Source: Mariano Giaquinta Stefan Hildebrandt, Calculus of Variations I 1st ed. 1996 . Corr. 2nd printing 2004, Springer-Verlag Berlin Heidelberg New York Page 132: 1. Already in his two papers on isoperimetric problems from 1732 and 1736. respectively, Euler introduced multipliers (which are nowadays called Lagrange multipliers))[10]. But these publications contain several serious mistakes and erroneous methods, which show that Euler had not yet mastered these questions." Only in his Methodus inveniendi (E65) from 1744, he obtained a basically sound theory (apart from the fact that Euler did not think of extremals which neither are maximizers nor minimizers). In Chapter 6 of this treatise, he developed the multiplier theory for isoperimetric side conditions, proving the existence of a constant multiplier, and in Chapter 5, he treated problems with several subsidiary conditions of isoperimetric type. For instance, he proved that, for two variational integrals F,G and two constants lambda,mu the functional lambda F + mu G yields the same extremals as the isoperimetric problem where A is to be given an extremum, keeping I fixed. In Chapter 3 of the Methodus inveniendi, Euler dealt with variational integrals int_a^b F(x,u,v,v',...,v^(k)) dx subject to nonholonomic constraints of the kind u'-G(x,u,v,v',.v^(k)) = 0 and he proved the existence of a multiplier function. Although Euler did not advance to the most general type of Lagrange problem, his analysis of the constraint problem is an achievement of first class that Caratheodory called "eine Spitzenleistung, wie sie auch einem Euler nicht allzu oft geglueckt ist." Later on, Lagrange has applied his delta-method to variational problems with nonholonomic constraints, thus obtaining a new approach to the multiplier rule. His name became attached to the multiplier rule by the applications to mechanics that he derived in his "Mechanique analitique" in 1708 (see Lagrange [1], Sect. 4, I). Moreover, he treated general nonholonomic conditions; yet his proof contained two serious gaps which were only Idled by A. Mayer (1886), A . Kneser (1900), and Hilbert (1906). For a brief historical account, we refer to Bolza [3], pp. 566 -569, and to Goldstine [I), pp. 148-150. The general Lagrange problem for simple integrals is carefully treated in: Bolza [3], Chapters I 1 and 12; Caratheodory [10], Chapter 18; Hestenes [5], Chapter 6; Rund [4], Chapter 5; L.C . Young [1], Vol. 2; Cesari [I]. [9] See also Bolza (3). pp . 198-20 1. [10] (E27) Problematis isoperimetrici in latissimo sensu accepti solutio generalis. Comm. acad. sci. Petrop. 6 (1732,3), 1738 , 123 -155; (E56) Curvarun maximi minimie proprietate gaudentium inventio nova et facilis, Comm. acad. sci. Petrop. 8 (1736), 1741 , 147 -158; cf. also: Opera omnia. Ser. I. vol. 25 . [11] Carathkodory, Schriften [16], Vol. 5, pp. 130 -135 . [12] "a major accomplishment that even an Euler did not achieve too often". [13] "This is no orthographic error, only the second edition was called Mecanique analytique. The first volume of this edition appeared in 1811, the second one in 1815, two years after Lagrange's death. The last three monographs also contain a detailed bibliography and describe the relations to optimization theory. Our treatment of the multiplier rule for the Lagrange problem is essentially taken from Bolza [3]. For multiple integrals, the state of art is much less satisfactory; some results can be found in Kloetzler [4] and in Bittner [1]. 2. The first proof of the isoperimetric property of the circle appears in the commentary of Theon to Ptolemy's Almagest and in the collected works of Pappus. The author of the proof is Zenodoros, who must have lived sometime between Archimedes (died 212 B.C.) and Pappus (about A.D. 340), since be quotes Archimedes and is cited by Pappus. Concerning the history of the isoperimetric problem, we refer to Gericke [1] and Blaschke [2]. The first complete proof of the isoperimetric property of the circle was provided by Weierstrass in his lectures at Berlin University. 3. The problem of the vibrating string was already considered by the Pythagorean. In modern times, experimental results were found by Mersenne (1625) and Galileo (1638). Theoretical results were obtained by Taylor (1713), Johann Bernoulli (1727), Daniel Bernoulli, Euler, d'Alembert, and Lagrange. We refer to Truesdell [1], [2]. The variational approach to eigenvalue problems was particularly stressed in the treatise of Courant-Hilbert [1-4], which had great influence on the development of mathematical physics and quantum mechanics. Also Rayleigh's celebrated treatise [I] on The theory of sound and the papers by W. Ritz [1] were very influential. 4. Hypersurfaces of constant mean curvature not only appear as soap bubbles, but also as capillary surfaces without gravity in equilibrium; cf. Finn [1], in particular Chapter 6. 5. The problem to determine the shape of a freely hanging rope (or chain) was posed by Jacob Bernoulli in 1690; the solution was found by Leibniz, Huygens, and Johann Bernoulli in 1691. Earlier, this problem had been considered by Galileo who wrote that the equilibrium configuration of a rope has the shape of a parabola. The notation catenary (Latin: catenaria) seems to be due to Leibniz. The treatment of the equilibrium configuration of a chain as an isoperimetric problem is due to Euler (cf. Methodus inveniendi, Chapter 5, nrs. 47 , 73 , 74). Recently it was disputed whether or not Galileo thought the catenary to be a parabola. The matter seems clear if we bear what Salviati says in the second day of Galileo's Discorsi:r` "Fix two nails at a wall at equal height above the horizontal, and at a distance from each other which is twice the length of the rectangle where the semi parabola is to be constructed. At both nails a fine chain is suspended which is so long that its lowest point is at a distance equal to the length of the rectangle below the horizontal through the nails: This