Nester, James M. "Invariant derivation of the Euler-Lagrange equation." Journal of Physics A: Mathematical, Nuclear and General 21.21 (1988): L1013-L1017.

Note that this is a physics paper, so the derivation is not completely rigorous.  I am convinced it can be made rigorous.

Edit. Here is a math reference:

Connections, Sprays and Finsler Structures by Jozsef Szilasi, Rezso L Lovas, David Cs Kertesz       ISBN: 9789814440097

It's in section 5.2.