# zbMATH — the first resource for mathematics

Classification of periodic orbits in the planar equal-mass four-body problem. (English) Zbl 1360.70017
Summary: In the $$N$$-body problem, many periodic orbits are found as local Lagrangian action minimizers. In this work, we classify such periodic orbits in the planar equal-mass four-body problem. Specific planar configurations are considered: line, rectangle, diamond, isosceles trapezoid, double isosceles, kite, etc. Periodic orbits are classified into 8 categories and each category corresponds to a pair of specific configurations. Furthermore, it helps discover several new sets of periodic orbits.

##### MSC:
 70F10 $$n$$-body problems 70F15 Celestial mechanics 49M15 Newton-type methods
##### Keywords:
four-body problem; periodic orbit; variational method
Full Text:
##### References:
 [1] R. Broucke, Classification of periodic orbits in the four- and five-body problems,, Ann. N.Y. Acad. Sci., 1017, 408, (2004) [2] K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses,, Arch. Ration. Mech. Anal., 170, 293, (2001) · Zbl 1028.70009 [3] K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem,, Erg. Thy. Dyn. Sys., 23, 1691, (2003) · Zbl 1128.70306 [4] L. Sbano, Periodic orbits of Hamiltonian systems,, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), 1212, (2011) [5] T. Ouyang, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,,
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.