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Euler's original derivation of elastica equation

Shigeki Matsutani

TL;DR

This paper reinterprets Euler's original elastica derivation in modern terms, showing that translational symmetry via a Noether current under an isometric constraint drives the elastica toward the static modified KdV (SMKdV) equation rather than the classical static sine-Gordon equation. It connects Euler's 18th-century variational approach to contemporary symmetry-based formulations, including the Goldstein–Petrich representation, and clarifies how SMKdV governs the isometric deformations of elastica and its excited-state structures. The work illuminates the deep geometric and elliptic-function structure underlying elastica, including its elliptic-integral representations and connections to Weierstrass-type functions, and argues that Euler's insight was ahead of its time. Overall, the paper provides a rigorous historical-physical interpretation of Euler's elastica, highlighting symmetry, conservation laws, and the natural emergence of SMKdV as the governing equation for isometric elastica.

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.

Euler's original derivation of elastica equation

TL;DR

This paper reinterprets Euler's original elastica derivation in modern terms, showing that translational symmetry via a Noether current under an isometric constraint drives the elastica toward the static modified KdV (SMKdV) equation rather than the classical static sine-Gordon equation. It connects Euler's 18th-century variational approach to contemporary symmetry-based formulations, including the Goldstein–Petrich representation, and clarifies how SMKdV governs the isometric deformations of elastica and its excited-state structures. The work illuminates the deep geometric and elliptic-function structure underlying elastica, including its elliptic-integral representations and connections to Weierstrass-type functions, and argues that Euler's insight was ahead of its time. Overall, the paper provides a rigorous historical-physical interpretation of Euler's elastica, highlighting symmetry, conservation laws, and the natural emergence of SMKdV as the governing equation for isometric elastica.

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.

Paper Structure

This paper contains 7 sections, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Euler's derivation of the elastica equation in 1744
  3. Discussion
  4. Appendix: static sine-Gordon equation and SMKdV equation
  5. Elastica equation by Lagrange and Gauss
  6. The first derivation of the SMKdV equation
  7. The second derivation of the SMKdV equation

Figures (3)

  • Figure 1: Shape of elastica: $s$ is the arclength and $\varphi$ is the tangential angle
  • Figure 2: Euler's diagrams of elastica: Euler illustrated the shapes of the elasticae by numerical computations E2
  • Figure 3: Lagrange's diagrams of elastica La2