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?