关于半长轴稳定性的Laplace-Lagrange定理的一个反例
A counterexample to the theorem of Laplace-Lagrange on the stability of semimajor axes
摘要：
长期以来，人们一直认为，在牛顿的行星问题中，半长轴是稳定的。在十九世纪的过程中，拉普拉斯，拉格朗日和其他人在这个方向上给出了越来越强的论据，从而最终形成了通常被称为第一个拉普拉斯-拉格朗日稳定性定理。在三行星问题中，我们证明了外行星的半长轴沿其发生大的随机变化的轨道的存在性，从而反驳了拉普拉斯-拉格朗日定理。不稳定的时间随着行星质量的负幂而变化。我们发现的轨道不属于涅霍罗舍夫-尼德曼理论的范围，因为它们不受角动量守恒的限制，而且哈密顿量相对于开普勒作用量不是（一致）凸的。
英文摘要：
A longstanding belief has been that the semimajor axes, in the Newtonian planetary problem, are stable. In 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 theorem of Laplace-Lagrange. 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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