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

TL;DR

This paper develops a new method using Weierstrass type variations to derive necessary conditions for minima in calculus of variation problems with degeneracies, refining existing results.

Contribution

Introduces a novel approach with numerical parameter variations to analyze minima under degeneracy conditions, providing strengthened necessary conditions.

Findings

Derived new equality and inequality type necessary conditions for minima.

Applied variations on both sides of points to improve analysis.

Provided examples demonstrating refinement over previous results.

Abstract

In the paper, we offer a method for studying an extremal in the classical calculus of variation in the presence of various degenerations. This method is based on introduction of Weierstrass type variations characterized by a numerical parameter. To obtain more effective results, introduced variations are used 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. The research is conducted under the assumption that along the considered extremal the Weierstrass condition and the Legendre condition degenerate, i.e. they are fulfilled as equalities at separate 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. Given specific examples and counterexample show that some of the necessary minimum conditions obtained in this article are strengthening and refining of the corresponding known results in this direction.