Abstract
Lagrange was not only the person who laid the analytic foundation for variational calculus, he was also willing to elaborate on the idea that the principle of least action could be the fundamental principle for all mechanics, including both statics and dynamics. In the 1750s he was enthusiastically hailed as the defender of this new approach to mechanics by Euler and Maupertuis. But in 1788 the same Lagrange wrote the classic work Méchanique analitique,2 in which the principle of least action appears only as a derivative theorem, subordinated to the principle of virtual velocities. Thus Lagrange would seem to personify the transformation of the principle of least action from a teleological principle to a mathematical theorem.
En réduisant les principes on les étendra.
Jean d’Alembert1
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References
D’Alembert, “Méchanique” (1765), 224.
The spelling here, which is no longer usual, is that used in the first edition. In the second and later edition, the title was spelt as Mécanique analytique.
Delambre, “Notice sur la vie et les ouvrages de Lagrange” (1867), x.
See section 5.3.4.
“Haberem fortas[s]is alia tibi mittenda (...) observationesque nonnullas circa maxima, et minima, quae in naturae actionibus insunt (...)” (Letter from Lagrange to Euler, 28 June [1754] (EOO IVa.5, 361–366, this passage at 362).
Lagrange hoped that his discovery of an analogy between the development of the power (a+b)m and the m derivative of (axb) would impress Euler, who was the most prominent mathematician of his day. However, this analogy had already been discovered in 1695 by Leibniz, and both published in an article in 1710 and described in a letter to Johann I Bernoulli (which was published around 1745). Lagrange only learned this a month after his letter (Jutkevii; and Taton, in EOOIVa.5, 365n.1). The fact that Euler only responded after Lagrange’s second letter may well be attributed to his knowledge of Leibniz’s prior claim to the discovery, which meant that he would have to share this rather deflating news with Lagrange.
Letter from Lagrange to Euler, 12 August 1755 (EOO IVa.5, 366–375).
Desideratur itaque Methodus a resolutione geometrica et lineari libera, qua pateat in tali investigatione maximi minimive, loco Pdp scribi debere -pdP” (“Thus what is required is a method, apart from a solution achieved by geometry and drawing lines, which will show that in such an investigation of the maximum and minimum, one may write -pdP instead of Pdp”) (Euler, Methodus inveniendi, 52 (Chapter 2, section 39). The word lineari’ constitutes a problem in translating this passage. Delambre’s translation is nonsense: “indépendante de la Géometrie et de l’Analyse” (Delambre, “Notice sur la vie,” xvi). Stäckel translates it as “unabhängig (…) von der geometrischen Lösung,” and simply omits lineari’ (Stackel ed., Abhandlungen über Variationsrechnung, 66), as do Jutkevié and Taton, when they translate the fragment in Lagrange’s letter to Euler, as “libre de considérations géométriques” (Jutkevii; and Taton, in EOO IVa.5, 370 and 374).
Fraser, “J.L. Lagrange’s Changing Approach to the Foundations of the Calculus of Variations” (1985), 163.
The term ‘indefinite integral’, expressed as jfdx, does not refer to a determined integral off—i.e., a number—but rather the set of primitive functions F off.
“(…) ad summum fere perfectionis fastigium erexisse videris (…)” Letter from Euler to Lagrange, 6 September 1755 (EOO IVa.5, 375–378, this passage at 375).
Euler realised the importance of this for the future power of what he was later to call variational calculus: “(…) quam ob causam etiam non dubito, quin tua analysis, si penitius excolatur, ad multo profundiora mox sit perductura” (“and for this reason I have no doubt that your [method of] analysis, when it is further refined, will take [us] to much deeper matters”) (ibidem). It is however remarkable that Euler evidently identifies the analytic character of Lagrange’s method with general variational calculus—although his own method could just as well be translated into an entirely analytic expression. In 1764 he was to repeat this position in the summary that introduces “Elementa calculi variationum” and “Analytica explicatio methodi maximorum et minimorum” (Carathéodory, “Einführung in Eulers Arbeiten über Variationsrechnung,” XXXI). According to Carathéodory, the fact that Lagrange’s method was used rather than Euler’s polygonal method until the end of the nineteenth century should be attributed not the more analytic character of the former, but to the resulting simplification and generalization of the procedure used to formulate the differential equations of variational calculus. This simplification was due to Lagrange’s use of the general variation of curves (ibidem, XXXII).
