Elliptic relative equilibria in the N-body problem
Elliptic relative equilibria in the N-body problem
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DOI：
10.1016/j.jde.2004.09.006
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发表时间：
2005-07
影响因子：
2.4
通讯作者：
K. Meyer;D. Schmidt
中科院分区：
文献类型：
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作者：
K. Meyer;D. Schmidt
N体问题的平面中心构型产生了一个解决方案，其中每个粒子在特定的开普勒轨道上移动，而粒子的整体在位形运动上移动。如果开普勒轨道是椭圆的，那么这个解在脉动坐标系中是一个平衡，所以我们称这个解为椭圆相对平衡。这种解决方案的整体形成一个四维辛子空间，我们给出了一个辛坐标系，这是适应这个子空间和它的辛补。在我们的坐标系中，这种解的线性变分方程解耦成三个子系统。一个子系统简单地给出了质心的运动，另一个是开普勒的问题，第三个确定非平凡的特征乘数。利用这些坐标，我们研究了三体问题的等边三角形中心位形所定义的椭圆相对平衡的线性稳定性。我们再现了G.罗伯茨我们还研究了四体和五体问题的线性稳定性，其中三个或四个单位质量的物体位于等边三角形或正方形的顶点，其余物体位于中心，具有任意质量μ。
A planar central configuration of the N-body problem gives rise to a solution where each particle moves on a specific Keplerian orbit while the totality of the particles move on a homothety motion. If the Keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium. The totality of such solutions forms a four-dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement. In our coordinate system, the linear variational equations of such a solution decouple into three subsystems. One subsystem simply gives the motion of the center of mass, another is Kepler's problem and the third determines the nontrivial characteristic multipliers. Using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three-body problem. We reproduce the analytic studies of G. Roberts. We also study the linear stability of the four- and five-body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or square and the remaining body is at the center with arbitrary mass μ.