If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle of the manifold.
I am curious how we would describe geometrically the Lagrangian function $L(u,\partial_x u,\partial_y u)$ for an embedded surface, $R^2$ to the manifold. Like for solving for a minimal surface. What is the domain of this Lagrangian function, is it $\Lambda^2 TM$? So it eats bivectors? No, that doesn't seem right. We don't expect $L(u,\partial_x u,\partial_y u) = -L(u,\partial_y u,\partial_x u)$ or any (anti)symmetry in the arguments. And not a tensor product either. What is its domain?