Saari's Conjecture for Elliptical Motions and Minimizing Solutions of the N-body Problem
Saari's Conjecture for Elliptical Motions and Minimizing Solutions of the N-body Problem
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DOI：
10.1137/140987663
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发表时间：
2016-02
期刊：
影响因子：
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通讯作者：
Xiang Yu;Shiqing Zhang
中科院分区：
文献类型：
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作者：
Xiang Yu;Shiqing Zhang
利用Kronecker的一个算术引理和Fourier级数中的一些标准展开式，证明了椭圆型牛顿N体问题的Saari猜想，并应用该结果证明了平面牛顿N体问题的变分极小解正是那些构型为极小中心构型的相对平衡解.我们注意到，我们不需要关于中心构型有限性的假设，这是一个远离完整证明的猜想，但在以前的研究中需要的问题。对于行星约束问题（忽略了所有行星间的引力相互作用），我们推出了相应的Saari猜想并给出了证明。
Using an arithmetic lemma of Kronecker's and some standard expansions in Fourier series, we prove Saari's conjecture for the elliptical type Newtonian N-body problem, and we apply the result to prove that the variational minimal solutions for the planar Newtonian N-body problem are precisely those relative equilibrium solutions whose configurations are minimizing central configurations. We notice that we do not need the hypothesis on finiteness of central configurations, which is a conjecture far from the complete proof but needed in previous studies of the problem. For the planetary restricted problem (which ignores all the mutual gravitational interactions between the planets), we deduce the corresponding Saari's conjecture and its proof.