In physics, many ODEs and PDEs are derived by applying a variational to some lagrangian density.
I more or less have a geometric interpretation of the quadratic terms of a lagrangian and the associated linear terms in the derived ODEs or PDEs. These linear terms usually involve differential forms, exterior derivatives, ..., i.e. a variety of mathematical objects associated to the vector spaces and linear operators usually found in differential geometry. But, I don't see how the higher order terms in the lagrangians (which usually account for interactions) can be geometrically interpreted.
As far as I can tell, the higher order terms are usually introduced in some sort of had-hoc manner, thinking in terms of perturbation theory. Namely, "free theories" (i.e. theories of non-interacting particles and waves) are modeled by the linear parts of ODEs and PDEs (and thus by the quadratic temrs of lagrangians) while interacting theories include higher-order terms into the lagrangians. How are these higher-order terms interpreted from the point of view of differential geometry?
Apologies if my question is rather vague. I am not sure how to rephrase it into more formal settings.