Editing Joseph-Louis Lagrange (section)

==== Variational calculus ====
Lagrange is one of the founders of the [[calculus of variations]]. Starting in 1754, he worked on the problem of the [[tautochrone]], discovering a method of maximizing and minimizing functionals in a way similar to finding extrema of functions. Lagrange wrote several letters to [[Leonhard Euler]] between 1754 and 1756 describing his results. He outlined his "δ-algorithm", leading to the [[Euler–Lagrange equation]]s of variational calculus and considerably simplifying Euler's earlier analysis.<ref>Although some authors speak of a general method of solving "[[isoperimetric]] problems", the eighteenth-century meaning of this expression amounts to "problems in variational calculus", reserving the adjective "relative" for problems with isoperimetric-type constraints. The celebrated method of [[Lagrange multipliers]], which applies to the optimization of functions of several variables subject to constraints, did not appear until much later. See {{cite journal | last = Fraser | first = Craig | title = Isoperimetric Problems in the Variational Calculus of Euler and Lagrange | journal = Historia Mathematica | volume = 19 | pages = 4–23 |year = 1992 | doi = 10.1016/0315-0860(92)90052-D | doi-access = free }}</ref> Lagrange also applied his ideas to problems of classical mechanics, generalising the results of Euler and [[Pierre Louis Maupertuis|Maupertuis]].

Euler was very impressed with Lagrange's results. It has been stated that "with characteristic courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claim the undisputed invention of the new calculus"; however, this chivalric view has been disputed.<ref>Galletto, D., ''The genesis of Mécanique analytique'', La Mécanique analytique de Lagrange et son héritage, II (Turin, 1989). Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 126 (1992), suppl. 2, 277–370, {{MathSciNet|id=1264671}}.</ref> Lagrange published his method in two memoirs of the Turin Society in 1762 and 1773.