论文标题
卫星的对称星座在大质量的中央体内移动
Symmetric constellations of satellites moving around a central body of large mass
论文作者
论文摘要
我们考虑一个$（1+n）$ - 身体问题，其中一个粒子具有质量$ m_0 \ gg 1 $，其余的$ n $具有单一的质量。我们可以假设，具有较大质量的身体（中央机构）处于静止状态，与质量较小（卫星）的$ N $体质量中心相吻合。两个粒子之间的相互作用力是通过形式的$ u \ sim \ frac {1} {r^α}的电势来定义的，其中$α\在[1,2）$中，$ r $是粒子之间的距离。施加对称性和拓扑约束，我们通过变异方法搜索该系统的周期性轨道。此外，随着中央机构的质量增加，我们使用$γ$合理的理论来研究这些轨道的渐近行为。事实证明，拉格朗日动作功能$γ$ - 符合在合适的循环中定义的开普勒问题的动作功能。在某些情况下，可以很容易地发现$γ$限制问题的最小化器，并且对于理解卫星的运动以$ m_0 $的大量值很有用。我们讨论了一些示例，其中对称性是由$ z_4 $，$ z_2 \ times z_2 $的组的动作定义的，而柏拉图polyhedra的旋转组则在一组循环中。
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.
