Second order necessary condition for a strong minimum in the classical problem of calculus of variations

The paper offers a second order necessary condition for a strong minimum in the standard problem of calculus of variations. No idea of such a result seems to have appeared in the classical theory. But a simple example given in the paper shows that the condition can work when all known conditions fail. At the same time, the proof of the proposition is fairly simple. It is also explained in the paper that the condition effectively works only for problems with integrands not convex with respect to the last (derivative) argument.