Elliptic curves for high school students

11 May 2009 at 8:08 pm Leave a comment

I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve…

Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE \ddot{x} + \sin x = 0 which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as
p^2 = q(q-q_0)(q-2l) = P(q)
where 0 and 2l are the extremal values of the altitude q, q_0 is the highest altitude which can be reached with a given energy (even if q_0 > 2l, which corresponds to the pendulum make full rotations around its axis), and p is the vertical momentum of the pendulum.

In this setting, there are classical Hamilton relations dq = p dt, dp = P'(q) dt, so the differential form dt = dq/p turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at t = t_1 + t_2 can be deduced from its position at times t_1 and t_2 by the classical secant-tangent law.

The notes for the talk (in French) are available here.

Entry filed under: algebraic geometry, calculus, curves, english. Tags: , , , , .

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