Hans Sagan - Introduction to the Calculus of Variations

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71> Hans SaganDOVER PUBLICATIONS, INC., New York Copyright 1969 by Hans Sagan. All rights reserved under Pan American and International Copyright Conventions. This Dover edition, first published in 1992, is an unabridged, corrected republication of the work first published by the McGraw-Hill Book Company, New York, 1969, in its International Series in Pure and Applied Mathematics. Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 Library of Congress Cataloging-in-Publication Data Sagan, Hans. p. cm. cm.

Originally published : New York : McGraw-Hill, 1969, in series : International series in pure and applied mathematics. Includes bibliographical references and index. eISBN: 9780486138022 1. Calculus of variations. I. Title.

QA315.S23 1992 515.64dc20 92-2356
CIP PREFACE The calculus of variations has attracted the attention of numerous eminent mathematicians, who have made significant and lasting contributions to its development since its inception 273 years ago. Early this century, when the problem of Bolza in its overwhelming generality was formulated and analyzed, most of the problems that were considered to lie within the province of the calculus of variations appeared to be solved, or at least solved in principle, and interest in the subject began to decline. Only in the last decade or so has a new upsurge of interest become noticeable. This is partly due to the novel demands of a new and sophisticated technology, in which space technology occupies a prominent position. Another factor in this revival of the calculus of variations is the availability of high-speed computing devices, which, together with newly developed mathematical algorithms, make it possible to solve many variational problems which heretofore could only be solved in principle. The modern theory of optimal control, which has been developed in recent years and is still in a process of growthin depth as well as in scopehas its roots in the calculus of variations, although many of its techniques are new and more easily adaptable to practical technological requirements than are the methods of the classical calculus of variations.

It is the purpose of this text to lay a broad foundation for an understanding of the problems of the calculus of variations and of its many methods and techniques, with their many-faceted subtleties, and to prepare the reader for the study of modern optimal control theory. The treatment is limited to a thorough discussion of single-integral problems in one or more unknown functions, where the integral is employed in the riemannian sense. The functions that are admitted are at worst sectionally smooth. Such a setting seems general enough for most practical purposes and, at the same time, keeps the formulation and the proofs on a simple enough level. The problem in two or more independent variables is given peripheral treatment in the appendices to ). The first three chapters deal with variational problems without constraints.

The theory of the first variation and the theory of fields are developed as far as seems practical and to a point beyond which the arguments would become tiresome and repetitious. . In Chap. is devoted to a discussion of its application to time-optimal control problems and to control problems of fixed duration that are associated with a nonautonomous system of differential equations. is devoted to a derivation of the multiplier rule for the problem of Mayer with fixed and variable endpoints and its application to the problem of Lagrange and the isoperimetric problem. , are derived within the framework of the theory of the second variation. , are derived within the framework of the theory of the second variation.

A new result that is derived in this chapter is Jacobis necessary condition for a weak relative minimum. I have resisted the temptation to include the customary chapter on direct methods because to do this subject justice, one would have to devote an entire book to it. Not only is the method of attack quite distinct from the methods employed in this book, but the variational problems and the concept of the minimum itself change their character substantially owing to a different concept of the space of competing functions. This book contains essentially the material I have developed during the past ten years in courses taught at the University of Idaho and North Carolina State University and in a series of lectures which I have delivered over the past five years to engineers and scientists at the Astromechanics Branch of the Space Mechanics Division of the National Aeronautics and Space Administration at Langley Field, Virginia. I have used elementary functional analytical methods in the discussion of the foundations of the theory of the first and the second variation, not to appear modern at any price, but principally because the functional analytical setting provides a much better and deeper understanding of the fundamental concept of the space of admissible variations and the concepts of a weak and a strong relative minimum of an integral. I have tried to avoid repetitive arguments whenever feasible by using different approaches to the same concept at various stages of generality.

Thus Bellmans fundamental partial differential equation is derived first by a method of Carathodory and then, in a more general setting, by Bellmans method of needle-shaped variations. The theory of fields and the Weierstrass excess function are introduced on three separate occasions by three different methods. Some of the interrelations of the various approaches to the solution of variational problems are spelled out in detail, some are merely hinted at, and some are left for the reader to discover. This book was written primarily for the student. To achieve greater clarity, care has been taken to spell out difficult steps in some detail, often at the expense of mathematical elegance. Every theorem is either preceded immediately by a derivation or succeeded by a proof, with one exception.

The exception is the fundamental theorem on underdetermined systems on which the multiplier rule for the Mayer problem is based. To give the reader a chance to familiarize himself with the notation and with the content of this theorem, the proof is postponed until after the theorem has been applied to a number of special cases. To illustrate a complicated situation with a complicated problem compounds the difficulties. Therefore, the examples discussed in the main body of the text (which are set aside from the main text by the use of smaller print) have been kept at a simple and technically uncomplicated level, uncluttered by extraneous matters and stripped of all unnecessary constants. There are 421 exercises interspersed throughout the text. Some of these exercises are intended to develop mastery of the formalism, some (which are marked by an asterisk) complement the text, and still others lead beyond the material presented in the text.

In order to read this book with profit or to take a course based on this book with a reasonable chance for success, the student should be able to use the implicit-function theorem in a variety of situations, he should be familiar with the Heine-Borel theorem, and he should know the standard existence and uniqueness theorems from the theory of ordinary differential equations. To put this in more conventional terms, the student should have had a course in advanced calculus and an intermediate course in ordinary differential equations. Some knowledge of the elements of matrix algebra would be helpful in . The material is arranged to give the user the greatest possible amount of flexibility. The material contained in the main portions of the chapters (excluding appendices) can be taught comfortably in a one-semester course, possibly by omitting .