Abstract:
We consider the problem of symmetry of the central configurations in the restricted 4+1 body problem when the four positive masses are equal and disposed in symmetric configurations, namely, on a line, at the vertices of a square, at the vertices of a equilateral triangle with a mass at the barycenter, and finally, at the vertices of a regular tetrahedron [1-3]. In these situations, we show that in order to form a non collinear central configuration of the restricted 4+1 body problem, the null mass must be on an axis of symmetry. In our approach, we will use as the main tool the quadratic forms introduced by A. Albouy and A. Chenciner [4]. Our arguments are general enough, so that we can consider the generalized Newtonian potential and even the logarithmic case. To get our results, we identify some properties of the Newtonian potential (in fact, of the function φ(s)=−skφ(s)=−sk, with k<0k<0) which are crucial in the proof of the symmetry.
Keywords:nn-body problem, central configurations, symmetry.
\Bibitem{VidSan07}
\by C.~Vidal, A. A. Santos
\paper Symmetry of the Restricted 4+1 Body Problem with Equal Masses
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 1
\pages 27--38
\mathnet{http://mi.mathnet.ru/rcd609}
\crossref{https://doi.org/10.1134/S1560354707010030}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350294}
\zmath{https://zbmath.org/?q=an:1229.37017}
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https://www.mathnet.ru/eng/rcd609
https://www.mathnet.ru/eng/rcd/v12/i1/p27
This publication is cited in the following 13 articles:
M. Alvarez-Ramírez, J. Lino Cornelio, Josep M. Cors, “Convex symmetric rectangular pentagon central configurations”, Communications in Nonlinear Science and Numerical Simulation, 138 (2024), 108250
Ya-Lun Tsai, “A Class of Symmetric Dziobek Configurations in Restricted Problems for Homogeneous Force Laws”, J Dyn Diff Equat, 2023
Jean Fernandes Barros, “Bifurcations and enumeration of central configurations of some planar restricted problems”, Celest Mech Dyn Astron, 135:2 (2023)
M. Corbera, J. Llibre, C. Valls, “Planar central configurations of some restricted (4 + 1)-body problems”, Journal of Mathematical Physics, 63:12 (2022)
Fernandes A.C., Mello L.F., Vidal C., “on the Uniqueness of the Isosceles Trapezoidal Central Configuration in the 4-Body Problem For Power-Law Potentials”, Nonlinearity, 33:1 (2020), 388–407
Hampton M., “Planar N-Body Central Configurations With a Homogeneous Potential”, Celest. Mech. Dyn. Astron., 131:5 (2019), 20
Antonio Carlos Fernandes, Braulio Augusto Garcia, Jaume Llibre, Luis Fernando Mello, “New central configurations of the (n+1)-body problem”, Journal of Geometry and Physics, 124 (2018), 199
Alan Almeida Santos, Marcelo Marchesin, Ernesto Pérez-Chavela, Claudio Vidal, “Continuation and bifurcations of concave central configurations in the four and five body-problems for homogeneous force laws”, Journal of Mathematical Analysis and Applications, 446:2 (2017), 1743
Martha Alvarez-Ramírez, Montserrat Corbera, Jaume Llibre, “On the central configurations in the spatial 5-body problem with four equal masses”, Celest Mech Dyn Astr, 124:4 (2016), 433
Martha Alvarez-Ramírez, Alan Almeida Santos, Claudio Vidal, “On Co-Circular Central Configurations in the Four and Five Body-Problems for Homogeneous Force Law”, J Dyn Diff Equat, 25:2 (2013), 269
Ya-lun Tsai, “Dziobek configurations of the restricted (N + 1)-body problem with equal masses”, Journal of Mathematical Physics, 53:7 (2012)
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