us consider the example

x1

L = √1 √1 + y2 dx.

This is a familiar integral; it is the formula for the length of any curve y = (x) between any two of its points. With respect to this integral the statement of the simplest probem of the Calculus of Variations is as follows: Given two fixed points Po (oo) and P, (x, y) in the xy plane; to determine that curve y = p(x) joining Po and P, for which the value of the integral L (ie., the length of the are PP) 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 PP., or

y = yo+(x − xo) (Y'1 −1'o)/(x, −x.).

It is at least plausible that any conditions which we may discover must, in this particular example, 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(x, y, y', y'', . . . y(n))dx. Another is that in which the integral involves several dependent variables:

Si(x,

f(x, y, z, . . ., y', z', . . .) dx. Finally, the integral considered may involve two (or more) independent variables and require two inte

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 "Lagrange's condition") in its general form, and developed the theory in several directions, solving incidentally many problems from the formal standpoint. Following Euler, Lagrange introduced many simplifications and generalizations in a series of important papers (cf. his Works, and his books Théorie des fonctions' and 'Calcul 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 development 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 variazioni Milan, 1897, German translation by Schepp, 1899); and Kneser, 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 subject and of publication may be found. notes concerning its history up to the dates It should be noted that only the latter of these books contains references to the important developments published since 1900.

Precise Statement of the Problem.-It is evident upon examination that the naive conception of the problem does not permit of exact matheFor definiteness, let us matical treatment. suppose that the function f(x, y, y') in (1) is an analytic function of its three arguments inside of a certain three-dimensional region R, which may be finite or infinite, but which ex

denote oz/0x and az/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.

Let us also restrict ourselves to curves of the type y= p(x), where p(x), together with its first derivative '(x), is a continuous, singlevalued function of x in the interval xxx1, and where (x) and p(x,) are equal, respectively, to the ordinates y, and y, of the fixed points P, and P1. We shall call these "curves of the class B." If there is a single one of these curves y (x), or K, for which I is less than [greater than] Ie for any other curve C [y = p(x)] of the class B, that curve K is said to render the given integral I an absolute minimum [maximum]. It is evident that this will rarely occur, as is also the case in extrema of functions of a single variable.

If we no..

(2) f(x) = f(x)+n(x), .

x。≤x≤x,,

'(5) x。+(i −2) • d ≤x¡ ≤x。 + (i − 1) · 0 ≤x1, i=2, 3,... (n+1),

where ndx,x,, the curve K is said to render I a limited weak [strong] minimum [maximum].

Geometrically these conditions mean that the curves compared to K must lie, in the case of a strong extremum (ie., 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, they must cut K at least once in every vertical strip of width o

It is easy to show that if K is to render I an extremum (of any sort), 4(x) must satisfy the equation

that this proof, which implicitly assumes the existence of the second derivative of (x), does not involve any restrictions. (Cf. Bolza, Lectures, chap. i).

Assuming the further details without proof, it becomes evident that any curve K, y = 4(x) 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 d2y/dx2 (=y''). The coefficient of y' is d2f/dy' (=fy'v'). If this coefficient f'y' 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:

Any one of these solutions, ie., 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 P1. Usually, however, there is only one of the extremals (12) which passes through two given points, for the equations, (13) Y。 = f(x。, α, ß), y1 = f(x1, α, B),

usually determine a and ẞ, and hence also determine a single extremal joining P, and P1. If this is actually the case, either that extremal is the required solution K, or else there is no solution of the problem.

A large number of special cases lead to differ

I

2

For example, if I = √1+y'2dx, we shall have f(x, y, y′) =√1+ y22, whence fx=fy = fxy' = fvv' =fxy = 0, fy' = y'/(1+ y'), fu' y = 1/(1+y'), and the equation (6) takes the form y = 0. The only solutions of this differential equation are the straight lines y=ax+b. It follows that if there is any curve of the class B in the plane along which the distance between two given fixed points is at a minimum, that curve is the straight line joining the two points. This result is independent of the Euclidean postulate, and depends only upon the definition of length by means of the preceding integral.

