This article is cited in 3 papers

Special Issue: On the 80th birthday of professor A. Chenciner

On the Uniqueness of Convex Central Configurations in the Planar 4-Body Problem

Shanzhong Sun^a, Zhifu Xie^b, Peng You<sup>c

a</sup> Department of Mathematics; Academy for Multidisciplinary Studies, Capital Normal University, 100048 Beijing, P. R. China
^b School of Mathematics and Natural Science, The University of Southern Mississippi, MS 39406 Hattiesburg, USA
^c School of Mathematics and Statistics, Hebei University of Economics and Business, 050061 Shijiazhuang Hebei, P. R. China

Abstract: In this paper, we provide a rigorous computer-assisted proof (CAP) of the conjec- ture that in the planar four-body problem there exists a unique convex central configuration for any four fixed positive masses in a given order belonging to a closed domain in the mass space. The proof employs the Krawczyk operator and the implicit function theorem (IFT). Notably, we demonstrate that the implicit function theorem can be combined with interval analysis, enabling us to estimate the size of the region where the implicit function exists and extend our findings from one mass point to its neighborhood.

Keywords: central configuration, convex central configuration, uniqueness, $N$-body problem, Krawczyk operator, implicit function theorem.

MSC: 70F10, 70F15

Received: 27.02.2023
Accepted: 20.06.2023

Language: English

DOI: 10.1134/S1560354723520076