Lagrange, “Essai d’une nouvelle méthode pour déterminer les maxima et les minima des formules intégrales indéfinies” and “Application de la méthode exposée dans le mémoire précédent it la solution de différents problèmes de dynamique” (both from 1760–1761).
Fraser, “J.L. Lagrange’s Early Contributions to the Principles and Methods of Mechanics” (1983), 233–234.
Lagrange’s correspondence with d’Alembert did not begin until 1759, and even then the principle of least action seems never to have been mentioned. They did not meet in person until 1763 (see page 207).
Fraser, “J.L. Lagrange’s Early Contributions to the Principles and Methods of Mechanics” (1983), 233–234.
The reference here is to a letter from Maupertuis to Lagrange on 4 January 1757, which was only discovered in 1986. The letter’s existence had been known previously, but it had been dated before 4 May 1756 because of the incorrect dating of Lagrange’s letter to Frisi in 1756 instead of in 1757 (see note 18). The letter is published in Taton, “Sur quelques pièces.”
The correspondence between Lagrange and Paolo Frisi, which was not yet available when Serret published Lagrange’s oeuvre (1867–1892), was published separately in Favaro, “Sette lettere inedite di Giuseppi Luigi Lagrange al P. Paolo Frisi” (1895–1896). Favaro dates the letter that is important here on 4 May 1756. The discovery of a letter from Maupertuis to Lagrange dated 4 January 1757 (see note 17) prompted René Taton to study Lagrange’s letter to Frisi more closely. He discovered that the date 1756 was a mistake in the manuscript—it should read 1757 (Taton, “Sur quelques pièces,” 12–16). Paolo Frisi (1728–1784) was professor of mathematics and physics in Pisa from 1756. He was also well known in Berlin, where he had won the competition for the department of mathematics for 1756 (see also Die Registres, 224 and 224n.2(361)).
This treatise and the accompanying letter have been lost. We only know its contents from a letter by Euler of 24 April 1756 (EOO IVa.5, 386–390), in which he responds to it. We also know that the treatise was presented in the Academy on 6 May 1756 by Euler himself (Die Registres, 223). The treatise and its covering letter must therefore have been sent before 24 April 1754, when Euler wrote his response. Since Euler had also given it to Maupertuis to read before he wrote his response, it is probable that the treatise and letter were sent as early as March 1754. The loss of both the letter and treatise may have occurred because Maupertuis took them with him on his definitive departure from Berlin at the end of May 1756 (Brunet, Maupertuis I. Etude biographique, 168–169).
The previous correspondence, consisting of four letters, seems to have been preserved in its entirety. Two letters from Lagrange to Euler, of 28 June 1754 (EOO IVa.5, 361–366, see pages 362 and 364) and 12 August 1755 (ibidem, 366–375, see pages 366 and 369–370) are of special importance here.
Die Registres, 223 (6 May 1756).
That is, that fmvds is an extreme. See section 5.3.4.
“[Maupertuis] tibi pro suscepto principii minimae [a]ctionis patrocinio maximas agit gratias (…)” (“[Maupertuis] thanks you very much for your defence of the principle of least [a]ction (…)”) (letter from Euler to Lagrange, 24 April 1756, in EOO IVa.5, 386–390, this passage at 387). Lagrange was in fact nominated as a non-resident member by Euler—on behalf of the absent president Maupertuis—on 26 August 1756. The Academy accepted the nomination on 2 September of that year (Die Registres, 225; letter from Euler to Lagrange, 2 September 1756 (EOOIVa.5, 394–396, see page 394)).
This conclusion does not contradict that of Fraser who, on the basis of various correspondences between Lagrange’s “Application de la méthode” of 1760 and Eulers “Harmonie entre les principes” of 1752, concludes that Lagrange must have known this work and been inspired by it, although he does not explicitly refer to it until the “Recherches sur la libration de la lune” (1764). However, there is no indication that Lagrange knew it as early as 1756 when he wrote his outlines (Fraser, “Lagrange’s Early Contributions,” 203–204 and 208–209).
“De principio minimae quantitatis actionis ego ita sentio, (…), omnium tam staticorum, quam dynamicorum problematum universalem veluti clavem haberi posse (…)” (Letter from Lagrange to Euler, 19 May 1756 (EOO IVa.5, 390–394, this passage at 391).
