CALCULUS OF VARIATIONS, The.
The Calculus of Variations is a natural growth of the Infinitesimal Calculus (q.v.)— in Particular of the Integral Calculus (q.v.) and of Diff erentiql Equations (q.v.). In the Integral Calculus, properly speaking, only integrals of the typeff(x, al, as . . . ) dx are considered, where f(x, a., ...) is a function of the variable of integration x and of several parameters al .. . which are independent of x. In solving differential equations of the type dy/dx=f (x, y) we are in one sense dealing with a new type of integral, f f (x, y) dx. Such integrals, in which y is to be replaced by a certain function of x, are called line integrals.
The integrals considered in the Calculus of Variations are essentially of this kind, but we shall see that the more interesting problems are those in which still another element is intro duced. The integral (1) / fix, y, where can be evaluated whenever y is known as a function of x. For if y= 0(x) be the known value of y in terms of x, and if 4(.2) and 4'(x)=40(x)/ be substituted for y and y' respectively under the integral sign, the inte grand becomes a function of x alone, and the integral itself has a definite numerical value, at least under certain very general restrictions which need not be stated here. Thus to every function of x which can be substituted for y there corresponds a definite number — the value of x and y. We shall denote the value of I relation y=*(x) defines a curve C in the plane ofx and y. We shall denote the value of I which corresponds to the function t(x) by the symbol I..
The central problem of the Calculus of Variations is the determination of a curve (x)], for which the value of I, I u, is less than [greater than] the value of I for any other curve C [y=4(x)], which satisfies the conditions of the particular example.
In most of the simpler examples it is speci fied or implied by the conditions of the problem that the curves C considered shall all pass through each of two given fixed points P.(x., y.) and Pi (xi, yi), whose abscissa:are respectively x. and xi, the limits of integration of the inte gral I. Hence only those functions of x, p(x) are to be considered for which cs (.20)=y. and • (xi) =Y1.
In order to clarify the general problem, let us consider the example L =-- y'l dx.
This is a familiar integral; it is the formula for the length of any curve (x) between any two of its points. With respect to this
integral the statement of the simplest prob lem of the Calculus of Variations is as fol lows: Given two fixed points P. (x.,y.) and P, (xi, y,) in the xy plane; to determine that curve y=.0(x) joining P. and Pi for which the value of the integral L (i.e., the length of the arc P.P,) is at a minimum. Accepting the Euclidean postulate that the shortest distance between two points is measured along the straight line joining them, it is evident a priori that the solution of this example is the straight line PoPi, or y=yri- (y,—y.)/(x,—x.).
It is at least plausible that any conditions which we may discover must, in this particular ex ample, be satisfied by this function.
It is easy to see how this simple problem may be generalized. For we might inquire what is the shortest path between a fixed point and a fixed curve, or between two fixed curves. Again, obstacles may be placed in the plane, and the shortest path then sought. This latter idea leads to an important application of the general theory: the determination of the shortest path between any two fixed points of a given surface, the surface being thought of as an obstacle placed in the plane. The most general problem of the kind mentioned above may be thought of as the determination of a certain shortest path.
An entirely distinct generalization of the preceding problem is that in which the integrand involves derivatives of higher order than the first, i.e., of the type:f f (x,y,y',y",. y(n)dx.
Another is that in which the integral involves several dependent variables: y, y, 1,...)dx. Finally, the in tegral considered may involve two (or more) independent variables and require two inte grations : fff(x, y, z, p, q) dx dy, where p and q denote az/ ax and as/ay, respectively, and where the function to be determined is a function of x and y which is to be substituted for z. Further generalizations are evident and would tend only to confuse if stated here. We shall return briefly to these generalized problems, but we shall state theorems principally for the simple integral I in one dependent and one independent variable. Many of these theorems can be generalized without essential difficulty to the other cases which have been mentioned.