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.