# Who established the current standard demonstration of Euler-Lagrange equation in calculus of variations?

Who established the current standard(*) demonstration of Euler-Lagrange equations in calculus of variations, that is, $$\displaystyle\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial \dot y} = 0$$?. As far as I know, the closest author could be JACOBI (On the theory of the Calculus of Variations and of differential equations, 1837(**). Really, he starts writing the exact equation and after tries to demonstrate it.

So, he starts writing:

"Let us consider the simplest case; let the integral is to have a maximum or minimum value be $$\displaystyle\int f(x, y, y') dx,$$ where $$y'$$ is put $$\displaystyle\frac{dy}{dx}$$. Then we know that $$y$$ is to be found from the differential equation $$\displaystyle\frac{\partial f}{\partial y}-\frac{d}{dx}\frac{\partial f}{\partial \dot y} = 0$$ "

Since then, the demonstrations Jacobi made have nothing to do with modern usuals ones, yet cited. I think they are more complex and, actually, I don't understand them at all.

Moreover, why or who put the so called name 'Euler-Lagrange' to the equation, authors whose works in calculus of variations are older and haven't nor to do with it? (***)

(*) E.g., Goldstein, Marion/Thornton, Taylor, etc.

(**) A english translation in Todhunter, History of the calculus of Variations, 1861. There's a recent text about this question: Goldstine, A History of the Calculus of Variations from the 17th through the 19th Century, 2011

(***) Euler, Elementa calculi variationum, 1766(there's english translation); Lagrange, Méthode de maximis e minimis, Miscellanea taurinensiea, T.I, 1759 (french translation in Ouvres de Lagrange, T.I); Sur la methode des variations, Miscellanea taurinensiea, t.IV, 1766(french translation in Ouvres de Lagrange, T.IV)

• Would History of Science and Mathematics be a better home for this question/ Commented Sep 24, 2021 at 10:56
• The site guidelines on history questions are here. This one seems to fall squarely on the History of Science and Mathematics side of the divide. Commented Sep 24, 2021 at 11:13
• I find this an interesting question. The physics community has gravitated to a particular way of demonstrating the Euler-Lagrange equation. I assume that when an author writes a new textbook most of the writing process is to reproduce the content of existing texbooks (either from memory or directly), updated where necessary. That is, I expect the uniformity isn't independent convergence, but reproduction. Commented Sep 24, 2021 at 13:07