We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.

中文翻译：

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Sharp Nekhoroshev 对围绕周期轨道的三体问题的估计

我们通过使用周期平均技术，在行星和受限圆形情况下，为平面三体问题构建了一个具有尖锐常数的类似涅霍罗舍夫的稳定性结果。我们的构造可以推广到任何近乎可积的哈密顿系统，其不受扰动的哈密顿量是准凸的。仔细考虑了常数对复汉密尔顿量的分析宽度的依赖性。这允许深入分析了解此类技术在确保涅霍罗雪夫稳定性高扰动方面的局限性，并提供有关如何在某些情况下克服此类障碍的提示。最后，考虑了两个具有具体值的例子，一个是行星情况，一个是受限情况。