Abstract
We study planar central configurations of the five-body problem where three of the bodies are collinear, forming an Euler central configuration of the three-body problem, and the two other bodies together with the collinear configuration are in the same plane. The problem considered here assumes certain symmetries. From the three bodies in the collinear configuration, the two bodies at the extremities have equal masses and the third one is at the middle point between the two. The fourth and fifth bodies are placed in a symmetric way: either with respect to the line containing the three bodies, or with respect to the middle body in the collinear configuration, or with respect to the perpendicular bisector of the segment containing the three bodies. The possible stacked five-body central configurations satisfying these types of symmetries are: a rhombus with four masses at the vertices and a fifth mass in the center, and a trapezoid with four masses at the vertices and a fifth mass at the midpoint of one of the parallel sides.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albouy A.: On a paper of Moeckel on central configurations. Regul. Chaotic Dyn. 8(2), 133–142 (2002)
Alvarez A., Delgado J., Llibre J.: On the spatial central configurations of the 5-body problem and their bifurcations. Discrete Contin. Dyn. Syst. Ser. S 1(4), 505–518 (2009)
Arribas M., Elipe A.: Bifurcations and equilibria in the extended n-body ring problem. Mech. Res. Comm. 31, 1–8 (2004)
Arribas M., Elipe A., Kalvouridis T., Palacios M.: Homographic solutions in the planar n + 1 body problem with quasi-homogeneous potentials. Celestial Mech. Dyn. Astron. 99(1), 1–12 (2007)
Arribas M., Elipe A., Palacios M.: Linear stability of ring systems with generalized central forces. Astron. Astrophys. 489(1), 819–824 (2008)
Buck G.: The collinear central configuration of n equal masses. Celestial Mech. Dyn. Astron. 51(4), 305–317 (1991)
Cors J.M., Llibre J., Ollé M.: Central configurations of the planar coorbitalsatellite problem. Celestial Mech. Dyn. Astron. 89(4), 319–342 (1991)
Fayçal N.: On the classification of pyramidal central configurations. Proc. Amer. Math. Soc. 124(1), 249–258 (1996)
Hadjifotinou K.G., Kalvouridis T.J.: Numerical investigation of periodic motion in the three-dimensional ring problem of n bodies. Int. J. Bifurcat. Chaos. 15, 2681–2688 (2005)
Hampton M.: Stacked central configurations: new examples in the planar five-body problem. Nonlinearity 18(5), 2299–2304 (2005)
Hampton M., Santoprete M.: Seven-body central configurations: a family of central configurations in the spatial seven-body problem. Celestial Mech. Dyn. Astron. 99(4), 293–305 (2007)
Kalvouridis T.J.: A planar case of the n + 1 body problem: the ring problem. Astrophys. Space Sci. 260, 309–325 (1999)
Kalvouridis T.J.: Zero velocity surface in the three-dimensional ring problem of n + 1 bodies. Celestial Mech. Dyn. Astron. 80, 133–144 (2001)
Lee T.L., Santoprete M.: Central configurations of the five-body problem with equal masses. Celestial Mech. Dyn. Astron. 104, 369–381 (2009)
Llibre J., Mello L.F.: New central configurations for the planar 5-body problem. Celestial Mech. Dyn. Astron. 100(2), 141–149 (2008)
Llibre, J., Mello, L.F., Perez-Chavela, E.: New stacked central configurations for the planar 5-body problem, preprint (2008)
Maxwell J.C.: On the Stability of Motions of Saturns Rings, pp. 381. Macmillan and Cia, Cambridge (1859)
Mello L.F., Chavesa F.E., Fernandes A.C., Garcia B.A.: Stacked central configurations for the spatial six-body problem. J. Geom. Phys. 59(9), 1216–1226 (2009)
Mioc V., Stavinschi M.: On the Schwarzschild-type polygonal (n + 1)-body problem and on the associated restricted problem. Baltic Astron. 7, 637–651 (1998)
Mioc V., Stavinschi M.: On Maxwell’s (n + 1)-body problem in the manev-type field and on the associated restricted problem. Physica Scripta 60, 483–490 (1999)
Moeckel R.: On central configurations. Math. Z. 205(4), 499–517 (1990)
Saari, D.G.: On the role and the properties of n-body central configurations. In: Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics (Math. Forschungsinst., Oberwolfach, 1978), Part I, vol. 21, pp. 9–20 (1980)
Salo H., Yoder C.F.: The dynamics of coorbital satellite systems. Astron. Astrophys. 205(1–2), 309–327 (1988)
Scheeres, D.J.: On symmetric central configurations with application to satellite motion about rings. Ph.D. thesis. University of Michigan (1992)
Scheeres D.J., Vinh N.X.: Linear stability of a self-gravitating ring. Celestial Mech. Dyn. Astron. 51, 83–103 (1991)
Sekiguchi M.: Bifurcation of central configuration in the 2n + 1 body problem. Celestial Mech. Dyn. Astron. 90(3–4), 355–360 (2004)
Smale S.: Mathematical problems for the next century. Math. Intelligencer 20, 7–15 (1998)
Wintner A.: The Analytical Foundations of Celestial Mechanics. Princeton University Press, Princeton, N.J. (1941)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gidea, M., Llibre, J. Symmetric planar central configurations of five bodies: Euler plus two. Celest Mech Dyn Astr 106, 89 (2010). https://doi.org/10.1007/s10569-009-9243-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10569-009-9243-0