×

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
PDF BibTeX XML Cite
Full Text: DOI
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,, <a href=
[6] T. Ouyang, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, <a href=
[7] D. Ferrario, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155, 305, (2004) · Zbl 1068.70013
[8] M. ҆uvakov, Three classes of Newtonian three-body planar periodic orbits,, Phy. Rev. Lett., 110, (2013)
[9] R. Vanderbei, New orbits for the n-body problem,, Ann. N.Y. Acad. Sci., 1017, 422, (2004)
[10] L. Bakker, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem,, Celestial Mech. Dynam. Astronom., 108, 147, (2010) · Zbl 1223.70029
[11] D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem,, J. Math. Anal. Appl. 388 (2012), 388, 942, (2012) · Zbl 1232.70009
[12] T. Ouyang, Periodic solutions with singularities in two dimensions in the n-body problem,, Rocky Mountain J. Math., 42, 1601, (2012) · Zbl 1263.70015
[13] D. Yan, New phenomena in the spatial isosceles three-body problem,, Inter. J. Bifurcation Chaos, 25, (2015) · Zbl 1325.70026
[14] D. Yan, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-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.