The problem of the brachistocrone, mentioned above, is to find the curve along which a particle with initial velocity , will descend

most quickly from a given initial point P, to another given point P,. It is easy to show tha the time of descent is given by the formula x1 Vi+12 xo (v62-28(x-xo)

t

=

S

dx.

dx which gives at once y'2 = c2(1+y'?) (vq+2g(xo−x)). It is easy to solve this equation in parameter form, and we find:

A + B

2

A+ B

2

-(w-sin w),

These extremals are cycloids on horizontal bases, the radius of the generating circle being (A+B)/2, and one cusp being at the point (A, C). Further investigation is necessary to decide just when a given pair of points can be connected by such a cycloid (cf. Bolza, Lectures,' p. 236). If such a cylcoid can be drawn, we can infer that it is the solution if there is any solution. If no such cycloid can be drawn, we can infer that there is no solution in the region R.

The problem of finding the geodetic lines on a given surface is that of minimizing the integral,

1 = ƒ ̃ ̃(E+2Fy +Gy2)}dx,

ΧΟ

fvv(x, f(x), f(x))>o[<o] for xxx, be satisfied along the supposed solution y = f(x) between the end points. We shall prove this, and we shall see that the same condition is actually a sufficient condition for a weak limited miminum if the sign be removed.

Jacobi then showed, by means of the second variation of the given integral, that a third necessary condition for a minimum [maximum] is that the quantity

4(x, x)=n,(x)2(xo) — 72(x) 71 (xo) should not vanish for any value of x in the interval x, <x<x,, where 4(x, x) is a solution of the equation

which vanishes for x=x,. The proof, which is omitted, can be found in Bolza, Lectures,' chap. ii. A beautiful geometrical interpretation of this condition exists: if we consider the one parameter family of extremals through (x, Yo). and call their envelope E, the extremal which joins x, to any point beyond its point of tangency with E cannot possibly render the integral a minimum [maximum] between those two points, i.e., the envelope of the extremals through (x, y) bounds all the points which can possibly be reached by a minimizing extremal from x.

It was long believed that Jacobi's condition, together with the previous two, was a sufficient condition. That such is not the case was first

pointed out by Weierstrass, who also showed that Jacobi's condition, while not sufficient for a minimum in general, is sufficient for a weak minimum (if the point (x, y) lies inside the envelope of the extremals through (xo, Yo). (Cf. Bolza, Lectures, chap. iii.)

cient is most readily seen by giving an actual That the preceding conditions are not suffi example in which the extremals, though all the above conditions are satisfied, do not minimize the integral. Such is the example (see Bolza 'Lectures,' p. 73),

Here the extremals are straight lines, but it is easy to join two points for which all the preceding conditions are satisfied by a simple broken line for which the value of the integra is less than that along the straight line extremal. Of course, the comparison line used varies considerably from the straight line extremal in direction, though not in position.

Weierstrass, in 1879, gave a fourth necessary condition. He defines a new function, E(x, y, y, p) = f(x, y, y') − f(x, y, p) − ( y − p)ƒ p(x, y, p). Then Weierstrass's (fourth) necessary condition for a minimum [maximum] is

E(x, y, y', po [≤o] x。≤x≤x1,

where x, y, p are the values of x, y, dy/dx along the extremal between the end points, and where is any finite number whatever. (Cf. Bolza, Lectures, chap. iii.)

Since we have

it follows that it is also necessary that fvvZo, which is precisely the second (Legendre's) necessary condition mentioned above. It is easy to show that if (a) the end points can be joined by an extremal K, (b) a one parameter family of extremals [y(x, a)] can be found, one of which is K itself, and one and only one of which passes through each point of the plane near K, so that yp(x, y) can be found, i.e., a function which gives the slope of the extremal of the family at any point (x, y) near K, then the integral

J = ['*' [f(x, y, p) + (y′ − p)¡p(x, y, p)]dx

Το

where C is any curve of the class B in the field about K, since I =JK=Je. It follows that the condition

E(x, y, y', p) ≥o, x。≤x≤x,,

for all x and y near K and for the function p(x, y) just mentioned and for any finite value of whatever, is a sufficient condition for a strong minimum, if the sign of equality holds only for p=y. (Cf. Osgood, Annals of Mathematics, II, 3: Bolza, 'Lectures, chap. iii.)

It is possible to show (cf. Hedrick, Bull. A. M. S., IX, 1) that for a limited minimum the conditions remain the same except that Tacɔbi' condition may be omitted. The conditions in the various cases may be summarized in the following scheme:

Sufficient.

Necessary.

Limited Variations.

Unlimited Variations.

Weak. Strong. Euler's, Euler's, Legendre's Legendre's

necessary.

Jacobi's necessary, Weierstrass's necessary.

Euler's Legendre's sufficient, Jacobi's necessary, Weierstrass's sufficinet.

