THE FIRST VARIATION

§1. INTRODUCTION

The Calculus of Variations deals with problems of maxima and minima. But while in the ordinary theory of maxima and minima the problem is to determine those values of the independent variables for which a given function of these variables takes a maximum or minimum value, in the Calculus of Variations definite integrals involving one or more unknown functions are considered, and it is required so to determine these unknown functions that the definite integrals shall take maximum or minimum values.

The following example will serve to illustrate the character of the problems with which we are here concerned, and its discussion will at the same time bring out certain points which are important for an exact formulation of the general problem:

Example I: In a plane there are given two points A, B and a straight line. It is required to determine, among all curves which can be drawn in this plane between A and B, the one which, if revolved around the line, generates the surface of minimum area.

We choose the line L for the x-axis of a rectangular system of coordinates, and denote the co-ordinates of the points A and B by x0, y0 and x1, y1 respectively. Then for a curve

y = f(x)

joining the two points A and B, the area in question is given by the definite integral

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where y' stands for the derivative f'(x). For different curves the integral will take, in general, different values; and our problem is then analytically: among all functions f(x) which take for x = x0 and x = x1 the prescribed values y0 and y1 respectively, to determine the one which furnishes the smallest value for the integral J.

This formulation of the problem implies, however, a number of tacit assumptions, which it is important to state explicitly:

a) In the first place, we must add some restrictions concerning the nature of the functions f(x) which we admit to consideration. For, since the definite integral contains the derivative y', it is tacitly supposed that f(x) has a derivative; the function f(x) and its derivative must, moreover, be such that the definite integral has a determinate finite value. Indeed, the problem becomes definite only if we confine ourselves to curves of a certain class, characterized by a well-defined system of conditions concerning continuity, existence of derivative, etc.

For instance, we might admit to consideration only functions f(x) with a continuous first derivative; or functions with continuous first and second derivatives; or analytic functions, etc.

b) Secondly, by assuming the curves representable in the form y = f(x), where f(x) is a single-valued function of we have tacitly introduced an important restriction, viz., that we consider only those curves which are met by every ordinate between x0 and x1 at but one point.

We can free ourselves from this restriction by assuming the curve in parameter-representation:

x = f(t), y = ?(t).

The integral which we have to minimize becomes then

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where x' = f'(t), y' = ?'(t), and where t0 and t1 are the values of t which correspond to the two end-points.

c) It is further to be observed that our definite integral represents the area in question only when y [??] 0 throughout the interval of integration. The problem implies, therefore, the condition that the curves shall lie in a certain region of the x, y-plane (viz., the upper half-plane).

d) Our formulation of the problem tacitly assumes that there exists a curve which furnishes a minimum for the area. But the existence of such a curve is by no means self-evident. We can only be sure that there exists a lower limit for the values of the area; and the decision whether this lower limit is actually reached or not forms part of the solution of the problem.

The problem may be modified in various ways. For instance, instead of assuming both end-points fixed, we may assume one or both of them movable on given curves.

An essentially different class of problems is represented by the following example:

Example II: Among all closed plane curves of given perimeter to determine the one which contains the maximum area.

If we use parameter-representation, the problem is to determine among all curves for which the definite integral

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has a given value, the one which maximizes the integral

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Here the curves out of which the maximizing curve is to be selected are subject — apart from restrictions of the kind which we have mentioned before — to the new condition of furnishing a given value for a certain definite integral. Problems of this kind are called "isoperimetric problems;" they will be treated in chap. vi.

The preceding examples are representatives of the simplest — and, at the same time, most important — type of problems of the Calculus of Variations, in which are considered definite integrals depending upon a plane curve and containing no higher derivatives than the first. To this type we shall almost exclusively confine ourselves.

The problem may be generalized in various directions:

1. Higher derivatives may occur under the integral.

2. The integral may depend upon a system of unknown functions, either independent or connected by finite or differential relations.

3. Extension to multiple integrals.

For these generalizations we refer the reader to C. JORDAN, Cours d' Analyse, 2e éd., Vol. III, chap. iv; Pascal-(Schepp), Die Variationsrechnung (Leipzig, 1899); and Kneser, Lehrbuch der Variationsrechnung (Braunschweig, 1900), Abschnitt VI, VII, VIII.

§2. AGREEMENTS CONCERNING NOTATION AND TERMINOLOGY

a) We consider exclusively real variables. The "interval (a b)" of a variable x — where the notation always implies a< b — is the totality of values x satisfying the inequality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The "vicinity (d) of a point x1 = a1, x2 = a2, ..., xn = an" is the totality of points x1, x2, ..., xn satisfying...