Euler's original derivation of elastica equation

TL;DR

This paper revisits Euler's original derivation of the elastica equation, revealing its connection to Noether's theorem and the modified KdV equation, thus clarifying its mathematical foundations.

Contribution

It provides a modern interpretation of Euler's derivation, linking it to Noether's theorem and the static modified KdV equation, which was previously unrecognized.

Findings

Euler's derivation used Noether's theorem before it was formally published.

The elastica equation is equivalent to the static modified KdV equation.

Euler's original approach can be understood through modern mathematical frameworks.

Abstract

Euler derived the differential equations of elastica by the variational method in 1744, but his original derivation has never been properly interpreted or explained in terms of modern mathematics. We elaborate Euler's original derivation of elastica and show that Euler used Noether's theorem concerning the translational symmetry of elastica, although Noether published her theorem in 1918. It is also shown that his equation is essentially the static modified KdV equation which is obtained by the isometric and isoenergy conditions, known as the Goldstein-Petrich scheme.