It is seen on glancing at the table that fro'n the simple conditions (Euler's and Legendre's) for limited weak variation we proceed to any other case by adding Weierstrass's conditions in the case of a strong minimum, and Jacobi's in case of an unlimited minimum, only. The above table represents substantially the present known conditions.

In special problems the irksomeness of these conditions can sometimes be circumvented. For 02f instance, given a problem in which ayo for all values of x, y, y', then the necessary and sufficient condition for a limited strong minimum is the possibility of finding a solution of Euler's quation joining the two given end points. Such is the case in the geodetic problem and also in the integral which leads to Hamilton's principle; and in each of these cases, fortunately, a limited strong minimum is all that is desired. Similar a2f simplification occurs in every case when for all x, y, y'. For then Legendre's and Weierstrass's conditions are always satisfied, and may be abstracted from the above table. this reason Hilbert has called a problem in a2f for all x, y, y' contained in a მ2 singly connected region R, in which the given end points lie, a "regular" problem of the Cal

which

culus of Variations.

2

For

Considering the example "Vi+y3 dx, we

2

see that f>o for all finite values of x, y, and y' whatever. Since (1 - p)2 E(x, y, y', p) = ·—†y'y'(x, y, §), y, it follows that such an example surely satisfies. Weierstrass's sufficient condition, provided that a field exist in the manner specified above. But in this case, since the extremals are all straight lines in the plane, it is obvious that all other conditions are satisfied. Hence the straight line joining any two points actually minimizes the given integral, i.e., the straight line is the shortest line between any two of its points if the preceding integral be the definition of length.

In the problem of the brachistochrone, mentioned above, it is shown that the extremals found (cycloids) actually render the integral of the problem a minimum provided no cusp lies between the end points (cf. Bolza, 'Lectures, chap. iv., pp. 126, 136, 146).

Returning to the integral which defines length, it is evident that some other integral might as well have been selected as the definition of length, if we are not to assume an intuitive knowledge of it. The variety of choice is limited only by the selection of those properties which we desire to have hold. This leads very naturally to The Inverse Problem of the Calculus of Variations: Given a set of curves which form a two-parameter family. What is the condition that they be the extremals of a problem of the Calculus of Variations? What are the conditions that they actually render the integral thus discovered a minimum? Let y= F(x, a, b) be the given family. Then (cf. Bolza, Lectures, p. 31) the integrand of any integral for which these are extremals must satisfy the equation

by dyuxaya=G(x, y, y') · бy'oy' where y"=G(x, y, y') is the differential equation of the given family. This equation for f(x, y, y′) always has an infinite number of solutions, of which only those are actually solutions of the given inverse problem which satisfy the relation f>o, and these are solutions in any region free from envelopes of one-parameter families of the given extremals. Some interesting conclusions for particular forms are to be found in a paper by Stromquist, Transactions of American Mathematical Society> (1905).

Such

Another interesting class of problems are the so-called isoperimetric problems. These are problems in which a further restriction is placed upon the solution by requiring that it shall give a second (given) integral a given value. is, for example, the problem of finding the curve of maximum area with a given perimeter. The problem is treated by means of the so-called method of multipliers, which is too long for presentation here. (See Bolza, 'Lectures, chap. vi.) of the details of the work for double integrals. This article is too short to give any account Suffice it to say that the known methods follow closely those given above for simple integrals. In the other possible problems mentioned above the same holds true. An interesting application of these other problems occurs in the well-known Problem of Dirichlet, which is fundamental in mathematical work. Another is Another is the well-known theory of mechanics the important problem of Minimum Surfaces. based upon Hamilton's Principle or one of the analogous mechanical principles. The modern methods have made these theories more rigorous.

Bibliography.-The following is a list of the more important works and articles published in America concerning the Calculus of Variations: Carll, Calculus of Variations' (New York, 1885); Osgood, Annals of Mathematics' (II, 3, and Trans. A. M. S., II); Whittemore, Annals of Mathematics' (II, 3); Hancock, (Various papers in Annals of Mathematics and Calculus of Variations' (Cincinnati, 1894); Bliss, 'Thesis' (Chicago, 1901); and various papers, 'Annals of Mathematics' and 'Transactions American Mathematical Society'; Hedrick, (Bulletin American Mathematical Society,' IX (1901-5); Bolza, various papers, (Bulletin American Mathematical Society,' (Transactions American Mathematical Society,' etc. (1901-5); Brochures published in the Chicago Decennial Publications,