Abstract

C. Jacobi found that in the general N-body problem (including N = 3), the negativity of the total energy of the system was necessary for the Lagrangian stability of any solution. For the restricted three-body problem, this statement is trivial, since a zero-mass body makes zero contribution to the system energy. If we consider only the equations describing the motion of a zero-mass point, then the energy integral disappears. However, if we average the equations over the longitudes of the main bodies, the energy integral appears again. Is the Jacobi theorem true in this case? It turns out that it is not. For arbitrarily large values of total energy, there are bounded periodic orbits. At the same time, the negativity of the energy turned out to be sufficient for the boundedness of an orbit in the configuration space.

中文翻译：

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Jacobi定理在单平均受限圆三体问题中有效吗？

摘要

C. Jacobi发现，在一般的N体问题（包括N = 3）中，系统总能量的负性是必要的对于任何解的拉格朗日稳定性。对于受限的三体问题，由于零质量的物体对系统能量的贡献为零，所以该陈述是微不足道的。如果仅考虑描述零质量点运动的方程，则能量积分消失。但是，如果我们对主体经度上的方程进行平均，则能量积分会再次出现。在这种情况下，雅可比定理是正确的吗？事实证明并非如此。对于总能量的任意大的值，存在有界的周期性轨道。同时，能量的负性足以满足配置空间中轨道的边界。