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The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. Sometimes in differential geometry, however, one runs into problems in which some function of second derivative is involved. One can, of course, introduce new variables and reduce the order of system and demand that a certain compatibility be satisfied. Namely, we can substitute $D^2 w$ by n vectors, each representing a row of the hessian matrix, and add a constraint, and possibly use the method of Lagrange multipliers.

My question is whether there is a more `intrinsic' approach to this kind of variational problem.

Addendum: I am mainly concerned with the analytical aspects, such as existence and regularity of minimisers. For example, in the usual theory, wherein the lagrangian density is a function of gradient, we know that under certain, say, convexity conditions on $F$, that is, ellipticity of the Euler-Lagrange equation, we have solutions in some appropriate space. Nevertheless, when we include $D^2u$ in the lagrangian, the equation we obtain is a 4-th order one and it won't be as straightforward as the second order case.

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    $\begingroup$ Higher order derivatives can be treated in an intrinsic manner using jet bundles. gmcnetwork.org/files/thesis/cmcampos.pdf Page 78 of this thesis describes a nice approach to variational calculus using this sort of machinery. $\endgroup$
    – Josh Burby
    Commented Oct 16, 2012 at 4:41

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You might find it helpful to study previous work on specific functionals like this. I suggest searching for journal articles that talk about extremal Kahler metrics (which usually means studying the integral of scalar curvature squared on a closed Kahler manifold) or Q-curvature (which is a curvature integral defined for a conformal structure).

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  • $\begingroup$ That was the original motivation, but I was curious as to how this problem is approached by analysts, probably in a more general setting. $\endgroup$
    – S.A.A
    Commented Oct 21, 2012 at 19:45
  • $\begingroup$ By 'that' I meant problems like extremal metrics. $\endgroup$
    – S.A.A
    Commented Oct 21, 2012 at 19:45
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Much has been written on higher order variational problems, but most of what I'm aware of concerns itself with geometric aspects like the generalization of the symplectic or Hamilton-Jacobi formalisms, so that might not be so relevant for you. You may want to have a look at Giaquinta and Hildebrand's treatise on the Calculus of Variations, vol I, part I, sec 1.5. They do discuss higher order problems there, though I cannot be sure that they have a sufficiently thorough discussion of the analytical aspects you are looking for.

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