Dynamics, Stability and Stochastic Analysis of Astrophysical Systems

该提案旨在开发几种重要的新技术和工具，以加深我们对随机动力学和经典轨道动力学的理解。这些具体问题是由天体物理学中的几个关键问题引发的，特别是对暗物质晕、银河核球和太阳系外行星系统动力学的分析。同时，该项目将针对当前应用数学感兴趣的几个经典问题开展新的研究方向，包括随机强迫希尔方程的分析、随机矩阵特征值的渐近分析、非牛顿势中的粒子动力学等。 该提案由三个相互关联的部分组成。第一个涉及对暗物质晕等扩展质量分布中测试粒子动力学中出现的轨道不稳定性的分析。 我们的分析得出了随机强迫希尔方程，可以通过分析随机矩阵的无限乘积来研究该方程。第二部分讨论了太阳系外行星系统中湍流的作用，并引发了随机摆问题的研究。这些系统的长期动态也可以通过具有随机参数的离散图来描述。 最后一部分考虑暗物质晕中的潮汐剥离现象。这项工作涉及研究小光晕落入大光晕时的轨道，包括动态摩擦及其对轨道动力学的影响。在这里，首要目标是了解物质密度分布的近乎普遍的形式，特别是它们如何受到较小光晕被同化到较大结构的影响。在过去的十年中，天体物理学领域经历了前所未有的大量观测发现和理论突破。这些发现包括银河系中心的超大质量黑洞、加速的宇宙、宇宙微波背景波动的测量、太阳系外行星和褐矮星。 观测的进步是通过探测器、望远镜和航天器等技术创新实现的。许多理论进展都是通过数值模拟实现的，而数值模拟又受益于计算机不断增长的功能。 然而不幸的是，这门科学的第三个支柱——对这些新发现的天文物体和物理现象的分析理解——却滞后了。该领域分析工作的困难之一是所研究的天体物理系统的巨大复杂性。特别是，混沌动力学和对初始条件的敏感依赖性在许多情况下都会出现，使得天体物理学家使用的传统分析方法难以取得进展。然而，本文提出的新数学工具的开发和应用将促进这些天体物理问题的进展，并将在许多其他应用中有用。 该项目将在多个方面产生教育影响，包括研究生的培训、研究生和本科生的教育，以及通过公开讲座向广大民众进行宣传。

This proposal seeks to develop several important new techniques and tools to further our understanding of stochastic dynamics and classical orbital dynamics. The specific problems are motivated by several key issues in astrophysics, in particular, the analysis of the dynamics of dark matter halos, galactic bulges, and extra-solar planetary systems. At the same time, this project will undertake new research directions concerning several classic problems of current interest in applied mathematics, including the analysis of the stochastically forced Hill's equation, analysis of the asymptotics of the eigenvalues of random matrices, and the dynamics of particles in non-Newtonian potentials. The proposal consists of three interrelated parts. The first pertains to the analysis of orbital instabilities that arise in the dynamics of test particles in extended mass distributions such as dark matter halos. Our analysis leads to a stochastically forced Hill's equation which can be studied by analyzing infinite products of random matrices. The second part discusses the role of turbulence in extra-solar planetary systems and leads to the study of stochastic pendulum problems. The long term dynamics of these systems can also be described by a discrete map with random parameters. The final part considers the phenomenon of tidal stripping in dark matter halos. This work involves the study of orbits of small halos as they fall into larger ones, including dynamical friction and its effects on orbital dynamics. Here, the overarching goal is to understand the nearly universal form found for the matter density profiles, and in particular how they are affected by smaller halos being assimilated into larger structures.The field of astrophysics has experienced an unprecedented number of observational discoveries and theoretical breakthroughs in the past decade. These discoveries include super-massive black holes in galactic centers, the accelerating universe, measurement of the fluctuations in the cosmic microwave background, extra-solar planets, and brown dwarfs. The observational progress has been made possible through technological innovations, including detectors, telescopes, and spacecraft. Much of the theoretical progress has taken place through numerical simulations, which in turn have benefited from the ever-growing capabilities of computers. Unfortunately, however, the third pillar of this science --- the analytic understanding of these newly discovered astronomical objects and physical phenomena --- is lagging behind. One of the difficulties associated with analytic work in this area is the enormous complexity of the astrophysical systems under study. In particular, chaotic dynamics and sensitive dependence on initial conditions arise in many contexts and render it difficult to make progress through the traditional analytic methods used by astrophysicists. However, the development and application of new mathematical tools, as proposed herein, will facilitate progress on these astrophysical issues, and will be useful in many additional applications. This project will have educational impacts on several fronts, including the training of graduate students, the education of graduates and undergraduates, and reaching out to the general population through public lectures.