In a book by Cash et al, I see the mention of the so-called Elastica problem (pg 221 in the link here).

The problem is presented as a system of ODEs,

$$ x' = \cos (\phi) $$

$$ y' = \sin (\phi) $$

$$ \phi' = \kappa $$

$$ \kappa' = F \cos (\phi) $$

and when I looked for where these ODEs come from I could not find a clear derivation anywhere. There are references to Bernoulli, Euler, but the formulation looked different. Does anyone know a book, reference that shows where these equations come from?


  • 3
    $\begingroup$ Reviewers: likely proof that the question is on-topic. $\endgroup$ – peterh - Reinstate Monica Aug 30 '19 at 20:34
  • $\begingroup$ So $\kappa'$ is just $x'$ times a constant, or is $F$ a function? $\endgroup$ – Carl Witthoft Sep 3 '19 at 13:36

A. G. Greenhill, The applications of elliptic functions, Macmillan, London & NY, 1892, pp. 87-88.

If you read French, a much clearer and more comprehensive discussion is in G.-H. Halphen, Traite de fonctions elliptiques et de leurs applications, 2-eme partie, Paris Gauthier-Villars et fils, 1888. Chapitre V.

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