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CHAPTER 1
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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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 = φ(t), y = ψ(t).
The integral which we have to minimize becomes then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where x' = φ'(t), y' = ψ'(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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
has a given value, the one which maximizes the integral
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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 (δ) of a point x1 = a1, x2 = a2, ..., xn = an" is the totality of points x1, x2, ..., xn satisfying the inequalities:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The word "domain" will be used in the same sense as the German Bereich, i. e., synonymous with "set of points" (compare E. II A, p. 44). The word "region" will be used: (a) for a "continuum," i. e., a set of points which is "connected" and made up exclusively of "inner" points; in this case the boundary does not belong to the region ("open" region); (b) for a continuum together with its boundary ("closed" region); (c) for a continuum together with part of its boundary. The region may be finite or infinite; it may also comprise the whole n-dimensional space.
When we say: a curve lies "in" a region, we mean: each one of its points is a point of the region, not necessarily an inner point.
For the definition of "inner" point, "boundary point" (frontière), and "connected" (d'un seul tenant) we refer to E. II A, p. 44; J. I, Nos. 22, 31; and Hurwitz, Verhandlungen des ersten internationalen Mathematikercongresses in Zürich, p. 94.
b) By a "function" is always meant a real single-valued function.
The substitution of a particular value x = x0 in a function φ(x) will be denoted by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
similarly
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Instead we shall also use the simpler notation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where it can be done without ambiguity, compare e).
We shall say: a function has a certain property IN a domain S of the independent variables, if it has the property in question at all points of the domain S, no matter whether they are interior or boundary points.
A function of, x1, x2, ..., xn has a certain property in the vicinity of a point x1 = a1, x2 = a2, ..., xn = an, if there exists a positive quantity δ such that the function has the property in question in the vicinity (δ) of the point a1, a2, ..., an.
If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we shall say: φ(h) is an "infinitesimal" (for Lh = 0); such an infinitesimal will in a general way be denoted by (h). Also an independent variable h which in the course of the investigation is made to approach zero, will be called an "infinitesimal."
c) Derivatives of functions of one variable will be denoted by accents, in the usual manner:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
For brevity we shall use the following terminology for various classes of functions which will frequently occur in the sequel. We shall say that a function f(x) which is defined in an interval (x0x1) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
with the understanding concerning the extremities of the interval that the definition of f(x) can be so extended beyond (x0x1) that the above properties still hold at x0 and x1.
If f(x) itself is continuous, and if the interval (x0x1) can be divided into a finite number of subintervals
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
such that in each subinterval f(x) is of class C'(C"), whereas f'(x)(f"(x)) is discontinuous at c1, c2, ..., cn-1, we shall say that f(x) is of class D'(D"). We consider class C'(C") as contained in D'(D"), viz., for n = 1.
From these definitions it follows that, for a function of class D', the progressive and regressive derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], exist, are finite and equal to the limiting values f'(cv + 0), f'(cv - 0) respectively.
d) Partial derivatives of functions of several variables will be denoted by literal subscripts (Kneser):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Also of a function of several variables we shall say that it is of class C(n) in a domain S if all its partial derivatives up to the nth order inclusive exist and are continuous in the domain S.
e) The letters x, y will always be used for rectangular co-ordinates with the usual orientation of the positive axes, i. e., the positive y-axis to the left of the positive x-axis. It will frequently be convenient to designate points by numbers: 0, 1, 2, ...; the co-ordinates of these points will then always be denoted by x0, y0; x1, y1; x2, y2; ... respectively; their parameters, if they lie on a curve given in parameter-representation, by t0, t1, t2,. ...
A curve (arc of curve)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
will be said to be of class C,C', etc., if the function f(x) is of class C,C', etc., in (x0x1). In particular, a curve of class D' is continuous and made up of a finite number of arcs with continuously turning tangents, not parallel to the y-axis. The points of the curve whose abscissæ are the points of discontinuity C1, C2, ..., Cn-1 of f'(x), ... will be called its corners. At a corner the curve has a progressive and a regressive tangent, and, (See Fig. 1.)
(f) The ITL∫ITL
J = ∫ F(x, y, y') dx
taken along the curve
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
from the point A(x0, y0) to the point B(x1, y1), i.e., the integral
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
will be denoted by Jg (AB) (more briefly Jg or J(AB)); or by Jμv, if the end-points are designated by numbers: μ, v.
g) The distance between the two points P and Q will be denoted by |PQ|, the circle with center O and radius r by (O, r) (Harkness and Morley). The angle which a vector makes with the positive x-axis will be called its amplitude.
§3. GENERAL FORMULATION OF THE PROBLEM
a) After these preliminary explanations, the simplest problem of the Calculus of Variations may be formulated in the most general way, as follows:
There is given:
1. A well-defined infinitude M of curves, representable in the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
the end-points and their abscissæ x0, x1 may vary from curve to curve. We shall refer to these curves as "admissible curves."
2. A function F(x, y, p) of three independent variables such that for every admissible curve G, the definite integral
(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
has a determinate finite value.
The set of values JG thus defined has always a lower limit, K, and an upper limit, G (finite or infinite). If the lower (upper) limit is finite, and if there exists an admissible curve G such that
JG = K, (JG = G),
the curve G is said to furnish the absolute minimum (maximum) for the integral J (with respect to M). For every other admissible curve [bar.G] we have then
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The word "extremum" will be used for maximum and minimum alike, when it is not necessary to distinguish between them.
Hence the problem arises: to determine all admissible curves which, in this sense, minimize or maximize the integral J.
b) As in the theory of ordinary maxima and minima, the problem of the absolute extremum, which is the ultimate aim of the Calculus of Variations, is reducible to another problem which can be more easily attacked, viz., the problem of the relative extremum:
An admissible curve G is said to furnish a relative minimum (maximum) if there exists a "neighborhood N of the curve G," however small, such that the curve G furnishes an absolute minimum with respect to the totality M1 of those curves of M which lie in this neighborhood; and by a neighborhood N of the curve G we understand any region which contains G in its interior.
According to Stolz, the relative minimum (maximum) will be called proper, if there exists a neighborhood N such that in (2) the sign > (<) holds for all curves [bar.G] different from 6; improper if, however the neighborhood N may be chosen, there exists some curve [bar.G] different from G for which the equality sign has to be taken.
A curve which furnishes an absolute extremum evidently furnishes a fortiori also a relative extremum. Hence the original problem is reducible to the problem: to determine all those curves which furnish a relative minimum; and in this form we shall consider the problem in the sequel.
We shall henceforth always use the words "minimum," "maximum" in the sense of relative minimum, maximum; and we shall confine ourselves to the case of a minimum, since every curve which minimizes J, at the same time maximizes — J, and vice versa.
c) In the abstract formulation given above, the problem would hardly be accessible to the methods of analysis; to make it so, it is necessary to specify some concrete assumptions concerning the admissible curves and the function F.
(Continues…)
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