On a calculus of variations problem

TL;DR

This paper analyzes the classical minimal surface problem of a soap film between two rings using an analytical approach linked to Sturm-Liouville theory, discussing energy interpretations and soliton potentials in the second variation analysis.

Contribution

It introduces an analytical method connecting the soap film problem to Sturm-Liouville theory and explores energy and soliton concepts in the stability analysis.

Findings

Relation to Sturm-Liouville problem established

Goldschmidt condition interpreted energetically

Soliton potential appears in second variation analysis

Abstract

The paper is of scientific-methodical character. The classical soap film shape (minimal surface) problem is considered, the film being stretched between two parallel coaxial rings. An analytical approach based on relations to the Sturm-Liouville problem is proposed. An energy terms interpretation of the classical Goldschmidt condition is discussed. Appearance of the soliton potential in course of the second variation analysis is noticed.