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Mathematical Physics

arXiv:2411.09227 (math-ph)
[Submitted on 14 Nov 2024 (v1), last revised 17 Nov 2024 (this version, v2)]

Title:Euler's original derivation of elastica equation

Authors:Shigeki Matsutani
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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.
Subjects: Mathematical Physics (math-ph); Differential Geometry (math.DG); Exactly Solvable and Integrable Systems (nlin.SI)
Cite as: arXiv:2411.09227 [math-ph]
  (or arXiv:2411.09227v2 [math-ph] for this version)
  https://doi.org/10.48550/arXiv.2411.09227

Submission history

From: Shigeki Matsutani [view email]
[v1] Thu, 14 Nov 2024 06:43:59 UTC (14 KB)
[v2] Sun, 17 Nov 2024 07:34:48 UTC (15 KB)
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