TITLE: Can calculus of varations be formalised with exterior calculus?
QUESTION [9 upvotes]: I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equations using exterior calculus. However, I do not know of how to reconcile the notion of a functional derivative with say an exterior derivative.

REPLY [4 votes]: The correct context for this problem is on infinite jet bundle and using the variational bicomplex.
The main reference for this is Ian Anderson's work: 
http://math.uni.lu/~michel/data/VARIATIONNAL%20BICOMPLEX.pdf