围绕大质量中心体运动的对称卫星星座
Symmetric constellations of satellites moving around a central body of large mass
摘要：
考虑一个$(1+N)$ -体问题，其中一个质点有质量$m_0 \gg 1$，其余的$N$有么正质量.我们可以假设质量较大的天体（中心天体）在原点静止，与质量较小的$N$天体（卫星天体）的质心重合。两个粒子之间的相互作用力通过$U \sim \frac{1}{r^α},$形式的势来定义，其中$α\in [1,2)$和$r$是粒子之间的距离。在对称性和拓扑约束条件下，我们用变分方法寻找了该系统的周期轨道。此外，我们使用$Γ$收敛理论来研究这些轨道的渐近行为，作为中心体的质量增加。事实证明，拉格朗日作用泛函$Γ$ -收敛到开普勒问题的作用泛函，定义在一组合适的循环。在某些情况下，$Γ$ -极限问题的极小值可以很容易地找到，并且它们对于理解$m_0$的大值的卫星的运动是有用的。我们讨论了一些例子，其中的对称性是由一个行动的组$Z_4$，$Z_2 \times Z_2$和旋转组的柏拉图多面体上的一组循环。
英文摘要：
We consider a $(1+N)$-body problem in which one particle has mass $m_0 \gg 1$ and the remaining $N$ have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the $N$ bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form $U \sim \frac{1}{r^α},$ where $α\in [1,2)$ and $r$ is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use $Γ$-convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional $Γ$-converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the $Γ$-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of $m_0$. We discuss some examples, where the symmetry is defined by an action of the groups $Z_4$ , $Z_2 \times Z_2$ and the rotation groups of Platonic polyhedra on the set of loops.
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