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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?

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    Thank you, Sam. I don't know much about jet bundles, though I've heard they're the natural way to handle higher derivatives. Now I'm even more intrigued. But since there isn't a more specific question, and you've pointed me in the right direction, I guess I will close this question. Thanks again!2012-02-06

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