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

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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.

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Archive for Rational Mechanics and Analysis, 2024, vol. 248, num.2

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Articles publicats en revistes (Matemàtiques i Informàtica)

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