Symplectic techniques in the restricted three body problem

天体力学的N体问题，即N个天体在相互引力作用下的运动，自牛顿时代以来一直是物理学和数学发展的主要推动力。例如，它导致了混沌的发现，以及阿诺德、科尔莫戈罗夫和莫泽的稳定性理论。虽然二体(或开普勒)问题是完全可积的，并且可以显式求解，但三体的情况已经表现出错综复杂的、有时甚至是混沌的动力学，远未被理解。由庞加莱提出的一个有趣的特殊情况是，当第三个天体(卫星)的质量与前两个天体(初级天体)相比可以忽略不计时。另外，如果两个重体在圆周上运动，并且所有三个物体在同一平面上运动，那么切换到旋转坐标系将问题转化为在与时间无关的两个自由度的哈密顿系统中设置的三维能级上的运动。这种情况称为(平面圆形)限制性三体问题。除了理论意义外，这个问题对空间轨道设计也具有实际意义，因为卫星在地球和月球影响下的运动，或航天器在太阳和地球影响下的运动，大致符合这一框架。这个项目的目的是利用现代辛拓扑技术，特别是全纯曲线理论来研究受限三体问题的动力学。更具体的目标如下：1.在第一临界值以下和以上产生平面受限三体问题的有限能量方程。研究空间受限三体问题的周期轨道族及其omega极限集。定义了双中心Stark-Zeeman系统的不变量，并将其应用于平面受限三体问题中的纤维凸性问题。建立拉格朗日容量的性质，特别是它与Ekeland--Hofer容量的关系。发展了空间旅行中燃料最小化的数学理论，并探索了它与辛拓扑中的概念之间的联系，如Floer同调和Mane临界值。

The N-body problem of celestial mechanics, i.e. the motion of N heavenly bodies under the mutual gravitational attraction, has been a major driving force in the development of physics and mathematics since the times of Newton. For example, it led to the discovery of chaos as well as to the stability theory of Arnold, Kolmogorov and Moser. While the 2-body (or Kepler) problem is completely integrable and can be explicitly solved, already the case of 3 bodies exhibits intricate and sometimes chaotic dynamics that are far from understood. An interesting special case, proposed by Poincare, arises when the mass of the third body (the satellite) is negligible compared to the first two (the primaries). If in addition the two heavy bodies move on circles and all three bodies move in the same plane, then switching to rotating coordinates transforms the problem into the motion on a 3-dimensional energy level set in a time-independent Hamiltonian system with two degrees of freedom. This case is known as the (planar circular) restricted 3-body problem. Besides its theoretical interest, this problem is also of practical relevance for space orbit design because the motion of a satelite under the influence of earth and moon, or a spacecraft under the influence of sun and earth, roughly fit into this framework. The goal of this project is to study the dynamics of the restricted 3-body problem using techniques of modern symplectic topology, in particular the theory of holomorphic curves. More specific goals are the following:1. Producing finite energy foliations for the planar restricted 3-body problem below and above the first critical value.2. Studying families of periodic orbits in the spatial restricted 3-body problem and their omega-limit sets.3. Defining invariants to two-center Stark--Zeeman systems and applying them to the question of fibrewise convexity in the planar restricted 3-body problem.4. Establish properties of the Lagrange capacity, in particular its relation to Ekeland--Hofer capacities.5. Developing a mathematical theory of fuel minimization in space travel and explore its connections to notions in symplectic topology such as Floer homology and Mane's critical value.