Ibidem, 396.
Letter from Lagrange to Paolo Frisi, 4 May 1757 (and not 1756!) (Favaro, “Sette lettere inedite,” 141–143, this passage at 142; Taton, “Sur quelque pièces,” 16–18, this passage at 18). There are two later mentions in his letters of 28 July and 4 August 1759 to Euler, in which he refers to the subject of the papers as “de applicatione principii minimae quantitatis ad mechanicam universam” (“on the application of the principle of least action to universal mechanics”) (EOOIVa.5 411–414, this passage at 411). In the letter of 4 August 1759 the term ‘universam’ is replaced by ‘totem’ (‘the whole of’) (ibidem, 414–417, this passage at 414).
From Maupertuis’s letter it is clear that Lagrange had read d’Arcy’s criticism, probably in his “Réplique à un Mémoire de Mr. de Maupertuis sur le principle de la moindre action,” which was printed in 1756 in the Mémoires de Paris. But it is unlikely that Lagrange would have drawn his enthusiasm from reading something so sharply critical. In his letter of 4 January, Maupertuis writes that he has sent one copy of the 1756 edition of his OEuvres to Ansaldi, a professor in Turin. He doubts whether he has in fact received it, and says that he will send two more exemplars, one of them for Lagrange. However, this does not exclude the possibility that Lagrange had already read his OEuvres by that time.
See note 27.
“plus générale et plus rigoureuse, et qui mérite seule l’attention des géomètres” (Delambre, “Notice sur la vie,” xvi).
Lagrange, Méchanique analitique,229 (II.I.17).
In his letter to Frisi of 4 May 1756, Lagrange writes that he has already got the two articles almost completely in order, but cannot find time to finish them because of the intense activity entailed by his chair at the Artillery school in Turin (Taton, “Sur quelques pièces,” 17; Favaro, “Sette lettere inedite,” 142). First Euler and later Maupertuis himself had by then assured him that they would publish his work—including the memoir he had already sent—in the Mémoires of the Berlin Academy (letter from Euler to Lagrange, 2 September 1756 (EOO IVa.5, 395); letter from Maupertuis to Lagrange, 4 January 1757 (Taton, “Sur quelques pieces,” 9)).
Letter from Euler to Lagrange, 2 October 1759 (EOO IVa.5, 418–423, this passage at 418).
See also Taton and Jutkeviè, [Introduction to Correspondance],42–43.
Letter from Lagrange to Euler, 24 November 1759 (EOO IVa.5, 429–432, this passage at 430).
Letter from Lagrange to Euler, 38 [28?] October 1762 (ibidem, 446–448, this passage at 447). Another event is also explained by this disappointment, namely a certain hostility on Lagrange’s part towards Euler. In the letter in which Euler tells him the sad news, he also writes about an analytic solution that he has developed himself, inspired by Lagrange’s treatise of early 1756, but that he does not intend to publish until Lagrange has published his own, so as not to steal his thunder. Euler’s articles (“Analytica explicatio methodi maximorum et minimorum” and “Elementa calculi variationum”) were in fact not published until 1766, in the Nouveaux Comm. Petrop., although they had been presented in the Berlin Academy on the 9th and 16th of September 1756, respectively. If Taton and JuIkeviè are right in their suggestion that Lagrange took the rejection personally (see note 34), it is quite possible that Lagrange developed a personal resentment towards Euler out of disappointment. This could plausibly explain his strange refusal to take a position in Berlin in 1764: “(…) I feel that Berlin would not suit me, while Monsieur Euler is there” (Lagrange’s letter to d’Alembert, 13 November 1764, OEuvres de Lagrange XIII, 20–23, this passage at 23).
“J’ai aussi composé moi même des elemens de Mécanique et de Calcul differentiel et integral à l’usage de mes ecoliers, et je crois avoir developpé la vraye metaphisique de leurs principes, autant qu’il est possible” (letter from Lagrange to Euler, 24 November 1759 (EOOIVa.5, 429–432, this passage at 430–431)).
Taton and JuIkevié, [Notes to Correspondance], 432n.14 and 15.
Fraser, “Lagrange’s Early Contributions,” 233.
Pulte, Das Prinzip der kleinsten Wirkung, 257–258 and 258n. 275.
