VARIATIONS, CALCULUS OF. 



VARIATIONS, CALCULUS OF. 



or""**; in our thin! there ii no function of x and y but v!.at U 

 o*|]^of being represented even by T^'-T a (xy) or 5^+6, Mid also 



by ;**+(* |f) +6 with a relation between a ami 6. Whatever 



function of JT, or of x and y, will solve these equation*, U sure to be 

 .. if the method be successful. Tbi< point would need a little 

 more development than we have here ipaoa to give. 



VARIATIONS, CALCULUS OK. The preceding word* might 

 mom fit to include every organised mode of dealing with the variation* 

 of value which algebraical quantities are made to receive ; the differ- 

 ential calculus, fur example : but they have a technical meaning, which 

 we proceed to explain. When a quantity is subject to one sort of 

 variation only, the consideration of that variation belongs to the simple 

 differential calculus: but when it U subject to two ..r more distinct 

 sorts of variation, suppose that of the differential calculus and another, 

 then the mode of dealing with the second sort of variation ia said to 

 Ix-long to the calculus of variations. In dynamics, for example 

 [ VIHTI-AL VELOCITIES], there are two distinct species of motion to 

 consider : one which, at the end of the time I, the system is about to 

 jtake during the ensuing time dt, in consequence of the velocities 

 acquired by its particle* ; and another which, without any considera- 

 i the first, must be impressed upon it for the examination of the 

 conditions which express the equivalence of the impressed and effective 

 force*. Here then is a case for the calculus of variations. 



Suppose a curve AH, with which is connected another, ab, infinitely 

 near to the first, and related to it by a given law, in such manner that 

 nny point r being given on the first, a corresponding point p can be 

 i.'iiii.l on the second. If the coordinates of p be x and y, and those of 

 u. (infinitely near to v) be x + dx and y + <ly, and if we signify the co- 

 ordinates of p by x + &r and y + Sy, we have two distinct notations, one 

 i'r the increment* which the coordinates receive in passing from point 

 t.. joint on the first curve, the other for those which they receive in 

 passing from a |>iut on the first curve to the corresponding point on 

 the second. Hence, p R being (/.r, and j- what dx becomes after varia- 

 tion, we have J(tAr)=^r-ru which ia obviously equal to <JN PM. 



But p u U Zs, and Q N in what 3.<- becomes wlicu s is cliangc.I into 

 X t <Lr, whence o, n pli = (/ (ir) ; orklx dls, and the some may bo 

 proved for y. We shall now recapitulate the results of the further 

 application of this method. It is quite beyond our limits to attempt 

 to prove them ; so that, referring to works on the differential calculus 

 for further information, we shall content ourselves with some remarks 

 on the loose manner in which this calculus is neatly always appliul 

 to questions of maxima and minima, and to a very few words on iU 



: 



1. The operations of differentiation and variation are interchangeable 



in order, as in MX dtx, tj vtir =J"t(\dx), Ac. 



2. If y be a function of x, and if if, y", to., stand for successive 

 differential coefficients of y with respect to x, the successive differential 

 coefficient* of ty y ' ir are J/ y'tjc, Sy " y'"tjc, Sy"' y ' is, &c. 



3. Let V be a function of x, y, y', y m , Ac., and \eif\-dx taken from 



*-aV,tox = xbe required, and let y , y",,, y" a , Ac., and y,', yY, y", , 

 Ac., be tho value* of y, y 1 , y", Ac., when x = J and x = x, : and let 

 moreover vty y'Sx, which becomes u a and >, at the two limits. 

 I>et the differential coefficients of V with respect to x, y, y', y", Ac., 

 separately made variable, be x, T, P, <j, Ac., and let the complete 

 diOerantiations of these with respect to x be denoted by accentuations, 

 and their limiting values by subscript ciphers and units as before : 

 then we shall have for iftdx the following formula : 



v i *, - v u *> 



+ (P, - o/, + H-, _ Ac.) , _ <p _p/ + B" -Ac.) 

