Who came up with the Euler-Lagrange equation?

Which man came up with the solution to the basic Calculus of Variations problem first?

http://en.wikipedia.org/wiki/Euler-Lagrange

Makes it sound like Lagrange got it first in 1755, then sent it to Euler.

But

http://en.wikipedia.org/wiki/Calculus_of_variations

makes it sound like Euler got it in the 1730s.

Could somebody please explain which man actually wrote out the equation first, and what the other then contributed to it?

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My wild guess, based on how these thing usually go, is that it was neither Euler nor Lagrange. –  Qiaochu Yuan Jul 31 '12 at 17:15
This has been asked also at math.stackexchange.com/questions/177243/… @Jacobi: it is usually best to wait a few days before posting a question in the other site and, when you do, you should provide a lnink to the other question. –  Mariano Suárez-Alvarez Jul 31 '12 at 18:57
Also, it was not Jacobi either ;-) –  Suvrit Jul 31 '12 at 19:39

According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi [2], Chapter 2, no. 21. Quite often, one speaks of Lagrange's differential equation, or the Euler-Lagrange equations. Yet Lagrange himself attributes this equation to Euler: 'Cette équation est quelle qu'Euler a trouvée le premier' (Oeuvres [12], Vol. 10, p. 397)."

NB: Euler's Methodus inveniendi was published in 1744.

Further down the same page in G&H, one finds "With Lagrange's letter of August 12, 1755 to Euler, the development of the variational calculus took a new turn. Lagrange explained the $\delta$-symbolism which quickly leads to Euler's equation. Euler was very impressed by the new method, and already one year later he lectured about it at the Berlin Academy."

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There is a very good reason for calling it "Euler--Lagrange". If we went around calling everything that Euler invented after him alone, we'd quickly lose track of which invention we're referring to. Similarly, I'm under the impression that Seiberg--Witten theory appeared in a paper of Witten's alone, although it is based on their joint work and Witten gives Seiberg the credit. –  Theo Johnson-Freyd Aug 20 '12 at 17:40
@Theo: I don't think anyone seriously wants to stop people from calling it the Euler-Lagrange equation (although, come to think of it, I think that Giaquinta and Hildebrandt actually do call it "Euler's equation"); the question was about historical priority, not true names. I once heard a not-to-be-named-but-distinguished mathematician joke(?) that we tend to name a mathematical concept for the next person to work on it after Euler. (I think the point was in sympathy with yours; if we named everything that Euler originated after Euler, no one could be sure what "Euler's [anything]" meant.) –  Robert Bryant Aug 20 '12 at 19:02
No, of course not. I was paraphrasing something I had heard about "Witten Theory" and "Witten Equations". In any case, we have these historical questions about names because it is well-known that the name attached to a result has very low probability of being the person who actually invented the result. –  Theo Johnson-Freyd Aug 25 '12 at 7:21

I think the main reference here is Euler's archive. You have to look at the two books on mechanics firstly presented on 1736. Euler often presented his works to the community before publication. Being these authors almost contemporary, it is possible that the name to these equations is indeed the right one.

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Since the brachistochrone problem was solved (by Newton first, but then some others), I would say that the germ of the idea goes to Newton's paper in 1697. In the linked article, Lagrange attributes the general formulation to Euler.

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