On sufficient conditions in the classical problem of the calculus of variations

TL;DR

This paper introduces new sufficient conditions for extrema in classical calculus of variations, expanding the applicability of known criteria and providing a framework for broader problem classes with demonstrated examples.

Contribution

It develops a novel concept of a set of integrands to derive new sufficient conditions for minima, including cases where the Weierstrass condition is also sufficient.

Findings

New sufficient conditions for weak and strong minima

Definition of a class of problems where Weierstrass condition is sufficient

Illustrative examples demonstrating the effectiveness of results

Abstract

This article is devoted to obtain new sufficient conditions for an extremum in problems of classical calculus of variations. The concept of a set of integrands is introduced. Using this concept, first and second order sufficient conditions for a weak and strong local minimum, as well as an absolute minimum were obtained. Also, this concept, in particular, allows us to define a class of variation problems for which the necessary Weierstrass condition is also sufficient condition. It is shown that the sufficient conditions for a minimum obtained here have new areas of application compared to the known sufficient conditions of the classical calculus of variations. The effectiveness of the obtained results is illustrated by examples.