La Lettre de l'Académie des sciences n°37-38

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La Lettre

Galileo (1564-1632) observing the oscillation of a

hanging lamp in the Pisa Cathedral (Fresco by Luigi

Sabatelli, circa 1840)

equations”. Nowadays, such secrets may seem childish but the invention of the differential calculus was an

unprecedented revolution, inasmuch as it made it possible to translate into equations all sorts of physical

problems based on tendencies and variations. Did Albert Einstein himself not declare this was the most

important step ever taken in physics?

The differential calculus: promises kept

One famous example of it is the stability of the solar system: knowing

the equations that describe the motions of stars, is it possible to predict

whether the solar system will stay as we know it or, on the contrary,

be devastated by a major cataclysm, such as a collision between

two planets? Thanks to differential equations and Newton’s law – the

sum of gravitational forces is equal to mass times acceleration – the

problem could now be translated into mathematics. From then on,

everything would go very fast. Ninety generations had passed since Thales and his disciples dreamt of

mathematizing the motions of planets; yet, once differential equations discovered, it would take less than

12 generations to send a human being on the moon; and two more for a machine to land on a comet

and transmit us a wealth of information. Each step of the way, though, was fraught with obstacles and

unexpected developments, and involved the parallel efforts of an ever-growing number of scientists.

Let’s consider one of those innumerable

stories that have been initiated by our heroes

of the 17

Century. It started with a familiar

object, the pendulum: a mass suspended at

the end of a thread. Pendulums have always

been there, under one form or the other; yet,

apparently, only circa 1600 did they start to be

really observed, with Galileo. The illustrious

Italian remarked, and rightly so, that the

oscillation period does not depend on mass,

but varies according to the amplitude of the

motion. When it came to using the regularity

of pendulum oscillations to build clocks,

such a variation limited accuracy. Huygens

then had a purely mathematical question:

is it possible to constrain the motion of a

pendulum by a well-chosen curve, so as to

make its oscillation period independent from

its energy? The solution was no other than the

famous cycloid, i.e. the curve described by a