CALCULUS OF VARIATIONS. When two points, A and B, are given in a plane, as shown in fig. 1, there is an infinity of arcs which join them. A simple problem of the calculus of variations is that of find ing in this class of arcs one which has the shortest length, the solution of the problem being of course a straight line segment. But we may also seek to find in the class of arcs joining A with B one down which a particle, started with a given initial velocity v, will fall in the shortest time from A to B ; or we may ask which one of these arcs, when rotated about the axis Ox, will generate a surface of revo lution of smallest area. These are typical problems of the calculus of variations of the so-called simplest type.
The notations which gave rise to the name, the calculus of variations, were originated by Joseph Louis Lagrange about the year 1762 and are still in use, though at the present time there is a tendency to replace them by others. If T represents. the time of descent of a particle falling along an arc E, then to get the cor responding time for a neighbouring arc E' a correction must be added to T. This correction is called a variation of ST and is designated in Lagrange's notation by ST. Similarly the vertical distances y in fig. 1 corresponding to the various points of E must be corrected by variations Sy in order to get the corresponding vertical distances for E'. The problem of finding an arc E such that the variation S T will be positive for all choices of the varia tions Sy is the problem of the curve of quickest descent men tioned in the last paragraph. Since the time of Lagrange the theory of such problems has been called the calculus of variations.
The Shortest Line from a Point to a Curve.—Some of the properties of mini mizing arcs of the calculus of variations are well illustrated by the problem of deter mining the shortest line joining a fixed point A with a fixed curve C, shown in fig. 2. Evidently the solution of the prob lem must be a straight segment AB, and it can further readily be seen that AB must cut the fixed curve C at right angles. For if C were in the position C' shown in fig. 2 the line AQ would evidently be shorter than AB.
It might be concluded that a straight line AB perpendicular to C is actually the shortest line joining A to C, but this is not always true. It is known that the straight lines cutting C at right angles are all tangent to a curve D, as shown in fig. 3, one of whose prop erties is that the length of the composite arc PQR is always equal to the length PB, whatever the position of Q at the right of P on D. This is the well-known string property of the curve D, so called because it means that a stretched string of length PB, at tached at P and allowed to wrap itself around the curve D, will describe the arc C with its movable end R. The length of the com posite arc APQR is equal to that of AB, as has just been indi cated, and the length of APQR will be less than that of AB when the curved arc PQ is replaced by a straight line. Evidently AB is not the shortest arc which can be drawn from A to the curve C if its point of contact P with the envelope D lies between A and B.
It is conceivable that other properties might be required of the segment AB in order to insure its minimizing property. But it can be proved, when the point A lies to the right of P in fig. 3, that there is a neighbourhood of AB in which that line is shorter than any other arc joining A with C.
A circular wire dipped in a soap solution and then withdrawn will have a disc of soap film stretched across it. If a smaller cir cular wire is made to touch this disc and is then withdrawn to a position shown in fig. 4 a film will be stretched between the two wires which is a surface of revolution about the common axis of the two circles. It is found by experiment that when the circle B is moved away from A in the direction of the dotted axis a posi tion is presently reached at which the film always becomes un stable. It contracts at the waist and separates into two plane discs through the two circles.
The determination of the shape of the soap film described in the last paragraph gives rise to a perfectly definite mathematical problem. It is that of finding among the arcs joining A with B, in the plane of the paper in fig. 4, one which when rotated about the dotted axis will generate a surface of revolution of smallest area. The curve which solves the problem is described by a mathematical equation, but its shape is that of a chain whose ends are fast but which otherwise hangs freely. The mathematicians call such a curve a catenary from the Latin word catena meaning a chain.
The catenaries of the problem which pass through the point A have an envelop ing curve D as shown in fig. 5. The critical - - point on one of them, beyond which its minimizing property ceases, is its point of tangency P with the envelope. A very interesting property of the curves in fig. 5 is that the surface of revolution generated by the composite arc AQP is always equal to that gen erated by the catenary AP whatever the position of Q below P on the envelope. This is the analogue of the string property of the envelope of the straight lines perpendicular to a curve, and an argument similar to the one described for that case shows that the area of the surface of revolution generated by a catenary arc APB can never be a minimum area, since the areas of the surfaces generated by the arcs AQPB are all equal to it.
When the point B lies above the envelope D there are always two catenary arcs join ing it to A, as shown in fig. 5, one of which has a critical point P on it. The other is the one which can be proved to furnish a minimum area. When B lies in the position B, then there is no catenary arc joining the points A and B and the minimum surface of revolution consists of the two discs gen erated by the broken line When the circular wire B in fig. 4 is moved away from A the catenary arc AB varies from one to another of the catenary arcs through A shown in fig. 5. The moment when the film decomposes into two discs, like those generated by the broken line is the moment when B reaches the enveloping curve D.
The isoperimetric problem of the Greeks imposes an additional property besides closure on the arcs of the class in which the mini mizing arc is sought, namely, that all of them shall have the same length. Similar restrictions may be put upon the class of arcs joining the points A and B described in the first paragraph of this article.
In the class of closed surfaces enclosing a given volume the one which has a minimum surface is a sphere. This agrees with the commonly observed fact that a soap bubble, enclosing a given volume of air, has spherical form. If, on the other hand, we turn the problem around and search in the class of closed surfaces having a given area, one which encloses a maximum volume, the solution is again a sphere. Every isoperimetric problem has associ ated with it in this way another one of the same sort and having the same solution surface or curve.
