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?



Extremals versus minima for variational problems
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.


