Calculus of Variations I
This 2-volume treatise by two of the leading researchers and writers in the field, quickly established itself as a standard reference. It pays special attention to the historical aspects and the origins partly in applied problems - such as
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Series editors A. Chenciner 5.5. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin 1. Hormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Y.G. Sinai N.J.A. Sloane J. Tits B. Totaro A. Vershik M. Waldschmidt
Editor-in-Chief M. Berger J.Coates
S.R.S. Varadhan
310
Springer-Verlag Berlin Heidelberg GmbH
Mariano Giaquinta Stefan Hildebrandt
Calculus of Variations I With 73 Figures
,
Springer
Mariano Giaquinta Scuola Normale Superiore Piazza dei Cavalieri, 7 56100 Pisa, Italy
Stefan Hildebrandt Universitiit Bonn Mathematisches Institut Wegelerstr. 10 53115 Bonn, Germany
1St ed. 1996. Corr. 21ld printing 1004
Library of Congress Cataloging-in-Publication Data. Giaquinta, Mariano, 1947- .Calculus ofvariationslMariano Giaquinta, Stefan Hildebrandt. p. cm.-(Grundlehren der mathematischen WlSsenschaften; 310-311) Includes bibliographical references and indexes. Contents: L The Lagrangian formalism-l. The Hamiltonian formalism. 1. Calculus of variations 1. Hildebrandt, Stefan. II. Title. III. Series. QA315.G46 1996 515'.64-dc1O 96-10419
Mathematics Subject Classification: 49-xx. 53-xx. 70-XXI.
ISBN 978-3-642-08074-6 ISBN 978-3-662-03278-7 (eBook) DOI 10.1007/978-3-662-03278-7 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law.
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@ Springer-Verlag Berlin Heidelberg 1004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004. Softcover reprint of the hardcover I st edition 2004 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: design 6- production GmbH, Heidelberg Printed on acid-free paper
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Preface
This book describes the classical aspects of the variational calculus which are of interest to analysts, geometers and physicists alike. Volume 1 deals with the formal apparatus of the variational calculus and with nonparametric field theory, whereas Volume 2 treats parametric variational problems as well as HamiltonJacobi theory and the classical theory of partial differential equations of first ordel;. In a subsequent treatise we shall describe developments arising from Hilbert's 19th and 20th problems, especially direct methods and regularity theory. Of the classical variational calculus we have particularly emphasized the often neglected theory of inner variations, i.e. of v
