author: Phillip A. Griffiths Department of Mathematics Harvard University Cambridge, MA 02138
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.G84 1982 515'.64 82-17878
CIP-Kurztitelaufnahme der Deutschen Bibliothek Griffiths, Phillip A.: Exterior differential systems and the calculus of variations 1 Phillip A. Griffiths. - Boston Basel ; Stuttgart : Birkhauser, 1982. (Progress in mathematics ; Vol. 25) NE:
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ISBN 978-0-8176-3103-1 ISBN 978-1-4615-8166-6 (eBook) DOI 10.1007/978-1-4615-8166-6 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. © 1983 Springer Science+Business Media New York Originally published by Birkhauser Boston in 1983
To the memory of my mother Jeanette Field Griffiths
TABLE OF CONTEHTS
INTRODUCTION
0.
a) b) c) d) e)
I.
15
PRELIMINARIES Notations from Manifold Theory The Language of Jet Manifolds Frame Manifolds Differentia! Ideals Exterior Differential Systems
EULER-LAGRANGE EQUATIONS FOR DIFFERENTIAL SYSTEMS INDEPENDENT VARIABLE
~liTH
ONE
32
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
II.
FIRST INTEGRALS OF THE EULER-LAGRANGE SYSTEM; NOETHER'S THEOREM AND EXAMPLES
1D7
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.
II I.
EULER EQUATIONS FOR VARIATIONAL PROBLEfiJS IN HOMOGENEOUS SPACES 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
161
IV.
ENDPOINT CONDITIONS; JACOBI EQUATIONS AND THE 2nd VARIATION; CONJUGATE POINTS; FIELDS AND THE HAMILTON-JACOBI EQUATION; THE LAGRANGE PROBLEt1 a) b) c) d) e)
APPEtlDIX: a) b)
199
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