The problems which have been described above are a few only of the large variety with which the calculus of variations is con cerned. By the use of the devices of mathematical analysis the scope of the theory has been steadily enlarged. Great progress has been made, for example, in generalizing the character of the re strictions upon the classes of curves and surfaces in which mini mizing elements are sought, as compared with the restrictions which have long been imposed for isoperimetric problems, and in generalizing the variable quantity dependent upon the curves or surfaces whose maximum or minimum value is to be obtained. The questions which have arisen are many of them only imper fectly answered and it is apparent to a student of the subject that the theory of the calculus of variations will always be a lively and growing one.
All of these were special problems solved by special methods. The foundations for a general theory were laid by Jacques Ber noulli in his solution in 1697 of the problem of the curve of quick est descent which had been proposed by his brother Jean. Leon hard Euler saw that the methods of Jacques Bernoulli were widely applicable and he deduced in the first general rule for the characterization of maximizing or minimizing arcs of the calculus of variations. In 1760-62 Lagrange devised the notations for vari ations which have given the theory its name, and greatly simpli fied and extended the results of Euler. In 1786 Adrien Marie Legendre studied for the first time what is called the second vari ation of the quantity to be minimized and found a criterion for distinguishing between maxima and minima. In his reduction of the second variation Legendre used a transformation which could not be justified in all cases. The difficulty was analysed in 1838 by Carl Gustav Jacob Jacobi, who in so doing discovered the ex istence of the critical point P beyond which the minimizing prop erties of an arc will fail.
The modern period in the development of the theory of the calculus of variations is characterized by much greater precision in the formulation of problems and in the methods and reasoning applied to their solution. It began with Karl Weierstrass, who in the decade preceding 1879 saw that one might continue indefinitely to seek for properties of minimizing arcs unless a proof could be made that a suitable set of properties would actually insure the minimum. He had himself found an important new necessary condition for a minimum which cannot easily be described here in non-technical language. He succeeded in proving in very in genious fashion that certain characteristics of an arc are suffi cient to insure its minimizing property. Between 1894 and 1898 the string property of the envelope of the normals to a curve was generalized for shortest lines on a surface by Jean Gaston Darboux, and for more general problems by E. Zermelo and Adolf Kneser. In 1899 David Hilbert stated an existence theorem which asserts that under certain circumstances a minimizing arc will surely exist. The theorem has been reproved more simply by other writers and extended by Leonida Tonelli, who in 1921-23 made it the basis for a new approach to the theory of the calculus of variations. The scope of the theory of the calculus of variations has been greatly extended by Adolf Mayer and Oskar Bolza, who formulated in 1878 and 1913, respectively, problems of very great generality to which the methods of the theory are applicable. For further historical data and accounts of the views of modern mathematicians on the subject, the reader should consult the references in the Bibliography.
a relatively elementary introduction to the calculus of variations one may read Edouard Goursat, Cours d'Analyse Mathematique, vol. iii. 3rd ed. pp. 545-660 (Gauthier-Villars 1923), or Gilbert Ames Bliss, The Calculus of Variations (Open Court Publishing Co., Chicago, 1925) . The most thorough presentations of modern theories are 0. Bolza, Lectures on the Calculus of Variations (University of Chicago Press, 1904), and Vorlesungen fiber Variations rechnung (Teubner, Leipzig, 5909) ; Jacques Hadamard, Lecons sur le Calcul des Variations (A. Hermann, 1910) ; and A. Kneser, Lehrbuch der Variationsrechnung, 2nd ed. (Vieweg, Braunschweig 1925) . L. Tonelli, Calcolo delle Variazioni (Zanichelli, Bologna, 1921, 1923), makes existence theorems the basis of a new attack upon the theory requiring analysis of the most modern sort.
For an excellent synopsis of the development of the theory of the calculus of variations with elaborate references see two articles by A. Knesee ; and by E. Zermelo and H. Hahn, Encyklopadie der Mathe matischen Wissenschaften, II. A 8 (1900), II. A. 8a (19o4). These have been translated into French and amplified importantly by Maurice Lecat Encyclopedie des Sciences Mathematiques, II. 31 (1913, 1916), pp. 1-288.
Books dealing with the history of the subject are M. Cantor, Geschichte der Mathematik, vols. i.-iv. (Teubner, Leipzig, 1892-1908) ; and I. Todhunter, A History of the Progress of the Calculus of Variations during. the Nineteenth Century (Macmillan, 1861) , treating the period from 1760 to 186o. An interesting historical sketch is contained in Pascal, Calcolo delle Variazioni (Hoepli, 1897), which has also been translated into German by Adolf Schepp (Die Variations rechnung (Teubner, Leipzig, An extensive bibliography has been published by M. Lecat, Bibliog raphie du Calcul des Variations, depuis les origines jusqu d 1850 (Hermann, 1916) ; 1850-1913 (ditto, 1913) . See also the additions in his Bibliographic des Series Trigonometriques (M. Lecat, 1921), p. 155; and in his Bibliographic de la Relativite (M. Lamertin, 1924), Ap pendix, p. 15. A brief bibliography of the principal treatises is given by G. A. Bliss in the book mentioned above. Two new books whose names do not appear there are G. Vivanti, Elementi del Calcolo delle Variazioni (Principato, Messina, 1923) ; and A. R. Forsyth, Calculus of Variations (Cambridge University Press, 1927) . (G. A. BL.)