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In the paper, we offer a method for studying an extremal in the classicalcalculus of variation in the presence of various degenerations. This method isbased on introduction of Weierstrass type variations characterized by anumerical parameter. To obtain more effective results, introduced variationsare used in two forms: in the form of variations on the right with respect tothe given point, and in the form of variations on the left with respect to thesame point. The research is conducted under the assumption that along theconsidered extremal the Weierstrass condition and the Legendre conditiondegenerate, i.e. they are fulfilled as equalities at separate points or on someintervals. Two types of new necessary conditions are obtained: of equality typeand of inequality type conditions for a strong and also a weak local minimum.Given specific examples and counterexample show that some of the necessaryminimum conditions obtained in this article are strengthening and refining ofthe corresponding known results in this direction.

Necessary conditions for a minimum in classical calculus of variation problems in the presence of various degenerations