Euler and the Calculus of Variations 243 Fig. 3. Figure 2 from De lina brevissima . [E9]. surface of revolution [T, pp. 117-131]. Notice at this time there was no concept of analytic function, and therefore no concept of analytic surface either. For a convex surface Euler gave a simple mechanical solution: fix a string at one point and pull it taut in the direction of the other. Obviously, this method fails in the case of non-convex surfaces and that is why Eu- ler developed an infinitesimal method for general surfaces, whereby in the tangent plane the line GMH composed of two straight lines GM and MH is to be minimized; see figure 3. This method is equivalent to a geometrical theorem on an osculating plane of a geodesic (i.e. extremal space curve on a surface) at a point P , developed but not published by Johann Bernoulli in 1698. This theorem states that the osculating plane intersects the tangent plane to the surface at P at a right angle. Shortest or geodesic lines can be characterized by this property of osculation. Obviously, Bernoulli did not teach this theorem to his disciple, and, more generally, we may infer that in Basel Euler was not yet involved in variational problems. Incidentally, it was a letter of Johann Bernoulli to his son Daniel in 1727 that drew Euler’s attention to this subject. In 1753, Euler used the theory of shortest lines for the foundation of spherical trigonometry, as well as for the extension of such a trigonome- try from the spherical surface to general surfaces [E214,E215]. Thanks to a theorem of Pierre Ossian Bonnet (1819-1892), geodesics on concave sur- LOL-Ch12-P9 of 20 244 R¨udigerThiele faces are always shortest lines. This is not the case in general and required further investigation (Jacobi theory). Moreover, in 1736 Euler published his Mechanica [E15,E16], in which he developed an analytic geometry of space and in which geodesics were characterized by the osculating plane. Moreover, Euler stated that inertial motion follows either straight lines in a plane or, more generally, geodesics on surfaces. In 1738 we have an important milestone on Euler’s way to the Methodus, the paper “Problematis isoperimetrici (Isoperimetric problems taken in the widest sense)” [E27], already written in 1732. Euler’s variational method has its roots in Jacob Bernoulli’s variational process, developed in 1697 in the study of isoperimetric problems. However, where the Bernoulli broth- ers solved specific problems, the disciple Euler mastered and extended the method and ultimately he began to look for a general theory. We see this intention in the methodical procedure he pursued in his investigations, for example he divided the problems into groups based on the side condition and laid down different kinds of variation for each group. Of course, be- cause side conditions depend on the choice of the coordinate system, such a classification is only relative. Quite naturally, having mastered the method of Jacob Bernoulli, Euler generalized the isoperimetric problems Jacob Bernoulli had dealt with, es- pecially those in which arc length also appeared among the independent variables. In Euler’s general investigation of isoperimetric problems, all of the variables x, y, s (s = arc length) enjoyed equal rights, which should not be the case because of the side condition ds2 = dx2 +dy2 and, moreover, be- cause of the fixed length of admissible curves (the isoperimetric condition). In the next paper “Curvarum maximi minimive proprietate gaudientium inventio nova et facilis (New and simple invention of curved lines which enjoy some property of maximum or minimum)” [E56], we will see, Euler partially corrected the mistake. We also find remarks on the independent integral, the integrand of which is a total differential and therefore depends not on the path of integration but only on the endpoints. Hilbert later used such integrals to give a very elegant two- or three-line sufficiency proof for the case of strong extrema - a royal road. 6 Here Euler correctly pointed out that no variational problems emerge from such integrands. Rather interestingly, he maintained that, in the case of a plane, for any differential form Ω = A(x, y) dx + B(x, y) dy 6 Mathematische Probleme, Lecture at the Paris Meeting in 1900, problem 13. In: Nachrichten der Akademie der Wissenschaften in G¨ottingen, 1900, pp. 253-297. En- glish translation by M. Winson in: Bull. AMS, 8 (1901/02), pp. 437-479, reprinted in Bull. AMS (New Series), 37 (2000), pp. 407-436. LOL-Ch12-P10 of 20 Euler and the Calculus of Variations 245 there exists an integrating factor F , an Euler multiplier, which generates a total differential F Ω = Π with dΠ = 0. In conclusion we remark that there is a similar paper by Alexis Claude Clairaut (1713-1765) “Sur quelque questions de maximis et minimis (On several question of maximum and minimum)” [Cl], independently written in 1733 in which the side conditions are delivered by a force field, related to his investigations on the shape of earth. 