Posts filed under ‘calculus’

Elliptic curves for high school students

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.

11 May 2009 at 8:08 pm Leave a comment

Rational approximations of √2

Introducing undergraduates to rational approximations of √2 can be an opportunity to insidiously tell them about many parts of mathematics they certainly don’t want to hear about. In a less pessimistic way, I would say this is a nice way to illustrate the use of several theories in abstract mathematics.

First, you may want to tell them it is not a rational number, which could be easy, unless they never heard about factoring integers.

Then, you could use classical sequences from high school classes: it is easy to check that iterating u_{n+1} = \frac{u_n+2}{u_n+1} converges to ±√2, and setting U_n = \frac{u_n-\sqrt 2}{u_n+\sqrt 2} they will even be able to give an explicit formula with a geometric sequence. There is of course a well-known algorithm which speeds up considerably the computation : Newton’s method. This can be illustrated geometrically by drawing the graph of a function having √2 as a root (for example x \mapsto x^2-2):

  1. take a rough approximation, such as x=1
  2. imagine the function is affine (replacing the graph by its tangent)
  3. use the approximation of the function to calculate an approximation of the solution
  4. use this newly found rough solution to iterate from step 1

This method requires iterating v_{n+1} = \frac 1 2 (v_n + 2/v_n), and converges considerably faster. Both methods yield sequences of rational numbers converging to √2, so should be considered as methods to obtain fractional numbers giving good approximations of √2. (more…)

23 February 2009 at 5:23 pm Leave a comment

Intégrales elliptiques et moyenne arithmético-géométrique

Le calcul de la moyenne arithmético-géométrique est un algorithme simple et très puissant découvert par Gauss permettant de calculer les intégrales elliptiques. L’excellent article de David Cox [1] dans L’enseignement mathématique expose l’historique des découvertes et recherches de Gauss sur le sujet.

La moyenne arithmético-géométrique a été définie par Lagrange, et est calculée de la manière suivante : si a et b sont deux réels positifs, on définit leur moyenne géométrique G = \sqrt{ab} et leur moyenne arithmétique A = \frac{a+b}{2}. Il est bien connu que G \leq A, et on a

A-G = \frac 1 2 (\sqrt b - \sqrt a)^2

Cette relation permet de montrer que si on définit des suites (a_n) et (b_n) telles que a_{n+1} et b_{n+1} sont les moyennes arithmétique et géométrique de a_n et b_n, ces suites convergent vers une même limite, notée M(a,b), la moyenne arithmético-géométrique.

Gauss remarqua à l’aide d’un changement de variable (très) astucieux que l’intégrale

\int_{0}^{\pi/2} \frac{d\theta}{\sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta}}

restait invariante en remplaçant a et b par leurs moyennes arithmétique et géométrique. Il en déduit la valeur \frac{\pi}{2M(a,b)}. Il montra aussi que la longueur d’un quart de lemniscate de Bernoulli est \pi / 2 M(1, \sqrt 2).
On sait également qu’il relia ces propriétés à celles des fonctions thêta.

6 January 2009 at 8:12 pm Leave a comment