CALCULUS OF VARIATIONS. CHAPTER XIII. EXPOSITION OF THE PRINCIPLES OF THE CALCULUS OF VARIATIONS. 290.*] The subjects of investigation in the preceding parts of our treatise have been functions whose forms are known and determinate; such as those symbolized by cos, tan-1, log, log-, * The authors and titles of the principal works on this branch of infinitesimal calculus are the following, and from them much assistance has been derived : Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne, 1744. Lagrange, Leçons sur le calcul des Fonctions, Paris, 1808. Lagrange, Théorie des Fonctions Analytiques, 3me edition, par J. A. Serret, Paris, 1847. Poisson, Mémoires de l'Institut de France, Tome XII, Paris, 1833. Ibid. Tome III. Sarrus, Mémoires presentés par divers savants à l'Académie des Sciences, Stegmann, Lehrbuch der Variationsrechnung, Cassel, 1854. Todhunter, History of the Progress of the Calculus of Variations ; Cambridge, 1861. Moigno et Lindelöf, Calcul des Variations ; Paris, 1861. and other such like: and the inquiry has been for the most part confined to the properties of these functions, which arise from the continuous and infinitesimal variation of their subject-variables ; we have had no occasion to consider the functions themselves as undergoing continuous change as to form; certain invariable relations have been shewn to exist between certain functions ; for by the process of derivation we pass from one function to another; but these are nevertheless determinate, and the relation arises from a continuous growth of the subject-variable, and not from a continuous and infinitesimal change of the function as to form. This distinction is important; for there is no conceivable reason why functions should not be continuously variable as to form, as well as numbers be as to magnitude. Thus for instance suppose the subject of investigation to be y = sin x; the value of y may manifestly be changed either by a change in the subject-variable x, or by a change of the functional symbol into any other, as tan-?; changes due to the former cause are considered in the Differential Calculus; but those arising from a continuous change in the form of the function require another mode of investigation; and whereas heretofore we have passed per saltum from one function to another, the new calculus requires a continuous passage: a wide extension then is opened before us, one the subject-matter of which is not number but functions: and as a functional symbol expresses the law of combination of its subject-variables, we shall have to consider laws, and not subjects of laws. Functions then, as they are the subject of this new calculus, are free from all concrete or applied signification, and express laws; and the proper end and object of such a calculus of functions is to investigate their origin and their principles, their growth and extent, their laws of combination, and to deduce from them properties with which they are pregnant. As differential calculus investigates properties of continuous number, so in this new calculus properties of continuous functions are discussed. 291.] Apart however from these general considerations, let us view the calculus in the light of an easy problem of that class, the attempt to solve which gave rise to it. Suppose that it is required to determine the form of the function connecting x and y which expresses the shortest distance between two given points: if the function were given, the problem would be one of rectification and would be solved by the integral calculus : also a posteriori we know that the required function is that which expresses a straight line : but the direct solution of the problem requires a different process; viz. the assumption of a general functional symbol undetermined as to form, and the expression for the distance between the two points in terms of it; so that if an infinitesimal variation of the distance due to an infinitesimal variation of the form of the function is calculated, the required form will be determined by equating to zero that variation : provided that the form so determined is such as to make the first variation change its sign: or what is equivalent, such as to make the second variation either positive or negative for all values of the determined function within the given points : for such an operation it is necessary (1) to calculate the infinitesimal change of the distance due to the infinitesimal change of the form of a function, (2) to be able to determine the form of the function by equating to zero the variation of the distance; in other words, we must be able to differentiate functions as to form, and to determine functions by means of given conditions; also if these conditions give many results, we must be able to discriminate according as one or another is taken. Such a process then requires a knowledge of functions as accurate and complete as that of number required in the differential calculus. It will be observed that, as the two points which are to terminate the line are given, the only variable quantity of the problem is the form of the function. Suppose however that the problem is, to determine the form of the function which expresses the shortest distance between two given curves in space; let the distance be expressed by