Global timerenormalization of the gravitational $N$body problem
Abstract
This work considers the {\em gravitational} $N$body problem and introduces global timerenormalization {\em functions} that allow the efficient numerical integration with fixed timesteps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate timerenormalization. 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 byproduct, a global power series representation of the solutions of the $N$body problem is obtained. Noteworthy, our global timerenormalizations remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new timerenormalization functions for some $N$body problems with close encounters.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.01221
 Bibcode:
 2020arXiv200101221A
 Keywords:

 Mathematics  Dynamical Systems;
 Astrophysics  Earth and Planetary Astrophysics
 EPrint:
 26 pages