三体“8”字形舞蹈转动惯量的演化
Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography
摘要：
研究了在相互作用势与r^alpha（alpha \neq 0）或log r成比例的情况下的三维三体运动，其中r表示物体之间的相互距离，满足以下条件：（I）转动惯量为非零常数，（II）角动量为零，（III）一个物体在某时刻位于质心上.证明了α \neq-2，2，4满足条件（Ⅰ）-（Ⅲ）的等质量运动是不可能的.并且对于α =2，4满足相同条件的运动被显式求解。这些轨道中有许多不是8字形的，并且这些运动存在碰撞。因此，证明了α \neq -2的8字形编排的转动惯量不守恒。我们还证明了在牛顿势α =-1下，满足条件（Ⅰ）-（Ⅲ）的具有一般质量的运动是不可能的。
英文摘要：
We investigate three-body motion in three dimensions under the interaction potential proportional to r^alpha (alpha \neq 0) or log r, where r represents the mutual distance between bodies, with the following conditions: (I) the moment of inertia is non-zero constant, (II) the angular momentum is zero, and (III) one body is on the centre of mass at an instant. We prove that the motion which satisfies conditions (I)-(III) with equal masses for alpha \neq -2, 2, 4 is impossible. And motions which satisfy the same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits are not figure-eight and these motions have collision. Therefore non-conservation of the moment of inertia for figure-eight choreography for alpha \neq -2 is proved. We also prove that the motion which satisfies conditions (I)-(III) with general masses under the Newtonian potential alpha=-1 is impossible.
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