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.


I will note that I asked this question a few days ago on math stackexchange. As of the time of this writing, it remains unanswered. This led me to believe that the question is specialized enough for mathoverflow. However, I am not a mathematician, so please be pedagogical when possible.


There is a large literature on this, and the roots go back more than one hundred years. Some of the modern work along these lines can be found by looking for papers containing the term 'variational bicomplex'. For example, look at the papers and books by Ian Anderson and his group.

You can also look at papers and books by Phillip Griffiths and his collaborators, such as Exterior Differential Systems and the Calculus of Variations and Exterior Differential Systems and Euler-Lagrange Partial Differential Equations.


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