 + (Q, - *', + ",-Ac.) /,-(%- ' + "- Ac.) <, 

 + (R, - ', + T", -Ac.) ', - (E.- ' + t" u - Ac.) " 



f ' (Y - i' -r q" - n" -r Ac.) tfitx, 

 Tlio uiot usual application of the preceding formula, in itu most 



' geometrical form, U as follows : v being a given function of 

 *. y> *" * c -. '< w required to draw a curve such that/* vdx shall be the 

 greatest possible or the least possible, provided that at one limit i 

 Integration x,, and y shall be coordinate* of one given curve, and that 

 at the other limit x. and y, shall be coordinates of another given curve, 

 Such a case arises when it is required to draw the shortest line between 

 two given curves, or to find in what form and position a flexible cnrxv 

 of given length will rest when its ends are supposed to slide upon given 

 curves. We have pointed out ( Differential Calculus, ' Library of 

 Useful Knowledge,' ch. xvi.) that the ordinary mode of treating tin 

 questions is not sufficiently general, and must in certain cam-- 

 lead to positive error. \\ o intend here to enforce this conclusion I y 

 showing that even in more ordinary questions of maim. and minim.i 

 the same want.of generality may lead to the same sort of fal.- 

 clusion. 



A maximum, or greatest value, mean* one which is greater than any 

 neighbouring value ; so that when a function is at its maximum, any 

 allowable slight change must be one of diminution. For greater read 

 less, and for diminution increase, and we have the definition of a 

 minimum. Now an ordinary question of maxima and minima is as 

 follows : $ix being a function of x, what are the real values of x which 

 make it a maximum or minimum ? There is a maximum when x = o, 

 provided that <f (< + A) and <t> (a A), when both are possible, ai 

 less than $o : but if one of the two <t> (a + A) and <t> (u A) be impos- 

 sible, there ia a maximum if Itatk raltuci of the other be less than <pa. 

 In all these coses it is supposed that h may be as small as we please. 

 Now 



1. When if (o + A) and <t> (a- A) are both real, the theory explained 

 in MAXIMA AND MINIMA ia perfectly sufficient : there is a maximum 

 when 4>'x changes from positive to negative in i>assiug through }>', and 

 there is not a maximum in any other cage, 



1. When $> (a + A) is impossible, there ia a maximum if both values 

 of if'x be positive from x=a A up to x=a : when ^ (o A) ia 

 impossible, there ia a maximum if both values of <f'x be negative from 

 x a to x = a + A. 



It is the neglect of the second case which has led to the oversight 

 in the calculus of variations which we shall presently mention. Wi- 

 shall now propose a case as follows : It is required to find the maxi- 

 mum value of y in the equation 



y = (l x)' + x* = ^x. 



The form of the curve which has this equation is as in thia diagram ; 

 o being the origin and o A (= A p) being unity. 

 Now it ought certainly to bo said that A p is tho 

 greatest ordinal* of the curve, but neither is <f'.r 

 here equal to nothing, nor does it change sign. In 

 fact when x = 1, we have x = 1, Q'x = 1'6. Tho 

 second criterion shows that A p in a maximum ; tho 

 firat shows nothing of the kind. 



Now we can easily imagine it said, that in such n 

 case as the preceding, A P, though unquestionably o 

 the greatest ordinate the curve con have, ia not what 

 is technically called a maximum : but it is meant 

 that the last term should be restricted solely to 

 denote those values of <f.c in which <t> (x + A) and 

 (x A) are both pouiole, and both less than pc. 

 To thia, ccltrit pariliia, there could be no objection : 

 it often happens that the technical use made of a 

 foreign term will not bear, and is not meant to 

 bear, translation into our own language. The word / , , n i:i 



its widest allowable use, and if all we ask for should be granted, will 

 not answer to yrcalttt : for there may be several maxima and minima, 

 and some of the minima may be greater than some of the maxima, 

 which cannot be true of the words when translated. Suppose, thru, 

 that the word maximum is so restricted as to apply to no value of tf-r 

 except when <f> (x + A) and <t> (x-A) are both possible : the disadvantage 

 will be twofold. First, in every problem of maxima and minima, or 

 in every problem which is reducible to one of maxima and mini 

 shall have to invent an additional term to fcU'liify. perhaps, t) 

 greatest or very least value of the function. Secondly, in applying tho 

 same limitation to tho calculus of variations we shall frequently be 

 obliged to forego tho solution of which we are in search, unless we 

 look for the very case, as an answer to a problem of maxima and 

 minima, to which we have previously refused to apply the term 

 maximum or minimum. 



In order to nuke/vtAr, as before described, a maximum, it is 

 generally presumed that !/v</x must=0, and that y must be found in 

 terms of x from this condition. Now the truth is thaty \dx, after the 

 variation, becomes Ji&x + 8/vrfx, and all that is absolutely 

 necessary is that 4/vrfx should be always negative, for all values of 

 !.< and ! v between the limits, and for all values which are con- 

 sistent with the limiting condition*, at the limits. It is oaaily 

 -how n that this requires, as to the indefinite integral port, the follow- 

 ing equation : 



_ __/._ n 11 '* 0* 



