Returning to the integral (1), let us consider the history of the problem very briefly. Al though a previous problem had been considered by Newton in 1687 nat. prin. Math.,' II, Sec. 7, Prop. 34), the first problem which gave rise to any general theory and encouraged investigation was the so-called problem of the brachistochrone — or curve of quickest descent — which we shall discuss as a particular exam ple. This problem was stated by Johann Bernouilli in 1698, solved by him in the ing year, by his brother Jacob in 1701 in an important memoir dealing with more general problems, and by Euler in 1744 in an important treatise (Methodus inveniendi lineas curvas... . It has remained of interest down to the present day, probably the last paper concerning it being that by Bolza, 'Bull. Amer. Math. Soc., 1904, No. 1,) in which a final solution is given. In the paper mentioned Euler first gave the first necessary condition (known as °Euler's condition' or less properly as condition)) in its general form, and developed the theory in several directions, solving inci dentally many problems from the formal stand point. Following Euler, Lagrange introduced many simplifications and generalizations in a series of important papers (cf. his 'Works,' and his books (Thione des fonctions' and 'Cal cul des fonctions'). In particular the Method of Multipliers for the treatment of problems of relative extrema, which we shall discuss briefly, is due to Lagrange. The other prominent names in the early history are Legendre, for whom the second condition is named; Gauss, who first studied double integrals with variable limits; Jacobi, who discovered the condition which bears his name; and Du Bois Reymond, who initiated the very modern critical develop ment of the theory. We shall restrict ourselves to a reference to Todhunter, 'A History of ... the Calculus of Variations...' (Cambridge 1861); and Pascal, 'Calcolo delle variazioru' (Milan 1897, German trans. by Schepp, 1899); and ICneser, (Variationsrechnung' and 'Ency. der Math. Wiss., II A 8, 1900' ; and Bolza, 'Lectures on the Calculus of Variations' (Chicago 1905). In these books exact and complete references to the literature of the sub ject and notes concerning its history up to the dates of publication may be found. It should be noted that only the latter of these books con tains references to the important developments published since 1900.
Precise Statement of the
It is evident upon examination that the naive con ception of the problem does not permit of exact mathematical treatment. For definiteness, let us suppose that the function f(x,y,y') in (1) is an analytical function of its three arguments inside of a certain three-dimensional region R, which may be finite or infinite, but which ex pressly does not include any points at infinity. Let us also restrict ourselves to curves of the type
where
together with its first derivative #'(x), is a continuous, single valued function of x in the interval x.
and if there exists a positive number d such that Is is less than [greater than] /e whenever the condition (3) 19(x)
Geometrically these conditions mean that the curves compared to K must lie, in the case of a strong extremum (i.e., maximum or minimum), close to the curve K; in the case of the weak extremum, they must lie close to K and vary only a little from K in direction; in the case of a limited extremum,
must cut K at least once in every vertical strip of width 6.
It is easy to show that if K is to render I an extremum (of any sort), '1(x) must in general satisfy the equation d cf of „ — — ' (6) dx u ay' ay Or _J 1 , al/ af Y y - - -0, ax ay' 8y which is known as Euler's (or less properly as Lagratsge's) equation. For we have (7) f ix, IP(x) +*(x), + fe(x)1dx, which must be a minimum [maximum] for a(x)=--.: O. Replacing e/(x) by e.a(x), where A(x) is an arbitrary function and e is a variable para meter, Ie will evidently be a function of the parameter c alone: (8) I.=--P (e) f(x tp + e . X, + e • ;Nu.
It is readily demonstrated that the ordinary rule applies and that we can have an extremum only if (9) dl 1p+ e•X , + e x.
+ ex, + O.') • X'Idx=0, __ when e=0, or • X + 0 , where flay, etc. Integrating the second term by parts, we get (10) F'(0) --=-{A(x)Mx, A.(4.1v(x. IP, V) — f P, 0') dx, or since X(x) evidently vanishes for x=x. and for x—xi, f lb, 1P' }d0. But ?(x) was itself any permissible function of x, and it is not hard to prove that the integral of such a product, of which one factor is arbitrary, can vanish only if the other factor vanishes. This gives precisely the equation (6). Certain further considerations are necessary to show that this proof, which implicitly assumes the existence of the second derivative of IP(x), does not involve any restrictions. (Cf. Bolza, 'Lec tures,' Chap. I).
Assuming the further details without proof, it becomes evident that any curve K, which is to render I a minimum (of any sort) must satisfy the differential equation (6). Since f and its derivatives are known functions, (6) is an ordinary differential equation of the second order, linear in d'y/dx'(=?). The coefficient of y" is er f yV). If this coefficient f a does not vanish, one and only one solution of (6) passes through a given point in a given direction. The general solution of (6) contains two arbitrary constants: (12) Y=f(x, a, g) • Any one of these solutions, i.e., any solution whatever of (6) is called an extremal. Hence the required curve K, if it exists, must be an extremal, and it is necessary to search for it only among the extremals. But K was to connect P. and Usually, however, there is only one of the extremals (12) which passes through two given points, for the equations, (13) y.=f(x.,a,/3) Y1-=1(xi, P), usually determine a and P, and hence also determine a single extremal joining Po and Pi. If this is actually the case, either that extremal is the required solution K, or else there is no solution of the problem.