牛顿引力中平面等质量三体问题的Saari单应性猜想
Saari's homographic conjecture for planar equal-mass three-body problem in Newton gravity
摘要：
牛顿引力下N体问题中的Saari单应性猜想是：转动惯量I=（\sum m_k）^{-1} \sum m_i m_j r_{ij} ^2的平方根与势函数U= \sum m_i m_j/ r_{ij}的乘积的构形测度μ= \sqrt{I} U是常数当且仅当运动是单应性的.其中m_k表示物体k的质量，r_{ij}表示物体i和j之间的距离。在这项工作中，我们使用三组形状变量。在第一步中，我们使用q_k =3q_3/（2（q_2-q_1）），其中q_k \in \mathbb{C}表示物体k的位置。在中间步骤中使用r_1= r_{23} / r_{12}和r_2= r_{31} / r_{12}，最后使用μ本身和ρ=I^{3/2}/（r_{12} r_{23} r_{31}）。形状变量μ和ρ使我们的证明变得简单。
英文摘要：
Saari's homographic conjecture in N-body problem under the Newton gravity is the following; configurational measure μ=\sqrt{I}U, which is the product of square root of the moment of inertia I=(\sum m_k)^{-1}\sum m_i m_j r_{ij}^2 and the potential function U=\sum m_i m_j/r_{ij}, is constant if and only if the motion is homographic. Where m_k represents mass of body k and r_{ij} represents distance between bodies i and j. We prove this conjecture for planar equal-mass three-body problem. In this work, we use three sets of shape variables. In the first step, we use ζ=3q_3/(2(q_2-q_1)) where q_k \in \mathbb{C} represents position of body k. Using r_1=r_{23}/r_{12} and r_2=r_{31}/r_{12} in intermediate step, we finally use μitself and ρ=I^{3/2}/(r_{12}r_{23}r_{31}). The shape variables μand ρmake our proof simple.
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