Central configurations in the spatial n -body problem for $$n=5,6$$ n = 5
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Central configurations in the spatial n-body problem for n = 5, 6 with equal masses Małgorzata Moczurad1 · Piotr Zgliczynski ´ 1 Received: 3 June 2020 / Revised: 30 October 2020 / Accepted: 3 November 2020 / Published online: 8 December 2020 © The Author(s) 2020
Abstract We present a computer assisted proof of the full listing of central configurations for spatial n-body problem for n = 5 and 6, with equal masses. For each central configuration, we give a full list of its Euclidean symmetries. For all masses sufficiently close to the equal masses case, we give an exact count of configurations in the planar case for n = 4, 5, 6, 7 and in the spatial case for n = 4, 5, 6. Keywords Central configurations · Symmetries · Interval arithmetic · Computer assisted proof
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . 2 Equations for central configurations . . . . . . . . . . . . . . . . . . . 3 The reduced system of equations for CC . . . . . . . . . . . . . . . . 3.1 Non-degenerate solutions of full and reduced systems of equations 3.2 The reduced system RS . . . . . . . . . . . . . . . . . . . . . . 3.3 Collinearity and coplanarity tests . . . . . . . . . . . . . . . . . . 3.3.1 Coplanarity test . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Collinearity test . . . . . . . . . . . . . . . . . . . . . . . . 4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Finding candidates for (R, σ ) . . . . . . . . . . . . . . . . . . . . 4.1.1 Initialization of σ . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Finding candidates for symmetric images of qn−2 and q0 . .
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M. Moczurad and P. Zgliczy´nski: Partially supported by the NCN Grants 2015/19/B/ST1/01454 and 2019/35/B/ST1/00655.
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Piotr Zgliczy´nski [email protected] Małgorzata Moczurad [email protected]
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Faculty of Mathematics and Computer Science, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland
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4.1.3 Constructing R+ and R− : the candidates for the symmetries . . . . . 4.1.4 Construction of σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometric description of (R, σ ) . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Rot
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