7 Euler made progress in the process of generalization and in 1736 he wrote the paper “Curvarum maximi minimive proprietate gautentium inventio nova et facilis (New and easy invention to find curves having a maximum or minimum property)” [E56], published in 1741, in which he tried to unify old results. In dealing with 40 problems he sought one general result including his 24 specific cases. As previously noted, he became aware that for some variational problems with constraints he was wrong, but he only noticed this fact after he had already written 33 paragraphs. So he briefly mentioned the fact in the introductory sentences and gave the corrections in detail, but only in the last four paragraphs. This was typical behavior for the busy Euler – the manuscript had probably already been sent to the printer and he only partially remembered the manuscript. Moreover, he had aready begun or was just about to begin the proof-reading of his textbook Methodus [E65]. Among the 24 expressions for the first variations there are only 9 (nos. I-VI, XIII-XIV) which are correct. Between the two papers E27 and E56 (i.e. between 1732 and 1736) Eu- ler was engaged with the Brachistochrone Problem in a resistant medium. He wrote “De linea celerrimi descensus in medio quocunque resistente (On the curve of fastest descent in whatever resistant medium)” [E42], which inspired him to allow even differential equations as constraints. Ultimately, in “Curvarum maximi minimive proprietate gautentium inventio (The find- ing of curves enjoying properties of maximum or minimum),” [E56] he gave corrected results. The Brachistochrone paper “De linea celerrimi descensus in medio quocunque resistente” was published in volume 7 of the Peters- burg Commentarii, incidentally the same volume in which Euler used the notation f(x) for a function f of x for the first time [E44]. In the last pages of E56 Euler made yet another very important remark. There had previously been no doubt concerning a principle used by Jacob Bernoulli in 1697 and later on by others to derive differential equations for solutions of variational problems: If any curve possesses a maximum or minimum prop- erty then each part of the curve (especially any infinitesimal part) enjoys 7 See also Daniel Bernoulli’s report to Euler on such knowledge in Paris, which he wrote there on his journey back to Basel on September 23, 1733; “dass dergleichen problemata den hiesigen Mathematicis nicht schwer fallen”. LOL-Ch12-P11 of 20 246 R¨udigerThiele this property too. However, Euler recognized that the principle is not, in general, true for variational problems with constraints. 5. Second period In 1744, at the age of 37, Euler published the Methodus inveniendi [E65], a landmark in the history of mathematics, with which he created the new branch of mathematics we now call the Calculus of Variations, although the name came later. Euler changed the subject from a discussion of special cases to that of very general classes of problems. Above all, in this textbook he set up a general analytic apparatus for writing down the so-called Euler or Euler-Lagrange differential equations, thus extending the methods of the Bernoulli brothers to a general theory of the first variation. The book consists of six chapters and two very important addenda. Be- fore sketching the contents of this book, I will mention the Scientia navalis [E110,E111] published in 1749 but already written in 1738. Some of the important results can already be found there, but this has not yet been investigated thoroughly. In the Methodus, Euler considered a general variational problem for one function y of one variable x, in which the integrand Z was allowed to involve derivatives of y of arbitrary order, Z J(y) = Z(x, y, y0, y00,...) dx. Euler regarded the variational integral J as an infinite sum so the variables and their variations could be inserted into the sum. He gave all changes under consideration in tables. Euler expressed the infinitesimal changes both of the functions (curves) under consideration and of their deriva- tives (i.e. extremal and admissible functions/curves) and then calculated the infinitesimal change of the variational integral, the valor differentialis formulae. It is noteworthy that Euler did not make a finite approximation and then carry out limiting processes. Rather, he operated completely in the spirit of the 18th century and its use of infinitesimals. A corresponding approximation that substituting finite quantities for infinitesimals is easily done and corresponds to our understanding. In 1907 Adolf Kneser showed that this procedure, which is preferable by modern standards, is indeed correct [K2].