Calculus of Variations

VARIATIONS, CALCULUS OF. The preceding words might seem fit to include every organised mode of dealing with the variations of value which algebraical quantities are made to receive; the differ ential calculus, for 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 is subject to two or more distinct sorts of variation, suppose that of the differential calculus and another, then the mode of dealing with the second sort of variation is said to belong to the calculus of variations. In dynamics, for example [I'iuTum. VELOCITIES], there are two distinct species of motion to consider : one which, at the end of the time I, the system is about to ,take during the ensuing time di, in consequence of the velocities acquired by its particles ; and another which, without any considers :Hon of the first, must be impressed upon it for the examination of the ronditions which express the equivalence of the impressed and effective lerces. Here then is a case for the calculus of variations.

Suppose a curve A II, with which is connected another, eh, infinitely near to the first, and related to it by a given law, in such manner that any point r being given on the first, a corresponding point p can be Tumid on the second. If the coordinates of r be x and y, and those of q (infinitely near to r) be x + dx and y + dy, and if wo signify the co ordinates of p by x+ ax and y + ay, wo have two distinct notations, one for the increments which the coordinates receive in passing from point to point on the first curve, the other for those which they receive in passing from a point on the first curve to the corresponding point on the second. Hence, P n being dx,andpr what dx becomes after eerie thou, we have b(dx)=pr —r it which is obviously equal to q N—r am.

But r u is ix, and GI N is what ax Locoman when x is changed into x dX, whence it a — P II= (bx); or tiehr= dbx, and the same may be 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 nearly always applied to questions of maxima and minima, and to a very few words on its history.

1. The operations of differentiation and variation are interchangeable In order, as in ddx = dbx, vdx fb(r d.r), &c.

2. If y be a function of x, and if y', y", &c., stand for successive differential coefficients of y with respect to x, the successive differential coefficients of by — y' ax are ay —y.a.r, ay.— y'"bx, — yi'bx, &c.

3. Let v be a function of x, y, y', &c., and let/vd.r taken from x es- to x = x, be required, and let y,„ y"„, &e. and y:, &c., be the values of y, y, , &c., when x x,„ and x x, : and let moreover co ss ay — fax, which become! te„ and (0, nt the two limits. Let the differential coefficients of v with respect to x, y, y', separately made variable, be X, T, P, 14, &c., and let the complete differentiations 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 bfvds the following formula : v', bx,— + — + — &c.) m, — + R",, + (Q, — a', + s",— &e.) (Q„,— n'„ + s"„— + (a, — s', + T", —dm.) — + T",,— 14c.) cf.

(T — + 4" — + &e.) todx.

The most usual spplicatiou of the preceding formula, in Ito most general geometriol form, is as follows :—v being a given function of y, ys, fie., It is required to draw a curve such that/yds shall be the greatest possible or the least possible, provided that at one limit of integration and y,„ shall be coordinates of one given curve, and that at the other limit x, i and be , shall coordinates of another given curve, Such a case arises when it is required to draw the shortest hne between two given curves, or to find in what form and position a flexible curve of given length will rest when its ends are supposed to slide upon given curves. We have pointed out (Differential Library of Useful Knowledge,' ch. xvi.) that the ordinary mode of treating these questions is not sufficiently general, and must in certain eased even lead to positive error. We intend here to enforce this conclusion by showing that even in more ordinary questions of maxima and minima the same want of generality may had to the same sort of false con clusion.