In the paper, we proposed an approach for studying strong and weak extremums in non-smooth vector problems of calculus of variation, namely, in classic variational problems with fixed ends and with a free right end, and also in a variational problem with higher derivatives. The essence of the proposed approach is to introduce a Weierstrass type variation characterized by a numerical parameter. Necessary conditions for minimum containing as corollaries the Weierstrass condition, its local modification and also the Legendre and transversality conditions are obtained. In the case when the Legendre condition degenerates, equality and inequality type necessary conditions are obtained for the weak local minimum. The examples showing the content-richness of the obtained main results are given.

在本文中，我们提出了一种研究变分非光滑向量问题中强弱极值的方法，即具有固定端和自由右端的经典变分问题，以及具有更高导数的变分问题. 所提出方法的本质是引入以数值参数为特征的 Weierstrass 型变体。获得了作为推论的最小值的必要条件，即 Weierstrass 条件、其局部修改以及勒让德和横向条件。在勒让德条件退化的情况下，得到弱局部极小值的等式和不等式的必要条件。给出了显示所获得的主要结果的内容丰富性的示例。