joining the fixed end points, and surround it by an area of finite
breadth, any other curve drawn within the area, and joining the
same end points, is called a variation of the original curve, or a
varied curve. The original curve is defined by specifying y as a
function of x. Necessary conditions for the existence of an extremum
can be found by choosing special methods of variation.
One method of variation is to replace y by y+εu, where u is a function of x, and ε is a constant which may be taken as small as we please. The function u is independent of ε. It is differentiable, and its differential coefficient is continuous within the interval of integration. It must vanish at x=x0 and at x=x1, . This method of variation has the property that, when the Weak variations. ordinate of the curve is but slightly changed, the direction of the tangent is but slightly changed. Such variations are called weak variations. By such a variation the integral is changed into
and the increment, or variation of the integral, is
In order that there may be an extremum it is necessary that the variation should be one-signed. We expand the expression under the sign of integration in powers of ε. The first term of the expansion contributes to the variation the termThe first variations.
xl BF 6F,
efxo <ayu -l- ay, u dx.
This term is called 'the first variation. The variation of the integral cannot be one-signed unless the first variation vanishes. On transforming the first variation by integration by parts, and observing that u vanishes at x =x0 and at x =x, , we find a necessary condition for an extremum in the form
x HF d 6F
-W udx-o.
It is a fundamental theorem that this equation cannot hold for all admissible functions u, unless the differential equation Li.. 'f§ f. Q§ =0
dx éy' 6y
is satisfied at every point of the curve along which the integral is taken. This is the principal equation for this problem. The curves that are determined by it are called the stationary curves, or the extremals, of the integral. We learn that curves the integral cannot be an extremum unless it is taken along a stationary curve.
A difficulty might arise from the fact that, in the foregoing argument, it is tacitly assumed that y, as a function of x, is one-valued; and we can have no a priori ground for assuming that this is the case for the sought curve. This difficulty might be met by an appeal to James Bernoulli's principle, according to which every arc of a stationary curve is a stationary curve between the end points of the arc-a principle which can be proved readily by adopting such a method of variation that the arc of the curve between two points is displaced, and the rest of the curve is not. But another method of meeting it leads to important developments. This is the method of parametric representation, introduced by K. Weier:;';:; c strass. According to this method the curve is defined mauled by specifying x and y as one-valued functions of a araP meter 0. The integral is then of the form 6 . .
0;f(x, y. x. y)d0,
where the dots denote differentiation with respect to 0, and f is a homogeneous function of x, y of the first degree. The mode of dependence of x and y upon 0 is immaterial to the problem, provided that they are one-valued functions of 0. A weak variation is obtained by changing x and y into x-i-eu, y-l-ev, where u and v are functions of 0 which have continuous differential coefficients and are independent of e. It is then found that the rinci al P P
equations of the problem are
i£! § =0 i f?I ?i 0
d<96x 6x 'd0 651 ay"
These equations are equivalent to a single equation, for it can be § roved wxthout difficulty that, when f is homogeneous of the first egree in x, y
1 I 6'f Bff
y deax axi ” x i do ay ayi “ayax"axay'H'(”" ) where
1 Qi 1 0'f I if
f1'=y2 ax” xy axay “E” ayf
The stationag curves obtained by this method are identical with those obtain by the previous method. The formulation of the problem by the parametric method often enables us to simplify the formation and integration of the principal equation. A very simple example is furnished by the prob, m problem: Given two points in the plane of (x, y) on the nh ° same side of the axis of x, it is required to find a curve ° the M joining them, so that this curve may generate, by revolu- ca °"° ° tion, about the axis of x, a surface of minimum area. The integral to be made a minimum is
§ ;y<x=+y2>ad@,
and the principal equation is
E & O
<10 lifsl-5'”)é T
of which the first integral is
3U@(i2+5'2)'i = C.
Ol
2' = Q 2 gi.
C 3 1+ (dx)and
the stationary curves are the catenaries y=c cosh{(x−a)/c}.
The required minimal surface is the catenoid generated by the revolution of one of these catenaries about its directrix. The parametric method can be extended without difficulty so as to become applicable to more general classes of problems. A simple example is furnished by the problem of forming the equations of the path of a ray of light in a variable medium. According to Fermat's principle, the integral ∫μds is a minimum, ds representing the element of arc of a ray, and n the refractive index. Thus the integral to be made a minimum is § ;u<x“+y2+2'>=d0.
The equations are found at once in forms of the type
dai i axfx -l'y +2)2-0,
and, since (x2+y”-I-22)§ dB=ds, these equations can be written in the usual forms of the type
d dx 6, uas
(Fda) ox- O
The formation of the first variation of an integral by means of a weak variation can be carried out without difficulty in the case of a simple integral involving any number of dependent variables and differential coefficients of arbitrarily high orders, and also in the cases of double and multiple integrals; and the quantities of the type eu, which are used in the process, may be regarded as equivalent to La range's éx, By, . . . The same process may not, however, be appliecf to isoperimetric problems. If the first variation of the integral which is to be made an extremum, R""" “f sub'ect to the condition that another integral has a pre- “fe "W" scribed value, is formed in this way, and if it vanishes, the “PH”curve is a stationary curve for this integral. If the prescribed value of the other integral is unaltered, its first variation must' vanish; and, if the first variation is formed in this way, the curve is a stationary curve for this integral also. The two integrals do not, however, in general possess the same stationary curves. We can avoid this difficulty by taking the variations to be of the form f, u1+e¢u», , where Q, and ez are independent constants; and we can thus obtain a completely satisfactory proof of the rule of the undetermined multiplier. A proof on these lines was first published by P. Du Bois-Reymond (1879). The rule had long been regarded as axiomatic. The parametric method enables us to deal easily with the problem of variable limits. If, in the First Problem, the terminal point (x, , y,) is movable on a given guiding curve ¢(x1, yr) =o, ' the first variation of the integral can be written § f if ~'0'[ $3.322 iaf Qi
“1ax+”ay x=, M=y1'~ 00 ideas E)x§ u+ld06 ji 6y]v§ d0 where (xl-l-ful, y, -l-fvl) is on the curve φ(x1, y1)=0, and u1, v1 denote the values of u, v at (x1, y1). It follows that the required curve must be a stationary curve, and that the condition
∂f∂x
must hold at (xi, yi). The corresponding condition in the case of the integral
F(x, y, y')dx A ' Ya' ° e Variable limits. is found from the equations
0{ F, 8F Bf 6F
Ev: y 57' 55:57
to be g
Fix: yr %'g7=o
