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
    Aug 30, 2019 at 20:34
  • $\begingroup$ So $\kappa'$ is just $x'$ times a constant, or is $F$ a function? $\endgroup$ Sep 3, 2019 at 13:36

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.