A Counterexample to the Theorem of Laplace–Lagrange on the Stability of Semimajor Axes

Abstract

A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. Our the course of the XIX century, Laplace, Lagrange and others gave stronger and stronger arguments in this direction, thus culminating in what has commonly been referred to as the first Laplace–Lagrange stability theorem. In the problem with 3 planets, we prove the existence of orbits along which the semimajor axis of the outer planet undergoes large random variations thus disproving the conclusion of the Laplace–Lagrange theorem. The time of instability varies as a negative power of the masses of the planets. The orbits we have found fall outside the scope of the theory of Nekhoroshev–Niederman because they are not confined by the conservation of angular momentum and because the Hamiltonian is not (uniformly) convex with respect to the Keplerian actions.