SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 4, Page 2658-2681, January 2020.
This work considers the gravitational $N$-body problem and introduces global time-renormalization functions that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-renormalization. In the new fictitious time $\tau$, it is then proved that any solution exists for all $\tau \in \mathbb{R}$ and that it is uniquely extended as a holomorphic function to a strip of fixed width. As a by-product, a global power series representation of the solutions of the $N$-body problem is obtained. Notably, our global time-renormalizations remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-renormalization functions for the numerical integration of some $N$-body problems with close encounters.

中文翻译：

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引力$ N $体问题的全局时间重整化

SIAM应用动力系统杂志，第19卷，第4期，第2658-2681页，2020年1月。
这项工作考虑了重力$ N $体问题，并引入了全局时间重新归一化函数，该函数允许使用固定的时间步长进行有效的数值积分。首先，得出原始方程解的收敛半径的下界，这表明适当的时间重新归一化。然后在新的虚拟时间$ \ tau $中，证明了所有\\ tau \ in \ mathbb {R} $中都存在任何解，并且它作为全纯函数唯一地扩展到固定宽度的条带。作为副产品，获得了$ N $体问题的解的全局幂级数表示。值得注意的是，当其中一个群众消失时，我们的全局时间重新归一化在极限内仍然有效。最后，