# How non-linear equations emerge in differential geometry?

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.

• In the first paragraph you talked about the linear term, which I understand. But you loss me at "higher order" term. By higher order term do you mean non-linear? Mar 6 at 23:17
• Thanks for the clarification. In PDE higher order means more derivatives (for example, the laplace equation $\Delta u =0$ is second order). Thus the confusion. Mar 6 at 23:22
• It seems to me that most of the interesting geometric objects in differential geometry involves non-linear quantity. For example, the Riemann curvature tensor R is a tensor which depends on the metric g in a nonlinear way. So any constraint on R would likely give you a non-linear PDE. For another example, the E-L equation for the volume functional of submanifolds is again non-linear. Mar 6 at 23:34
• Linear object (say, from Hodge theory) ultimately gives you only topological properties, but not geometric one. Mar 6 at 23:35
• For a nice example, see here: en.wikipedia.org/wiki/… Relevant to string theory (insofar as that is considered physics), as well as mathematics. Mar 7 at 8:45