chain has the shape of a parabola If we puncture the line formed by the chain onto the wall, we have described a complete parabola." However, the historian M. Fierz [I ], p. 63, pointed out that, on the fourth day of the Discorsi, Salviati gives a somewhat different description of the catenary: [14] "Discorsi e dimostrazioni matematiche, Elsevir, Leiden, 1638; cf. giornata seconda, p. 146: Ferminsi ad alto due chiodi in un parete equidistanti ail' Orizonte, e trd di loro lontani it doppio della larghezza del rettangolo, su 'I quale vogliamo not are la semiparabola, e da questi due chiodi penda una catenella sottile, e canto lunga, the la sum sacca si stenda quanta a la lunghezza del Prisma: questa catenella si piega in figura Parabolica. Si the andando punteggiando sopra'1 muro la strada, the vi fd essa catenella, haremo descritto un inter Parabola. " We feel amazement and joy, if the little or tightly stretched rope approaches the parabolic form, and the similarity is such that if, in a plane, you draw a parabola, and you consider it invertedly, the vertex below and the base parallel to the horizontal, and you suspend a chain at the endpoints of the base of the parabola, then the chain is more or less curving and will attach itself to the said parabola, and the attachment is the more precise, the less the parabola is curved, that is, the more it is stretched; in parabolas with an elevation [slope] of 45 degrees, the chain covers the parabola almost perfectly."" In the opinion of Fierz, this second statement of Galileo shows clearly, that he was well aware of the fact that the parabola is merely an approximation for the catenary. Fierz goes even further by pointing out that, for I 1, the parabola y = I + }x2 is a good approximation toy = cosh x. He could be right, but at least this reasoning seems a bit far fetched; in any case it does not explain why, in the second dialogue, Galileo described the chain as a "complete parabola" ("intera parabola"). Of course, he could have discovered later that the parabola is merely an approximation to the true catenary and he might have forgotten to erase what he had written in the first place. This, in fact, is the speculation of A. Herzig and 1. Szabo [1], pp. 53 -54 . As the present authors are no historians of mathematics, they have to ask the reader to form his own opinion. 6. The notion of a harmonic mapping was introduced by S. Bochner [1] in 1940, and in full generality by Fuller [I]. Three surveys of the development till 1988 were given by Eells and Lemaire [I], [2], [3]. Harmonic mappings have become an important too] in differential geometry. 7. The first published paper on shortest lines on a surface is due to Euler.16 However, it now seems to be certain that Johann Bernoulli possessed the "law of the osculating plane" in 1698." 8. The modem theory of geodesics in Riemannian manifolds began with Riemann's lecture on occasion of his Habilitationskolloquium June 10, 1854: "Ober die Hypotheses, welche der Geometrie zugrunde liegen. ` 9. The history of the principle of least action has often been described. Yet the matter is still controversial, and there seems to be no general agreement who invented the principle, Leibniz. Euler, or Maupertuis. A popular account of the controversy between Maupertuis and Euler on the one hand, and S. Konig/Voltaire on the other is given in Hildebrandt Tromba [1]. A more pro- found discussion can be found in Carathbodory [16], Vol. 5, pp. 160 -165, Fleckenstein [I], and F. Klein [3], Vol. 1, pp. 191-207; see also Pulte [1]. We mention that the first mathematical treatment of the action principle was given by Euler (1744) in the Additamentum 11 of his Methodus invendiendi. Moreover, there are different mathematical versions of the least action principle which some- times are emphatically distinguished from each other. We do not want to participate in these controversies and just refer the reader to Klein's opinion ([3], Vol. 1, pp. 192-193),19 and to Prange [2], pp. 565 -566 and footnote 153 on p. 607. [15] "Recandovi insieme maraviglia, e diletto, the to corda cosi tesa, e poco, d motto tirata, si piega in tine, le quasi assai Si avvicinano alle paraboliche, c la similitudine a tarsa the se voi segnerete in una superficie piana, & eretta all'Orizonte una linen parabolica, e ienendola inversa, cioe cot aertice in giti, e con la base parallela all' Orizonte, facendo pendere una catenella sostenuta nelle estremitd delta base delta segnata parabola, vedrete allentando pin, d meno la detta catenuzza incurvarsi, e adattarsi alla medesimaparabola: e tale adattemento Canto pin esser preciso, quarto Ia segnata parabola sar d men' curva, clod pin distesa; si the nelle parabole descritte con elevazioni sotto d i grad. 45 la catenella camina quasi ad unguem sopra la parabola". Cf. Discorsi, giornata quarto, p. 284. [16] (E9) De linea brevissima in superfine quacunque duo quaelibet puncta jungente, Comm. acad. sci. Petrop. 3 (1728) 1732, 110-124; cf. also: Opera omnia, Ser. I , Vol. 25, 1 -12. "Johann Bernoulli's Opera omnia, Vol. 4, p. 108, Nr. 166 . 16 Werke [3], second ed., or. 13, p. 272. [19] "The Lagrange equations arise immediately from the variational problem delta int L dt = 0 (the limits of the integral kept fixed). Remarkably, this idea appears in Lagrange only between the lines; therefore we find the strange fact that this relation - mainly by the influence of Jacobi - is generally known in Germany, and therefore also in France, as "Hamilton's principle", whereas in England no one will understand this terminology, There one denotes this equation by the correct though unintuitive name of the "principle of stationary action" ... The widely-known "princple of least action" is another, by Lagrange preferred concept which he found at the beginning of his studies in 1759. In the 18th century this principle drew great interest, in particular from philosophical quarters .. Here Maupertiuis is especially to be mentioned.