具有刚体转动的平面引力多体问题的混沌动力学
Chaotic dynamics in the planar gravitational many-body problem with rigid body rotations
摘要：
在过去的十年中，冥王星小卫星的发现引起了人们对矮行星卫星动力学的关注。考虑到这样的系统，我们研究了一个平面$N$体系统，其中所有的机构是点质量，除了一个单一的刚体。然后，我们提出了一个简化的模型，包括一个平面$N$体问题的刚体作为一个一维连续体（即身体被视为一个杆与任意质量分布）。这种模型提供了一个很好的近似高度不对称的几何形状，如最近观察到的星际小行星'Oumuamua，但也适合分析。通过Melnikov方法，我们解析地证明了当其中一个轨道是近圆轨道时同宿混沌的存在性，并给出了当轨道较复杂时混沌的数值证据.我们表明，在参数空间中的混沌的程度是密切相关的偏离一个纯粹的圆形轨道。这些结果表明，混沌是普遍存在的多体问题时，一个或多个刚体表现出非球形和高度不对称的几何形状。混沌旋转的激发似乎不需要潮汐耗散、潮汐变化或轨道共振。这样的动力学给出了一个可能的解释路线观察到的混沌动力学$N$体系统，如冥王星系统，其中一些机构是高度非球形的。
英文摘要：
The discovery of Pluto's small moons in the last decade brought attention to the dynamics of the dwarf planet's satellites. With such systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid 'Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits non-spherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly non-spherical.
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