Page images

similarly dx dy dz is the volume-element referred to rectangular coordinates, and is independent of the particular form of the bounding surface.

Now these and similar general expressions for infinitesimal elements are made the subjects of investigation; and we calculate their variations according to processes which will be developed hereafter. If an integral function is the subject of inquiry, it is considered as the integral or sum of elements; and to this sum we apply the conditions, so far as they are applicable, for determining the unknown function. By this artifice we avoid the difficulty of making to vary the function in its general form. Thus, for instance, in the problem of finding the shortest line between two given points, (ans Yn) and (xo, Yo); instead of assuming the distance to be F(Xm, Yn, Xo, yo), where F represents some undetermined function, and then determining F by equating to zero the change of the distance due to the variation or change of form of F, we assume ds = {dx2 + dya}to be an element of the distance, so that the distance =* ("{dx2 +dy}t; and we make the latter sum the subject of investigation.

And in the most general case; suppose that we have to investigate the form of the relation y=f(x), where f is the symbol of some unknown function, so that a given condition should be satisfied, when that condition is the sum or integral of a series of elements, each of which is a given function of x, dx, dx,...d"X, y, dy, dạy,...d"y, neither x nor y being equicrescent; then, if the element = F(x, dx, dax, ...d"x, y, dy, dạy,...d"y), where F represents a given function, and if x1, 91, X, Yo, are the given limiting values of x and y, the unknown function

= | F(x, dx, dạx, ... d"x, y, dy, dy, ... d"y); (6) and the relation which exists between y and x, that is, the form of f, is determined by means of conditions to which (6) is subject. A similar method is applicable if the element of the unknown function involves more variables and their differentials. · Now the principle by which these and similar problems are solved is of the greatest importance; and is that of which the calculus of variations is the development. It may be formally stated in the following terms;

* Instead of expressing the limits of integration at length, we have merely put their subscript letters, as we have already explained in Art. 247.

[ocr errors]

If a quantity depends on a certain unknown function, and the determination of the form of that function depends on certain given conditions, which it has to satisfy; in the general case our knowledge of functions and of their laws is insufficient for the determination of the function, and especially when the conditions require an infinitesimal variation of it: but as the form of many infinitesimal elements is known, and is the same whatever is the unknown continuous function, we may consider the quantity which depends on the function to be the sum of certain elements between given limits, and may make the quantity in its latter form the subject of inquiry.

295.) When the problem has been put into the above form, the following is the most convenient method of effecting and of symbolizing the necessary operations : the unknown function is made to assume a new form by an infinitesimal change of the variables and their differentials which are involved in the given element-function, the infinitesimal variations being functions of the variables to which they are applied ; and as hereby the element-function will have changed value, so will also the sum of all these; and as these infinitesimal changes are not made subject to the conditions of an original given function, they may be, and generally will be, inconsistent with it, and thus a new law will be introduced which will be expressed by a new functional symbol. Or to employ the language of geometry : suppose a certain curve to be expressed by the undetermined function : and suppose each point of the curve to be shifted, and thereby each of the length-elements and each of the successive differentials to change value; the curve in its new position will generally have taken a new form, and so will require a new function to express it. Thus, suppose the curve under consideration to be a curve of double curvature, and let the position and form of it be changed ; then if ox, dy, ôz are the variations of the coordinates, these being functions of x, y, z, the point (x, y, z) becomes (x+ox, y+oy, 2+ôz); observe then the change; the point on the old curve infinitesimally near to (x, y, z) is (x + dx, y + dy, 2+dz), whereas *+8X, y +øy, 2+òz refer to the same point as x, y, 2, but to it in a new position, and on a new curve, and when the form of the function has varied. Similarly also 8.dx, 8.dy,, 8.dx,...8.d"X express variations which the several successive differentials un


[ocr errors]
[ocr errors]

dergo, and which are due to the change of the form of the function.

As however the element-function which will be the subject of variation will generally contain the differentials of the variables as well as the variables themselves, it is necessary to consider with great precision the relations between variations and differentials. And as the operations of which these infinitesimal quantities are the results are evidently of the same nature though of different species, the order in which they are effected on a given subject is evidently indifferent; and consequently they are subject to the commutative law; so that we have

8.dx = d.dx ;
8.dx = d.8.dx = .8x ;

[ocr errors]

8.d"y = d".df (w). And similar results are also true for other variables.

Now as d denotes an operation subject to the index law, and which is true for negative as well as for positive values of the index, the results of the operations effected on x and on f (x) being true for positive integral values of n will also be true for negative integral values; that is, as they are true for differentiation, so will they also be true for integration. Thus

8.s dx = fo.da ;
8.[svydo = fo.flwyda ;

(9) 0.85862, y)dy dx = Sfos«,y) dy dx. ] Similar results are also true for successive variations; so that we have generally

god" f(x) = dugmf(x); gm. SI... V.... dz dy dr = SI....".( dy dr). S

ļ (10)

[subsumed][ocr errors]

occur; and the change of value in the integral, of which these are the limits, is the sum of the quantities which have been determined in (86) and (87), Art. 96. If the limits are fixed, there are of course no variations of them; and if the limits are constrained to be on certain curves, their variations are not arbitrary, but must be in agreement with the equations to these curves. This illustration indicates how the total variation of a quantity expressed in the form of a definite integral involves partial variations due to changes of the limits, as well as of those due to the change of form of the function.

297.] Problems within the range of this calculus may involve either a single infinitesimal element, or the integral of such elements between given limits of integration. In the latter case a finite function is the subject of investigation; but as it is, and is expressed as, the sum of a series of infinitesimals, and as the order in which the operations of variation and of integration are effected is indifferent in the result, here also the infinitesimal elements are the first subjects of investigation. It is necessary therefore in the first place to investigate the effects of the operations of the calculus of variations on these elements; and I shall take those which have arisen in the preceding chapters.

Let (x, y, z) (x + dx, y+dy, z+dz) be two points in space infinitesimally near to each other; and let ds be the distance between them ; so that

ds2 = dx2 + dy? + dza. Let (x, y, z) be shifted to (x + 8x, y + oy, z+öz), where 8x, dy, 8z are arbitrary functions of x, y and z; so that the displacement of the point (x, y, z) may be most general; then, taking the variation of (14) according to the theorem given in (5), we have

ds 8.ds = dx 8.dx + dy 8.dy+dz ;

.:. .ds = 4.dx + 5% 8.dy + p ; (15) which is the total variation of ds, and shews that it is equal to the sum of the projections on the tangent to the curve at (x,y,z) of the several variations of the coordinates.

The following however is another proof of this theorem, and as it involves quantities which will be of great use in the sequel, it is desirable to insert it at once. Let (f, n, 5) be the varied place of (x, y, z); so that

§ = x +oX, = y + Øy, Ś = 2 + oz.



« PreviousContinue »