Sarton, “Lagrange’s Personality (1736–1813)” (1944), 477–478; Itard, “Lagrange,” 569.
“Je suis bien sûr que vous n’avez pas été consulté. Tout le monde se moque de ce programme, et l’Académie n’a pu s’empêcher d’en rire quand M. de Condorcet l’a lu” (letter from d’Alembert to Lagrange, 22 September 1777 (OEuvres de Lagrange XIII, 330–332, this passage at 332)).
“Vous avez bien raison de croire que je n’ai eu aucune part au programme de Métaphysique. Cette science, si c’en est une, n’est nullement de mon gibier. Il me semble que chaque pays a presque sa Métaphysique particulière comme sa langue, et la question proposée est de Métaphysique allemande et leibnitzienne” (letter from Lagrange to d’Alembert, 27 January 1778 (ibidem, 334–336, this passage at 336)).
See the quotation on page 215.
D’Alembert, Traité de dynamique, 50–51 (see also section 4.4.3.2). Strangely enough, there is no mention of the principle of virtual velocities in the extensive correspondence between d’Alembert and Lagrange. Lagrange has also not given us any insight into this transition in other places (Fraser, “Lagrange’s Early Contributions,” 220 and 233–235).
This formulation is a paraphrase of Lagrange, Méchanique analitique, 18. Lagrange also gives a general formulation on page 20. In mathematical form the principle then reads: Pdp + Qdq + Rdr +… = 0 (ibidem, 26). A general description of ‘virtual velocities’ is: “by virtual velocities is meant that speed that a body in equilibrium is disposed to receive if its equilibrium is destroyed, that is, the velocity that the body would actually assume in the first instant of its motion” (“On doit entendre par vitesse virtuelle, celle qu’un corps en équilibre est disposé it recevoir, en cas l’équilibre vienne it être rompu, c’est-à-dire la vitesse que ce corps prendrait réellement dans le premier instant de son mouvement”) (ibidem, 17–18). A second definition of ‘virtual velocities’ is given apart from the general formulation of the principle: the infinitesimal distance dx that an element in a mass system moves, if an infinitesimal arbitrary motion is given to that system (ibidem, 20). Notice that Lagrange makes a silent transition from velocity to distance.
Lagrange, “Recherches sur la libration,” 10–12.
See section 5.3.1 above.
Lagrange is referring here to Euler’s “Harmonie entre les principes généraux” (see above, Chapter 5, page 167).
For d’Alembert’s derivation see Chapter 4, page 129.
“(…) renferme la solution de tous les Problèmes qui regardent le mouvement des corps” (Lagrange, “Recherches sur la libration,” 12).
Lagrange does not ask himself whether this derivation could also be reversed, which is an argument for the suggestion that he wrote the piece with d’Alembert in mind but it is not an argument against my interpretation of the way he ranks principles, using an axiomatic and deductive ranking based on mathematical principles.
Lagrange, “Théorie de la libration de la lune et des autres phénomènes qui dépendent de la figure non sphérique de cette planète” (1780).
In which he introduces a mathematical variable V which would later be called ‘potential energy’: V= E {mJ(Pdp + Qdq + Rdr +…)} (ibidem, 24).
“Mais la combinaison de ces deux principes est un pas qui n’avait pas été fait, et c’est peut-être le seul degré de perfection qui, après la découverte de M. d’Alembert, manqait encore à la Théorie de la Dynamique” (ibidem, 11).
“Ces principes doivent être regardés plutôt comme des résultats généraux des lois de la Dynamique (…)” (Lagrange, Méchanique analitique, 225).
Ibidem, 26. See also note 46.
Lagrange’s ‘Whig history’ has left a considerable mark on the historiography of eighteenth century mechanics, especially through the work of Ernst Mach. Mach’s original inspiration for Die Mechanik, historisch-kritisch dargestellt (1883) came from Lagrange’s historical introductions in the Méchanique analitique. See also Chapter 1 above, pages 10–11.
] Lagrange, Méchanique analitique, 2–10, and 10–17 respectively.
] Ibidem,17–23.
] Ibidem, 20, 19 and 21 respectively.
