On the instability of the planetary problem

SpeakerJacques Fejoz
Date: 3.05.2024 at 17:00

The planetary problem is a nearly integrable approximation of the Solar System, where planets move around a very massive Sun. In the first approximation, planets describe Kepler ellipses. Because of the mutual attraction of planets, elliptical elements slowly vary in time (it is the so-called variation of constants). But do these variations have zero average, as the stability theorems of Laplace-Lagrange tend to assert, or can they pile up?

“Arnold’s theorem” shows that a set of positive Lebesgue measure of initial conditions leads to quasi periodic motions, with no collisions and no ejections, and along which adiabatic invariants cannot drift. Yet, this theorem does not preclude large instabilities, as foreseen by PoincarĂ©, as conjectured by Arnold in 1964, and as seen in long term numerical simulations by Laskar and others from the 1990s on.

Indeed, it turns out that, the problem with three planets (or more planets, conjecturally), a set of positive Lebesgue measure of initial conditions leads to random motions, along which a planet may for example switch its direction of revolution, or the major semi-axis of another planet may grow in an arbitrary ratio. We will describe some ideas of the proof of existence of such motions. This is a joint work with Andrew Clarke and Marcel Guardia.