Calculus of Variations

Robert Weinstock

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486630694
• eBook ISBN: 9780486141060
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians.

The author begins slowly, introducing the reader to the calculus of variations, and supplying lists of essential formulae and derivations. Later chapters cover isoperimetric problems, geometrical optics, Fermat's principle, dynamics of particles, the Sturm-Liouville eigenvalue-eigenfunction problem, the theory of elasticity, quantum mechanics, and electrostatics. Each chapter ends with a series of exercises which should prove very useful in determining whether the material in that chapter has been thoroughly grasped.
The clarity of exposition makes this book easily accessible to anyone who has mastered first-year calculus with some exposure to ordinary differential equations. Physicists and engineers who find variational methods evasive at times will find this book particularly helpful.
"I regard this as a very useful book which I shall refer to frequently in the future." J. L. Synge, Bulletin of the American Mathematical Society.

CHAPTER 1

INTRODUCTION

The definite integral

 

is a well-defined quantity—a number—when x₁ and x₂ have definite numerical values, when the integrand f is given as a function of the arguments x, y, (dy/dx), and when y is given as a function of x. The “first” problem of the calculus of variations involves comparison of the various values assumed by (1) when different choices of y as a function of x are substituted into the integrand of (1). What is sought, specifically, is the particular function y = y(x) that gives to (1) its minimum (or maximum) value. Explicit examples of this type of problem are given detailed treatment in Chap. 3. These include the problems of “the shortest distance between two points on a given surface,” “the curve of quickest descent between two points,” and “the surface of revolution of minimum area.”

Generalization of the first problem is effected in many directions. For example, the integrand of (1) may be replaced by a function of several dependent variables, with respect to which a minimum (or maximum) of the definite integral is sought. Further, the functions with respect to which the minimization (or maximization) is carried out may be required to satisfy certain subsidiary conditions. Explicit examples of various aspects of these generalizations are handled in Chaps. 3 and 4. An important special case is the problem of “the maximum area bounded by a closed curve of given perimeter.”

Another line along which generalization is pursued is the replacement of (1) by a multiple integral whose minimum (or maximum) is sought with respect to one or more functions of the independent variables of integration. Thus, for example, we seek to minimize the double integral

 

carried out over a fixed domain D of the xy plane, with respect to functions w = w(x, y). Such problems are dealt with in the opening sections of Chaps. 7 and 9.

The techniques of solving the problems of minimizing (or maximizing) (1), (2), and related definite integrals are intimately connected with the problems of maxima and minima that are encountered in the elementary differential calculus. If, for example, we seek to determine the values for which the function y = g(x) achieves a minimum (or maximum), we form the derivative (dy/dx) = g′(x), set g′(x) = 0, and solve for x. The roots of this equation—the only values of x for which y = g(x) can possibly achieve a minimum¹ (or maximum)—do not, however, necessarily designate the locations of minima (or even of maxima). The condition g′(x) = 0 is merely a necessary condition for a minimum (or maximum); conditions of sufficiency involve derivatives of higher order than the first. The vanishing of g′(x) for a given value of x implies merely that the curve representing y = g(x) has a horizontal tangent at that value of x. A horizontal tangent may imply one of the three circumstances: maximum, minimum, or horizontal inflection; we call any one of the three an extremum of y = g(x).

The treatment of many of the problems cf the calculus of variations in this volume is analogous to the treatment of maximum and minimum problems through the use of the first derivative only; quite often we merely derive a set of necessary conditions for a minimum (or maximum) and rely upon geometric or physical intuition to establish the applicability of our solution. In other cases our interest lies only in the attainment of an extremum; in these it is immaterial whether we have a maximum, minimum, or a condition analogous to a horizontal inflection in the elementary case. The methods involved in establishing the conditions sufficient for a minimum (or maximum)—and in proving the existence of a minimum (or maximum)—are extremely profound and intricate; such investigations are found elsewhere in the literature.²

The chief purpose of the present work is to illustrate the application of the calculus of variations in several fields outside the realm of pure mathematics. Such applications are found in the chapters following Chap. 4. By no means can the treatment here of any special field be considered exhaustive in its relationship to the calculus of variations; each of several of the later chapters is amenable to expansion to the length of a volume the size of the present one.

The reader is expected to have as a part of his (or her) permanent knowledge most of the concepts and techniques learned in a first-year calculus course, including a smattering of ordinary differential equations. Furthermore, he (or she) must be familiar with many of the matters encountered in a short course in advanced calculus. Practically all the required results from this latter category are collected in Chap. 2; the corresponding proofs may be found in texts listed in the Bibliography.³ With one brief exception (11-2), no use is made of the methods of vector analysis. The same statement holds for the use of complex numbers; in the absence of a statement to the contrary, all quantities that appear are to be assumed real.

The wider the reader’s knowledge of physics, quite naturally, the fuller will be his (or her) appreciation of several of the results achieved in later chapters. Only the barest acquaintance with the concepts of elementary physics is presupposed, however; the reader to whom the study of physics is completely foreign will experience difficulty in following the development at only a very few points.

With respect to purpose the exercises at the end of each chapter may be divided, roughly, into three categories: (i) filling in of details in the development of the text, (ii) illustration of methods and results treated in the text, and (iii) extension of the results achieved in the text. In nearly all cases adequate hints are given; often these hints appear only as final answers.

Study should begin with Chap. 3. The material of Chap. 2 should be referred to only as it is required in the work following.

---

¹ It is clear that here “minimum” (or “maximum”) refers to relative minimum (or relative maximum).

² For example, see Bliss (1, 2), Bolza, and Courant (1) listed in the Bibliography.

³ Goursat, Franklin, and Kellogg.

CHAPTER 2

BACKGROUND PRELIMINARIES

2-1. Piecewise Continuity, Piecewise Differentiability

(a) Let

denote “x approaches x₀ from the left” and x → x₀⁺ denote “x approaches x₀ from the right.” In this volume we consider only those functions f(x) for which

f(x) and

f(x) both exist for all x₀ interior to the interval (x₁

x

x₂) in which f(x) is defined. At the respe...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Chapter 1 Introduction
• Chapter 2 Background Preliminaries
• Chapter 3 Introductory Problems
• Chapter 4 Isoperimetric Problems
• Chapter 5 Geometrical Optics: Fermat’s Principle
• Chapter 6 Dynamics of Particles
• Chapter 7 Two Independent Variables: The Vibrating String
• Chapter 8 The Sturm-Liouville Eigenvalue-Eigenfunction Problem
• Chapter 9 Several Independent Variables: The Vibrating Membrane
• Chapter 10 Theory of Elasticity
• Chapter 11 Quantum Mechanics
• Chapter 12 Electrostatics
• Bibliography
• Index