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? 
 A: 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."
A: 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.
A: 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.
