Bruce van Brunt
     The Calculus of
     Variations
     With 24 Figures
             Bruce van Brunt
             Institute of Fundamental Sciences
             Palmerston North Campus
             Private Bag 11222
             Massey University
             Palmerston North 5301
             New Zealand
             b.vanbrunt@massey.ac.nz
             Editorial Board
             (North America):
             S. Axler                                              F.W. Gehring
             Mathematics Department                                Mathematics Department
             San Francisco State University                        East Hall
             San Francisco, CA 94132                               University of Michigan
             USA                                                   Ann Arbor, MI 48109-1109
             axler@sfsu.edu                                        USA
                                                                   fgehring@math.lsa.umich.edu
             K.A. Ribet
             Mathematics Department
             University of California, Berkeley
             Berkeley, CA 94720-3840
             USA
             ribet@math.berkeley.edu
             Mathematics Subject Classification (2000): 34Bxx, 49-01, 70Hxx
             Library of Congress Cataloging-in-Publication Data
             van Brunt, B. (Bruce)
                  The calculus of variations / Bruce van Brunt.
                    p. cm. „ (Universitext)
                  Includes bibliographical references and index.
                  ISBN 0-387-40247-0 (alk. paper)
                   1. Calculus of variations.  I. Title.
               QA315.V35 2003
               515′.64„dc21                              2003050661
             ISBN 0-387-40247-0               Printed on acid-free paper.
             ©2004 Springer-Verlag New York, Inc.
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             987654321                        SPIN 10934548
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                To Anne, Anastasia, and Alexander
    Preface
    Thecalculus of variations has a long history of interaction with other branches
    of mathematics such as geometry and differential equations, and with physics,
    particularly mechanics. More recently, the calculus of variations has found
    applications in other fields such as economics and electrical engineering. Much
    of the mathematics underlying control theory, for instance, can be regarded
    as part of the calculus of variations.
      This book is an introduction to the calculus of variations for mathemati-
    cians and scientists. The reader interested primarily in mathematics will find
    results of interest in geometry and differential equations. I have paused at
    times to develop the proofs of some of these results, and discuss briefly var-
    ious topics not normally found in an introductory book on this subject such
    as the existence and uniqueness of solutions to boundary-value problems, the
    inverse problem, and Morse theory. I have made “passive use” of functional
    analysis (in particular normed vector spaces) to place certain results in con-
    text and reassure the mathematician that a suitable framework is available
    for a more rigorous study. For the reader interested mainly in techniques and
    applications of the calculus of variations, I leavened the book with numer-
    ous examples mostly from physics. In addition, topics such as Hamilton’s
    Principle, eigenvalue approximations, conservation laws, and nonholonomic
    constraints in mechanics are discussed. More importantly, the book is written
    on two levels. The technical details for many of the results can be skipped
    on the initial reading. The student can thus learn the main results in each
    chapter and return as needed to the proofs for a deeper understanding. Sev-
    eral key results in this subject have tractable analogues in finite-dimensional
    optimization. Where possible, the theory is motivated by first reviewing the
    theory for finite-dimensional problems.
      The book can be used for a one-semester course, a shorter course, or in-
    dependent study. The final chapter on the second variation has been written
    with these options in mind, so that the student can proceed directly from
    Chapter 3 to this topic. Throughout the book, asterisks have been used to
    flag material that is not central to a first course.