Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy
Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy
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DOI：
10.1007/s00023-010-0026-7
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发表时间：
2010-03
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影响因子：
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通讯作者：
Yuecai Han;Yong Li;Y. Yi
中科院分区：
文献类型：
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作者：
Yuecai Han;Yong Li;Y. Yi
研究了高阶真简并的近可积Hamilton系统中准周期不变环面的存在性，哈密顿量的可积部分涉及几个时间尺度，并且在每个时间尺度上，对应的哈密顿量仅取决于部分作用量变量。这样的哈密顿系统经常出现在天体力学问题中，例如，在受扰开普勒问题中，如限制和非限制三体问题和空间月球问题，其中几个质量很小的物体与两个大质量物体耦合，并且几乎可积的哈密顿系统自然涉及不同的时间尺度。利用KAM方法，我们将证明在Bruno-Rüssmann型的某些高阶非退化条件下，与可积部分相关的大多数准周期不变环面在不可积扰动之后将继续存在。这实际上得出了这样一个适当退化的哈密顿系统的KAM度量稳定性的结论。
We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.