Abstract:

Humankind has scrutinized the sky since the origins of time. The first traces come from Babylonian clay tablets from 2000 B.C. where relative positions of stars are recorded in a mysterious antique unit system. Numerous clay tablets from the period 700 B.C. to the Christian Era records the position of the moon and planets. The accuracy of the data provides evidence that Babylonians were able to predict astronomical ephemeris as eclipses. The Greeks geometrized Babylonian astronomy and astronomers, as Hipparcus and Ptolemy, defined the astronomical theory. Ptolemy's Almagest is the oldest written preserved account on astronomy. The genius Newton created the modern Gravitational Theory and invented simultaneously Infinitesimal Calculus.

Despite the progress over the centuries, the method of prediction of astronomical ephemeris by the Babylonians has remained unjustified. Why the bodies in the solar system do have quasi-periodic motions t The main progress (after more than 4000 years !) on this problem was achieved in the last century by K.A.M. theory and the theory of Small Divisors.

After a historical survey, we will go over the mathematics of modern theories and present the challenging open problems for the next 4000 years in this ever-young subject that has shaped modern Science.