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A geodesic on a Riemannian manifold is generally not the shortest among nearby curves with the same endoints. But it can always be divided into parts each the shortest among nearby curves between its endpoints.

How much does that generalize? Is there some natural description of Lagrangians with the following property: for each solution $f$ to the Euler-Lagrange equation on a given domain, that domain can be covered by subdomains such that each restriction of $f$ to one of those subdomains actually minimizes (or maximizes) the Lagrangian integral on that subdomain?

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I don't think this "local minimality" is really general. I'm thinking of the variational problem that defines Reeb orbits for a given contact form: it takes the form $\gamma \mapsto \int_\gamma \alpha$, where $\alpha$ is the contact $1$-form and so pieces of $\gamma$ don't seem to be maxima nor minima. For $1$-dimensional problems at least convexity/ellipticity of the integrand seems to play a role. Note: I'm writing this off the top of my head so please check. – alvarezpaiva Apr 10 '12 at 18:54
On the other hand, I think if the energy functional satisfies an appropriate ellipticity or convexity condition, then any critical point is "locally minimal". I think the same argument used for geodesics can be used more generally. – Deane Yang Apr 10 '12 at 19:26
I think one also needs to be careful about the regularity of the class of solutions. For instance if $\Sigma$ is a branched minimal surface in $\Real^3$ then $\Sigma$ locally minimizes area away from the branch set but cannot minimize area in any neighborhood of a branch point. – Rbega Apr 10 '12 at 21:36
Yes, I agree about the need to assume sufficient regularity for the solution. – Deane Yang Apr 10 '12 at 22:00
up vote 5 down vote accepted

You are asking a very classical question, which is the question of sufficient conditions for a minimum in the calculus of variations. Quite a lot is known about conditions on Lagrangians that ensure minimality for local solutions.

For example, it is known that the Lagrangian for $k$-dimensional area in a Riemannian manifold has this property for all $k$. (The case $k=1$ is the case of geodesics.) The technique that works in this case is the technique of calibrations, developed to a very high degree in the fundamental paper Calibrated Geometries by Harvey and Lawson.

For the more general case, you should take a look at Volume II of Giaquinta and Hildebrandt's Calculus of Variations. There you will find methods due to Weyl, de Donder, Caratheodory, etc.

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Just a comment on terminology: in classic texts (and in Giaquinta-Hildebrandt) calibrations are referred to as "null Lagrangians". – alvarezpaiva Apr 12 '12 at 16:36
@alvarezpaiva: That's true, but, I, personally, don't like the term because it's a little misleading. It's not the Lagrangian itself that is 'null'; instead, it's the Poincaré-Cartan form of the Lagrangian that's null, i.e., the Euler-Lagrange equations of the Lagrangian vanish identically. – Robert Bryant Apr 12 '12 at 18:41
@Robert: I don't like the term either, but there was a risk that someone following the classic references you mentioned would not find the term "caibration" anywhere. – alvarezpaiva Apr 13 '12 at 6:53

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