Necessary conditions for the extremum in non-smooth problems of variational calculus

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.