The stability of vertical motion in the N-body circular Sitnikov problem
The stability of vertical motion in the N-body circular Sitnikov problem
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DOI：
10.1007/s10569-009-9194-5
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发表时间：
2009-03
影响因子：
1.6
通讯作者：
T. Bountis;K. E. Papadakis
中科院分区：
文献类型：
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作者：
T. Bountis;K. E. Papadakis
本文研究了Sitnikov限制N体问题中垂直运动的稳定性及其分叉成三维周期轨道族的问题。特别地，我们考虑ν =N− 1个等质量的质点绕圆周旋转，而第N个质点（质量可以忽略）垂直于质点平面运动。因此，我们将以前对四体Sitnikov问题的工作推广到N体情形，其中N = 5，9，15，25及更大。我们发现，对于N ≥ 4的所有情形，Sitnikov族都只有一个稳定区间（在z轴上），而不像N = 3的情形那样有无穷多个这样的区间.我们还表明，对于N = 5，9，15，25，分别有14，16，18，20个临界Sitnikov周期轨道，从3D家庭（不再直线）分叉。我们还研究了物理上有趣的问题，即有界动力学远离z轴的程度，在x，y平面上取初始条件，在常数z（0）= z0值，其中z0位于稳定直线运动的区间内。我们进行了类似的研究的动态附近的一些成员的3D家庭的周期性的解决方案，并发现，在适当选择庞加莱表面的部分，“岛”的有序运动，而远离他们大多数轨道变得混乱，并最终逃逸到无穷大。最后，我们解决了一个小质量的运动方程的存在下，一个均匀旋转的环。研究这种情况下垂直轨道的稳定性，我们再次发现了一个单一的稳定区间，当N增长时，当环的密度和半径等于相应的N − 1个主质量系统的密度和半径时，这个稳定区间趋于与N体问题的稳定区间一致。
We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restrictedN-body problem. In particular, we consider ν =N− 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to theN-body case, withN= 5, 9, 15, 25 and beyond. We find, for all cases we have considered withN≥ 4, that the Sitnikov family hasonly onestability interval (on thez-axis), unlike theN= 3 case where there is an infinity of such intervals. We also show that forN= 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from thez-axis, taking initial conditions onx,yplanes, at constantz(0) =z0values, wherez0lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, asNgrows, tends to coincide with the stability interval of theN-body problem, when the values of the density and radius of the ring equal those of the corresponding system ofN− 1 primary masses.