means of a general undetermined function, as in the former case, and in terms of the current coordinates of the two curves which it is to meet; then it becomes dependent on the form of the function, and on the coordinates of these two curves; and as these quantities are independent of each other, they may be considered as independent variables, and their variations may be taken separately; that arising from a change in the form of the function may be estimated as in the former case, and thence may be deduced the form that gives the least distance: and those which arise from the coordinates of the points on the given curves at which the required curve is to meet them must be calculated according to the rules of the differential calculus, and by equating them to zero we shall be able to determine the points of meeting. In the solution of this problem therefore two kinds of variations will be required, one arising from a change in the form of the function, and the other from the differentiation of the equations of the given curves. 292.] The infinitesimal variations therefore of the calculus of functions and of the differential calculus are essentially distinct in kind : in the former they result from a change of form of an undetermined function; in the latter from a change of the subjectvariables of a determinate function: and to use language borrowed from the geometry of curves, a variation of the former kind leads from a point on a curve to a point on another curve infinitesimally near to it; a variation of the latter from a point on a curve to a point on the same curve infinitesimally near to it. It is convenient therefore to have different names for quantities so different, and to express them by different symbols : in the former calculus they are called variations, in the latter differentials : hence arises the name " calculus of variations,” and so henceforth we shall employ the term “variation” in a technical sense, to indicate the particular infinitesimal change of this calculus : we also shall use d to express differential, and 8 to express variation : consequently d indicates a passage from one system of variables to another, both of which satisfy a given determinate function; è indicates a passage from a system which satisfies one function to a system satisfying a function infinitesimally different from the former one: thus a variation as applied to a function may be defined as the infinitesimal change of the value of the function due to its change of form ; and variation as applied to a variable is the infinitesimal arbitrary increment of it. 293.] The symbols in relation to their subjects stand as follow: let u be an undetermined function; then ou is the change of value of u due to its change of form. Now let a certain operation symbolized by F be performed on u; it may be differentiation or integration ; and let then 8v = 8.F(u); (1) and 8v is the change in v, = F(u), due to a change of form of u. As in the differential calculus there are partial and total differentials of functions of many variables, according as one or all of the variables change value; so if a function, whose variation is to be calculated, involves many undetermined and independent functions, it is susceptible of different variations according as one or more or all of these undetermined functions vary, and therefore in the present calculus there will be partial and total variations ; and, by the principle of such infinitesimal changes, the total variation is equal to the sum of the several partial variations. Thus let 47, 4a, ... Un denote n undetermined functions, and let denote an operation performed on a certain combination of them; and let v = P(U7, U2, ... Un); (2) then Öv =.) ou + (W) ous+...+(ou. Jouni (3) using brackets to denote partial variations. Now, and this remark is important, so long as the relation between F and u, remains the same, the ratio of the infinitesimal changes of F and U, must be independent of the particular species of them, that is, must be the same, whether the changes are of magnitude or of form; and consequently (4) and similarly for the others ; and thus (5) whence it follows that the variations of finite quantities and of finite functions follow the same laws as the differentials of similar quantities. 294.] Thus far as to the general principles of the Calculus of Variations : we proceed to investigate methods by which it may be applied to the solution of problems which are of the greatest importance in the present state of mathematical science, and which the Differential Calculus fails to solve. Of functions in their integral and determinate forms our knowledge is too scanty for the attainment of the present object; but there are certain general expressions for infinitesimal elements, independent of the functions of which they are elements, and therefore the same for all, provided that the functions satisfy the law of continuity within the range for which they are considered; thus ds={dx2 + dy%}# is the distance between two points (x, y) (x +dx, y+dy) on a plane curve, whatever is the form of the function y = f(x), which is the equation to the curve. Thus also {dyo dz+ dz2 dx? + dæ2 dy?) is the surface-element, whatever is the form of f(x, y, z)=c, which is the equation to the surface: |