On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to...

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Bibliographic Details
Published in:arXiv.org 2018-01
Main Authors: Fenucci, Marco, Gronchi, Giovanni Federico
Format: Article
Language:English
Subjects:
Algorithms
Interval arithmetic
Motion stability
Orbits
Polyhedra
Stability
Symmetry
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Summary:In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic.
ISSN:2331-8422
DOI:10.48550/arxiv.1801.06429