In this paper we provide estimates for mutual distances of periodic solutions for the Newtonian \(N\)-body problem. Our estimates are based on masses, total variations of turning angles for relative positions, and predetermined upper bounds for action values. Explicit formulae will be proved by iterative arguments. We demonstrate some applications to action-minimizing solutions for three- and four-body problems.

中文翻译：

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$$N$$ 身体问题的行动最小化解决方案的距离估计

在本文中，我们提供了牛顿\(N\)体问题的周期解的相互距离的估计。我们的估计基于质量、相对位置的转动角度的总变化以及动作值的预定上限。显式公式将通过迭代论证来证明。我们演示了三体和四体问题的行动最小化解决方案的一些应用。