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.
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 $it$.)