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?