This proof is based on a thought experiment with a pulley system. The suggestion is that the principle of virtual velocities can be based on what is in effect a generalised principle that the centre of gravity tends to the lowest point. The latter is called the ‘principle of pulleys’ here, and was later elaborated further. The arbitrariness of this suggestion can be seen from the fact that d’Alembert had earlier used the argument in the opposite direction: his principle was supposed to explain that the centre of gravity in a mechanical system tends to the lowest point. See ibidem, 21–22; Lagrange, “Sur le principe des vitesses virtuelles” (1797); d’Alembert, Traité de dynamique, 93–95.
Lagrange, Méchanique analitique, 20.
] Ibidem, 20.
“expression générale des lois de l’équilibre”(ibidem, 21).
As was noted in section 7.3, something like this may have been what Lagrange meant, ironically, by his reference to ‘true metaphysics’.
For the way in which d’Alembert introduced these principles in his Traité de dynamique see section 4.4.
Lagrange, Méchanique analitique, 209.
Ibidem, 210 and 213 respectively. In the previous section, article 4 (ibidem, 213) Lagrange says that the measure of forces of impact requires a new principle, because the forces are in this case unknown, but he then introduces a measure that is analogous to the measure of accelerating force. However, in this case the measure is not related to acceleration, but to imposed velocity (“mouvement imprimé”).
Ibidem,208.
Ibidem,231. When Lagrange refers here to ‘reducing’ such magnitudes, he is pointing to the need to introduce a measure for the various magnitudes, by taking a known force, distance, etc. as a unit and relating the others to it, “(…) thus the forces, spaces, times, and speeds will be no more than simple ratios, that is, ordinary mathematical quantities” (“(…) les forces, les espaces, les temps, et les vitesses ne seront que des simples rapports, des quantités mathématiques ordinaires”) (ibidem).
“(…) pour appliquer au mouvement d’un système de corps la formule de son équilibre, il suffira d’y introduire les forces qui proviennent des variations du mouvement de chaque corps, et qui doivent êtres détruites” (ibidem, 231).
“Il faut, dans la Mécanique, prendre les effets simples des forces pour connus; et l’art de cette science consiste uniquement à en déduire les effets composés qui doivent résulter de l’action combinée et modifiée des mêmes forces” (ibidem, 224–225).
Ibidem, 224. For d’Alembert’s treatment of this transition see section 4.4, especially page 126. Despite Lagrange’s reformulation of the principle, it is known as ‘d’Alembert’s principle’.
Ibidem, 267–274 and 274–281 respectively.
French: ‘disposition’; German: ‘Zwangsbedingung’. An example of a constraint is the connection of two masses by a weightless rod of length I. We can express this mathematically as the condition that for the coordinates of the positions of mass 1 and mass 2, xl and x2 respectively, 011_x2)2 = l2.
Ibidem, 268. The formula Pdp + Qdq + Rdr +… represents the sum of the products of forces P, Q, R… with displacements dp, dq, dr… in the direction of the forces. Lagrange calls this the “intégrable,” rather than the total differential, which amounts to the same thing: the integration here is independent of the path, and so depends only on the beginning and end points.
For the different ideas on both heads Lagrange still had in 1760, when the principle of least action promised to be the key for all mechanics, see note 27.
For a contrary opinion, see Fraser, “Lagrange’s Early contributions,” 234. Fraser interprets the change in status from universal to derivative as a rejection of the metaphysical status of the principle of least action. D’Alembert’s influence would then be very plausible. In my interpretation, Lagrange was only concerned with universality, and for that, he did not need d’Alembert.
Pulte, Das Prinzip der kleinsten Wirkung, 236–238.
Lagrange, Méchanique analitique, 1.
See pages 221–222.
This could be the reason for Duhem’s agreement with Fourier’s accolade to Lagrange’s work as a “philosophical mechanics” (Duhem, L’évolution de la mécanique, 23).
The evolution of mathematics itself is not considered here. Mathematics was originally qualitative in nature. In Pythagorean number theory, ratios were associated with psychic relationships such as harmony and jealousy. Therefore the Pythagorean motto, ‘everything is number’, is not simply the reduction of reality to quantity. With this in mind, one might, as Lanczos suggests, draw a parallel continued on next page continuation of former page between the transition from geometry to analysis in eighteenth century mathematics and the transition in ideas about force (see Lanczos, The Variational Principles).
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Boudri, J.C. (2002). Lagrange’s Concept of Force. In: What was Mechanical about Mechanics. Boston Studies in the Philosophy of Science, vol 224. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3672-5_7
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