Necessary conditions for a minimum in classical calculus of variations in the presence of various types of degenerations

In the paper, a new approach is offered for studying an extremal in the problems of classical calculus of variations in the presence of various types of degenerations. This approach: firstly, is based on using Weierstrass type variations in two forms: in the form of variations on the right with respect to the given point, and in the form of variations on the left with respect to the same point; secondly, allows to get new and stronger results. The research is conducted under the assumption that along the considered extremal the Weierstrass and Legendre conditions as well as other necessary conditions for minimum degenerate, i.e. they are fulfilled as equalities at points or on some intervals. Two types of new necessary conditions are obtained: of equality type and of inequality type conditions for a strong and also a weak local minimum. Specific examples and a counterexample show that some of the necessary conditions for minimum obtained in this article are strengthening and refining of the corresponding known results in this direction.