An elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system Arnold in his essay On teaching mathematics made the following statement:

The de-geometrisation of mathematical education and the divorce from physics sever these ties. For example, not only students but also modern algebro-geometers on the whole do not know about the Jacobi fact mentioned here: an elliptic integral of first kind expresses the time of motion along an elliptic phase curve in the corresponding Hamiltonian system.

I admit that I find myself among those who didn't know this fact about elliptic integrals. Can anyone elaborate on that please? A layman's explanation is most welcome.
 A: Arnold is being a bit cryptic here, but I'll try to guess what he refers to. Earlier in the text, he says:
"Jacobi noted, as mathematics' most fascinating property, that in it one and the same function controls both the presentations of a whole number as a sum of four squares and the real movement of a pendulum."
The pendulum equation is
$$
d^2 \theta/dt^2 = -\omega^2 \sin\theta
,
$$
where $\theta$ is the pendulum's angle to the vertical (with $\theta=0$ downwards), $t$ is time, and $\omega=\sqrt{g/l}$ ($g$=gravitational acceleration, $l$=length of the pendulum).
If we look at solutions whose energy is low enough so that the pendulum doesn't swing over the top, there is a maximal angle of deflection $\theta_{\max}$. These solutions will form closed curves in the phase portrait (see picture in this blog post, for example), and I believe Arnold is referring to these curves, though I don't know why he calls them "elliptic", since they are not ellipses. Perhaps he's thinking of their relationship to elliptic functions? Namely, such a solution (if we take $\theta=0$ at time $t=0$) is given explicitly by the formula
$$
\theta(t) = 2 \arcsin(k \operatorname{sn}(ωt,k))
, 
$$
where
$k = \sin(\theta_{\max}/2)\in(0,1)$,
and $\operatorname{sn}(z,k)$ is the Jacobi elliptic function "sinus amplitudinis" with parameter $k$.
This function is doubly periodic in $z$ with a real period $4K(k)$ and an imaginary period $2iK'(k)$, where $K(k)$ is the complete elliptic integral of the first kind,
and $K'(k)$ is shorthand for $K(\sqrt{1-k^2})$.
(I've written a little about this here.)
So the period of the pendulum is $4K(k)/\omega$, which I think is what Arnold is talking about.
(Fun fact: the imaginary period of the $\operatorname{sn}$ function corresponds to the period of oscillation that one would get if gravity were pointing upwards, since replacing $g$ by $-g$ in the pendulum equation is mathematically equivalent to replacing $t$ by $it$.)
