预辛系统及多尺度哈密顿系统的KAM-型定理

Presymplectic dynamical systems arise naturally in many phsical problem, such as degenerate Lagrangian systems, Hamiltonian systems with constrains, time-dependent Hamiltonian systems and in control theory. Hamiltonian systems with high order proper degeneray freguentaly root in celestial mechanics problem, for instence, in spatial lunar problem s in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonians systems naturally involve different time scales.. Kolmogorov-Arnold-Moser theory, which was proved in 1960's, is the most important breakthrough in the stability problem of dynamical systems. After that, one has proved the persistence of invariant tori, the persistence of lower dimensional tori for generalized Hamiltonian systems, the existence of quasi-periodic solution for infinite dimensional Hamiltonian systems and so on. Now, KAM-tye theorem is developed as one of the most important probelm in the stability of dynamical systems. Our project is focused on this problem, specifically, we concentrate on the existence of whiskered tori for presymplectic systems and the persistence lower dimensional tori for Hamiltonian systems with high order proper degeneracy. These two results are positive in both developing of the KAM-type theory and solving of the physical problem.

预辛系统，即附有预辛结构的动力系统，具有广泛的应用背景。例如，退化的拉格朗日系统、带有限制的哈密顿动力系统、时间t相关的哈密顿系统，都可以化简为相应的预辛动力系统，而多尺度近可积哈密顿系统，是天体力学中的多体问题为背景抽象出来的数学模型。. Kolmogorov-Arnold–Moser 理论，是上世纪动力系统稳定性研究领域的重大突破。此后，人们运用这一思想，证明了有限维保守系统的不变环面保持性、低维环面保持性；无穷维哈密度系统的拟周期解存在性等重要结果。发展至今，KAM环面保持性定理，已经成为动力系统稳定性研究领域的重要课题之一。本项目将致力于这方面的研究，集中考虑预辛动力系统的whiskered环面的存在性，以及多尺度、高阶退化哈密顿系统的低维环面保持性问题。这两个问题的研究，无论是对KAM-型理论体系自身的发展与完善，还是对解决实际的物理问题，都是很有意义的。

本项目研究内容及结果共分为两个部分。. 第一部分，我们关注的是以天体力学中的开普勒问题为背景的多尺度近可积哈密顿系统。类似于经典近可积系统，我们研究了多尺度系统的共振环面，并且证明了大多数的庞加莱非退化的低维环面在小扰动下依然具有保持性。该证明运用了标准型化简和修正的KAM迭代，确切的说，为克服多尺度造成的迭代序列不收敛等技术问题，我们首先进行了有限次迭代，将系统扰动阶数推高，再用KAM迭代格式证明低维环面的存在性。. 第二部分，我们证明了在恰当预辛映射下，系统 whiskered 环面的存在性。证明过程以后验型迭代格式为基础，这种迭代并不依赖于作用-角变换理论，而是先假设系统有近似不变环面，再通过迭代修正误差项。迭代的收敛性依赖于系统的几何结构。由于迭代过程中涉及到的参数少，这类迭代格式更容易得到数值实现结果。