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)
