Lagrange Solutions to the Discrete-time General Three-body Problem
Lagrange Solutions to the Discrete-time General Three-body Problem
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离散时间一般三体问题的拉格朗日解
DOI：
10.1088/0004-6256/145/3/64
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Yukitaka Minesaki
中科院分区：
文献类型：
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作者：
Seiichi Iwamoto;Yutaka Kimura;Toshiharu Fujita;澤 正憲;山本卓宏;Naofumi Muraki;澤 正憲;Naofumi MUraki;N. Muraki;Masakazu Jimbo;O. Saeki;Masakazu Jimbo;Masakazu Jimbo;Yukitaka Minesaki
对于一般的三体问题（G3BP），目前还没有已知的能产生精确轨道的积分器。很难验证数值过程是否产生G3BP的正确解，因为这样做需要了解所有11个守恒量，而只有6个已知。不需要跟踪所有的守恒量，就可以证明离散一般三体问题（d-G3BP）产生与G3BP的拉格朗日解对应的正确轨道。我们证明了d-G3BP在等边三角形和共线构型两种特殊情况下给出了G3BP的正确解。对于三角形解，我们利用三体情况的解是三个两体情况的解的叠加这一事实，我们证明了三个物体在任何时候都保持相同的相对距离。为了得到共线解，我们假设三个物体沿一条直线旋转排列，并证明d-G3BP与G3BP保持相同的两个物体之间的距离比。证明这些情况下的d-G3BP解与G3BP解是等价的，就有可能在其他情况下d-G3BP和G3BP解是等价的。据我们所知，这是第一个证明离散解和拉格朗日轨道等价的工作。
There is no known integrator that yields exact orbits for the general three-body problem (G3BP). It is difficult to verify whether a numerical procedure yields the correct solutions to the G3BP because doing so requires knowledge of all 11 conserved quantities, whereas only six are known. Without tracking all of the conserved quantities, it is possible to show that the discrete general three-body problem (d-G3BP) yields the correct orbits corresponding to Lagrange solutions of the G3BP. We show that the d-G3BP yields the correct solutions to the G3BP for two special cases: the equilateral triangle and collinear configurations. For the triangular solution, we use the fact that the solution to the three-body case is a superposition of the solutions to the three two-body cases, and we show that the three bodies maintain the same relative distances at all times. To obtain the collinear solution, we assume a specific permutation of the three bodies arranged along a straight rotating line, and we show that the d-G3BP maintains the same distance ratio between two bodies as in the G3BP. Proving that the d-G3BP solutions for these cases are equivalent to those of the G3BP makes it likely that the d-G3BP and G3BP solutions are equivalent in other cases. To our knowledge, this is the first work that proves the equivalence of the discrete solutions and the Lagrange orbits.