We prove for the $N$-body problem the existence of hyperbolic motions for any prescribed limit shape and any given initial configuration of the bodies. The energy level $h>0$ of the motion can also be chosen arbitrarily. Our approach is based on the construction of global viscosity solutions for the Hamilton-Jacobi equation $H(x,d_xu)=h$. We prove that these solutions are fixed points of the associated Lax-Oleinik semigroup. The presented results can also be viewed as a new application of Marchal's theorem, whose main use in recent literature has been to prove the existence of periodic orbits.

中文翻译：

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粘度解和双曲线运动：一种用于N体问题的新 PDE 方法

我们证明 $N$-body 问题对于任何规定的极限形状和任何给定的物体初始配置都存在双曲线运动。运动的能级$h>0$也可以任意选择。我们的方法基于为 Hamilton-Jacobi 方程 $H(x,d_xu)=h$ 构建全局粘度解。我们证明这些解是相关的 Lax-Oleinik 半群的不动点。所呈现的结果也可以看作是马歇尔定理的新应用，其在最近的文献中的主要用途是证明周期轨道的存在。