Angular Momentum为零的三体轨道同步相似三角形
Synchronised Similar Triangles for Three-Body Orbit with Zero Angular Momentum
摘要：
研究了零角动量三体轨道的几何性质。如果沿轨道的转动惯量也是恒定的，那么以物体的位置为顶点的三角形和以物体的动量为周长的三角形总是相似的（"同步相似三角形"）。这种相似性产生了相互距离和动量大小之间的运动学等式。此外，如果轨道是齐次势下运动方程的解，则轨道有一个新的包含动量的常数。对于零角动量和非常数转动惯量的轨道，我们引入标度变量，位置除以转动惯量的平方根，动量来自标度位置的速度。那么相似性和运动学等式对于尺度变量成立。利用这一相似性，我们证明了在次数小于2的齐次势作用下，任何角动量为零的有界三体轨道都有无穷多个共线构型（合合或食）或碰撞.
英文摘要：
Geometrical properties of three-body orbits with zero angular momentum are investigated. If the moment of inertia is also constant along the orbit, the triangle whose vertexes are the positions of the bodies, and the triangle whose perimeters are the momenta of the bodies, are always similar (``synchronised similar triangles''). This similarity yields kinematic equalities between mutual distances and magnitude of momenta. Moreover, if the orbit is a solution to the equation of motion under homogeneous potential, the orbit has a new constant involving momenta. For orbits with zero angular momentum and non-constant moment of inertia, we introduce scaled variables, positions divided by square root of the moment of inertia and momenta derived from the velocity of the scaled positions. Then the similarity and the kinematic equalities hold for the scaled variables. Using this similarity, we prove that any bounded three-body orbit with zero angular momentum under homogeneous potential whose degree is smaller than 2 has infinitely many collinear configurations (syzygies or eclipses) or collisions.
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