Foundations of the calculus of variations in generalized function algebras

TL;DR

This paper develops a framework using generalized function algebras to analyze highly singular variational problems, deriving conditions for extremals, symmetries, and applications in physics and geometry.

Contribution

It introduces a novel approach employing generalized functions for variational calculus, extending classical methods to singular problems and establishing new theoretical connections.

Findings

Derived Euler-Lagrange equations in generalized function setting

Established a version of N"other's theorem for singular problems

Applied the framework to problems in mechanics and geometry

Abstract

We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we introduce the first and second variation of a variational problem. We then derive necessary (Euler-Lagrange equations) and sufficient conditions for extremals. The concept of association is used to obtain connections to a distributional description of singular variational problems. We study variational symmetries and derive an appropriate version of N\"other's theorem. Finally, a number of applications to geometry, mechanics, elastostatics and elastodynamics are presented.