Global time-renormalization of the gravitational $N$-body problem
Abstract
This work considers the {\em gravitational} $N$-body problem and introduces global time-renormalization {\em 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. Noteworthy, 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 some $N$-body problems with close encounters.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2020
- DOI:
- 10.48550/arXiv.2001.01221
- arXiv:
- arXiv:2001.01221
- Bibcode:
- 2020arXiv200101221A
- Keywords:
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- Mathematics - Dynamical Systems;
- Astrophysics - Earth and Planetary Astrophysics
- E-Print:
- 26 pages