Part 9 (2/2)
METHOD OF LAGRANGE.
Lagrange, in endeavouring to bring all the different problems of isoperimeters to depend upon a common a.n.a.lysis, organized into a distinct calculus, was led to conceive a new kind of differentiation, to which he has applied the characteristic d, reserving the characteristic _d_ for the common differentials. These differentials of a new species, which he has designated under the name of _Variations_, consist of the infinitely small increments which the integrals receive, not by virtue of a.n.a.logous increments on the part of the corresponding variables, as in the ordinary transcendental a.n.a.lysis, but by supposing that the _form_ of the function placed under the sign of integration undergoes an infinitely small change. This distinction is easily conceived with reference to curves, in which we see the ordinate, or any other variable of the curve, admit of two sorts of differentials, evidently very different, according as we pa.s.s from one point to another infinitely near it on the same curve, or to the corresponding point of the infinitely near curve produced by a certain determinate modification of the first curve.[11] It is moreover clear, that the relative _variations_ of different magnitudes connected with each other by any laws whatever are calculated, all but the characteristic, almost exactly in the same manner as the differentials. Finally, from the general notion of _variations_ are in like manner deduced the fundamental principles of the algorithm proper to this method, consisting simply in the evidently permissible liberty of transposing at will the characteristics specially appropriated to variations, before or after those which correspond to the ordinary differentials.
[Footnote 11: Leibnitz had already considered the comparison of one curve with an other infinitely near to it, calling it ”_Differentiatio de curva in curvam_.” But this comparison had no a.n.a.logy with the conception of Lagrange, the curves of Leibnitz being embraced in the same general equation, from which they were deduced by the simple change of an arbitrary constant.]
This abstract conception having been once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems of _Isoperimeters_ to the simple ordinary theory of _maxima_ and _minima_.
To obtain a clear idea of this great and happy transformation, we must previously consider an essential distinction which arises in the different questions of isoperimeters.
_Two Cla.s.ses of Questions._ These investigations must, in fact, be divided into two general cla.s.ses, according as the maxima and minima demanded are _absolute_ or _relative_, to employ the abridged expressions of geometers.
_Questions of the first Cla.s.s._ The _first case_ is that in which the indeterminate definite integrals, the maximum or minimum of which is sought, are not subjected, by the nature of the problem, to any condition; as happens, for example, in the problem of the _brachystochrone_, in which the choice is to be made between all imaginable curves. The _second_ case takes place when, on the contrary, the variable integrals can vary only according to certain conditions, which usually consist in other definite integrals (which depend, in like manner, upon the required functions) always retaining the same given value; as, for example, in all the geometrical questions relating to real _isoperimetrical_ figures, and in which, by the nature of the problem, the integral relating to the length of the curve, or to the area of the surface, must remain constant during the variation of that integral which is the object of the proposed investigation.
The _Calculus of Variations_ gives immediately the general solution of questions of the former cla.s.s; for it evidently follows, from the ordinary theory of maxima and minima, that the required relation must reduce to zero the _variation_ of the proposed integral with reference to each independent variable; which gives the condition common to both the maximum and the minimum: and, as a characteristic for distinguis.h.i.+ng the one from the other, that the variation of the second order of the same integral must be negative for the maximum and positive for the minimum. Thus, for example, in the problem of the brachystochrone, we will have, in order to determine the nature of the curve sought, the equation of condition
d?_{_z2_}^{_z1_}v([1 + (_f'(z)_) + (p'(_z_))]/(2_gz_))_dz_ = 0,
which, being decomposed into two, with respect to the two unknown functions _f_ and p, which are independent of each other, will completely express the a.n.a.lytical definition of the required curve. The only difficulty peculiar to this new a.n.a.lysis consists in the elimination of the characteristic d, for which the calculus of variations furnishes invariable and complete rules, founded, in general, on the method of ”integration by parts,” from which Lagrange has thus derived immense advantage. The constant object of this first a.n.a.lytical elaboration (which this is not the place for treating in detail) is to arrive at real differential equations, which can always be done; and thereby the question comes under the ordinary transcendental a.n.a.lysis, which furnishes the solution, at least so far as to reduce it to pure algebra if the integration can be effected. The general object of the method of variations is to effect this transformation, for which Lagrange has established rules, which are simple, invariable, and certain of success.
_Equations of Limits._ Among the greatest special advantages of the method of variations, compared with the previous isolated solutions of isoperimetrical problems, is the important consideration of what Lagrange calls _Equations of Limits_, which were entirely neglected before him, though without them the greater part of the particular solutions remained necessarily incomplete. When the limits of the proposed integrals are to be fixed, their variations being zero, there is no occasion for noticing them. But it is no longer so when these limits, instead of being rigorously invariable, are only subjected to certain conditions; as, for example, if the two points between which the required curve is to be traced are not fixed, and have only to remain upon given lines or surfaces. Then it is necessary to pay attention to the variation of their co-ordinates, and to establish between them the relations which correspond to the equations of these lines or of these surfaces.
