论文标题
现实行星系统中轨道的数值搜索中的符号方法
Symplectic methods in the numerical search of orbits in real-life planetary systems
论文作者
论文摘要
本文的目的是说明用于实际目的的符号几何形状中的方法的使用。我们的意图受众是对哈密顿系统轨道感兴趣的科学家（例如，三体问题）。本文追求的主要指示是：（1）给定两个定期轨道，决定何时可以由正规家庭联系； （2）使用浮子理论中的数值不变性，这些不变性有助于预测分叉存在的轨道的存在； （3）将符号+/-连接到封闭对称轨道的每个椭圆形或双曲线浮子乘数，从而将经典的krein-moser符号推广到还包括双曲线外壳； （4）以视觉，易于实现和资源有效的方式进行以上所有操作。数学框架由第一和第三作者提供，其中``Broucke稳定图''重新发现了``Broucke稳定图''，但通过上述符号进一步完善，并且根据符号组的GIT商的代数进行了重新完整。上述框架的优点是它适用于对任意哈密顿系统的闭合轨道的研究。此外，如果系统以``反射''的形式接纳对称性的情况，即反透明接头，对于许多感兴趣的系统来说，这种情况是这种情况，对称的轨道提供的信息更富裕，并且可以区分更多的对称轨道。太空任务设计行业中的几个知名家庭，例如光环轨道，在现实生活太空任务中无处不在。我们将根据木星 - 欧洲和土星 - 塞拉德斯系统的细胞映射方法进行数值工作。这些目前是感兴趣的系统，属于NASA等太空机构的议程，因为这些冰冷的卫星被认为是携带适合外星生命的条件的候选者。
The intention of this article is to illustrate the use of methods from symplectic geometry for practical purposes. Our intended audience is scientists interested in orbits of Hamiltonian systems (e.g. the three-body problem). The main directions pursued in this article are: (1) given two periodic orbits, decide when they can be connected by a regular family; (2) use numerical invariants from Floer theory which help predict the existence of orbits in the presence of a bifurcation; (3) attach a sign +/- to each elliptic or hyperbolic Floquet multiplier of a closed symmetric orbit, generalizing the classical Krein--Moser sign to also include the hyperbolic case; and (4) do all of the above in a visual, easily implementable and resource-efficient way. The mathematical framework is provided by the first and third authors, where the ``Broucke stability diagram'' was rediscovered, but further refined with the above signs, and algebraically reformulated in terms of GIT quotients of the symplectic group. The advantage of the above framework is that it applies to the study of closed orbits of an arbitrary Hamiltonian system. Moreover, in the case where the system admits symmetries in the form of ``reflections'', i.e. anti-symplectic involutions, which is the case for many systems of interest, the information provided for orbits which are symmetric is richer, and one may distinguish more symmetric orbits. This is the case for several well-known families in the space mission design industry, such as the Halo orbits, which are ubiquitous in real-life space missions. We will carry out numerical work based on the cell-mapping method, for the Jupiter-Europa and the Saturn-Enceladus systems. These are currently systems of interest, falling in the agenda of space agencies like NASA, as these icy moons are considered candidates for harbouring conditions suitable for extraterrestrial life.
