Arnold said the following in a talk on teaching:
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.
What's the meaning of this? Is the function he mentioned the theta function or something else? What's the relation with the real movement of a pendulum?
He means elliptic functions.
If, instead of using the harmonic oscillator equation for a pendulum, you use the expression involving the actual force, you get an elliptic integral for the time as a function of the angle (see for instance John Baez's example sheet http://math.ucr.edu/home/baez/classical/pendulum.pdf).
Taking the inverse to get the angle as a function of the time you get an elliptic function, by Jacobi's definition as inverse functions of elliptic integrals. Jacobi made that definition in an analogy to how you get sin cos and tan as inverse functions of certain integrals.
As for decompositions into squares, I recall that in Hardy and Wright's Introduction to the Theory of Numbers there are three proofs of the four square theorem. One is "elementary", one uses the integral quaternions/Hurwitz integers, and the third uses elliptic functions. This last one is (a newer version of?) Jacobi's proof.