_A more general consideration._ This essential consideration is only the final complement of a more general and more important consideration relative to the variations of different independent variables. If these variables are really independent of one another, as when we compare together all the imaginable curves susceptible of being traced between two points, it will be the same with their variations, and, consequently, the terms relating to each of these variations will have to be separately equal to zero in the general equation which expresses the maximum or the minimum. But if, on the contrary, we suppose the variables to be subjected to any fixed conditions, it will be necessary to take notice of the resulting relation between their variations, so that the number of the equations into which this general equation is then decomposed is always equal to only the number of the variables which remain truly independent. It is thus, for example, that instead of seeking for the shortest path between any two points, in choosing it from among all possible ones, it may be proposed to find only what is the shortest among all those which may be taken on any given surface; a question the general solution of which forms certainly one of the most beautiful applications of the method of variations.
_Questions of the second Cla.s.s._ Problems in which such modifying conditions are considered approach very nearly, in their nature, to the second general cla.s.s of applications of the method of variations, characterized above as consisting in the investigation of _relative_ maxima and minima. There is, however, this essential difference between the two cases, that in this last the modification is expressed by an integral which depends upon the function sought, while in the other it is designated by a finite equation which is immediately given. It is hence apparent that the investigation of _relative_ maxima and minima is constantly and necessarily more complicated than that of _absolute_ maxima and minima. Luckily, a very important general theory, discovered by the genius of the great Euler before the invention of the Calculus of Variations, gives a uniform and very simple means of making one of these two cla.s.ses of questions dependent on the other. It consists in this, that if we add to the integral which is to be a maximum or a minimum, a constant and indeterminate multiple of that one which, by the nature of the problem, is to remain constant, it will be sufficient to seek, by the general method of Lagrange above indicated, the _absolute_ maximum or minimum of this whole expression. It can be easily conceived, indeed, that the part of the complete variation which would proceed from the last integral must be equal to zero (because of the constant character of this last) as well as the portion due to the first integral, which disappears by virtue of the maximum or minimum state.
These two conditions evidently unite to produce, in that respect, effects exactly alike.
Such is a sketch of the general manner in which the method of variation is applied to all the different questions which compose what is called the _Theory of Isoperimeters_. It will undoubtedly have been remarked in this summary exposition how much use has been made in this new a.n.a.lysis of the second fundamental property of the transcendental a.n.a.lysis noticed in the third chapter, namely, the generality of the infinitesimal expressions for the representation of the same geometrical or mechanical phenomenon, in whatever body it may be considered. Upon this generality, indeed, are founded, by their nature, all the solutions due to the method of variations. If a single formula could not express the length or the area of any curve whatever; if another fixed formula could not designate the time of the fall of a heavy body, according to whatever line it may descend, &c., how would it have been possible to resolve questions which unavoidably require, by their nature, the simultaneous consideration of all the cases which can be determined in each phenomenon by the different subjects which exhibit it.
_Other Applications of this Method._ Notwithstanding the extreme importance of the theory of isoperimeters, and though the method of variations had at first no other object than the logical and general solution of this order of problems, we should still have but an incomplete idea of this beautiful a.n.a.lysis if we limited its destination to this. In fact, the abstract conception of two distinct natures of differentiation is evidently applicable not only to the cases for which it was created, but also to all those which present, for any reason whatever, two different manners of making the same magnitudes vary. It is in this way that Lagrange himself has made, in his ”_Mechanique a.n.a.lytique_,” an extensive and important application of his calculus of variations, by employing it to distinguish the two sorts of changes which are naturally presented by the questions of rational mechanics for the different points which are considered, according as we compare the successive positions which are occupied, in virtue of its motion, by the same point of each body in two consecutive instants, or as we pa.s.s from one point of the body to another in the same instant. One of these comparisons produces ordinary differentials; the other gives rise to _variations_, which, there as every where, are only differentials taken under a new point of view. Such is the general acceptation in which we should conceive the Calculus of Variations, in order suitably to appreciate the importance of this admirable logical instrument, the most powerful that the human mind has as yet constructed.
The method of variations being only an immense extension of the general transcendental a.n.a.lysis, I have no need of proving specially that it is susceptible of being considered under the different fundamental points of view which the calculus of indirect functions, considered as a whole, admits of. Lagrange invented the Calculus of Variations in accordance with the infinitesimal conception, and, indeed, long before he undertook the general reconstruction of the transcendental a.n.a.lysis. When he had executed this important reformation, he easily showed how it could also be applied to the Calculus of Variations, which he expounded with all the proper development, according to his theory of derivative functions.
But the more that the use of the method of variations is difficult of comprehension, because of the higher degree of abstraction of the ideas considered, the more necessary is it, in its application, to economize the exertions of the mind, by adopting the most direct and rapid a.n.a.lytical conception, namely, that of Leibnitz. Accordingly, Lagrange himself has constantly preferred it in the important use which he has made of the Calculus of Variations in his ”a.n.a.lytical Mechanics.” In fact, there does not exist the least hesitation in this respect among geometers.
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