Abstract:The main result to be presented at the talk contains a second order necessary condition for a strong minimum in the standard problem of calculus of variations. Novelty of this result is emphasized by the fact that no idea of a possibility of such a condition has ever appeared in the classical theory. A simple example shows that the theorem can work when other known necessary conditions fail. Originally the theorem was obtained as a consequence of a corresponding result for a sufficiently general class of optimal control problems (Calculus of Variations and PDEs (2020), 59:83) which will be brie y explained in the second part of the talk. But an independent proof of the theorem is noticeably simpler and will be described in sufficient details. The key element of the proof is a version of the convexication theorem proved in 1930 by N.N. Bogolyubov (which is really a fundamental result closely connected with the classical Weierstrass necessary condition but, strangely, typically not mentioned in courses on calculus of variations).