# American Institute of Mathematical Sciences

2015, 2015(special): 1115-1124. doi: 10.3934/proc.2015.1115

## Classification of periodic orbits in the planar equal-mass four-body problem

 1 School of Mathematics and System Science, Beihang University, Beijing 100191, China 2 Department of Mathematics, Brigham Young University, Provo, Utah 84602 3 Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806

Received  September 2014 Revised  April 2015 Published  November 2015

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.
Citation: Duokui Yan, Tiancheng Ouyang, Zhifu Xie. Classification of periodic orbits in the planar equal-mass four-body problem. Conference Publications, 2015, 2015 (special) : 1115-1124. doi: 10.3934/proc.2015.1115
##### References:
 [1] R. Broucke, Classification of periodic orbits in the four- and five-body problems,, Ann. N.Y. Acad. Sci., 1017 (2004), 408. Google Scholar [2] K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses,, Arch. Ration. Mech. Anal., 170 (2001), 293. Google Scholar [3] K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem,, Erg. Thy. Dyn. Sys., 23 (2003), 1691. Google Scholar [4] L. Sbano, Periodic orbits of Hamiltonian systems,, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), (2011), 1212. Google Scholar [5] T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , (). Google Scholar [6] T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , (). Google Scholar [7] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155 (2004), 305. Google Scholar [8] M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits,, Phy. Rev. Lett., 110 (2013). Google Scholar [9] R. Vanderbei, New orbits for the n-body problem,, Ann. N.Y. Acad. Sci., 1017 (2004), 422. Google Scholar [10] L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem,, Celestial Mech. Dynam. Astronom., 108 (2010), 147. Google Scholar [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 (2012), 942. Google Scholar [12] T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the n-body problem,, Rocky Mountain J. Math., 42 (2012), 1601. Google Scholar [13] D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem,, Inter. J. Bifurcation Chaos, 25 (2015). Google Scholar [14] D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., (). Google Scholar

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##### References:
 [1] R. Broucke, Classification of periodic orbits in the four- and five-body problems,, Ann. N.Y. Acad. Sci., 1017 (2004), 408. Google Scholar [2] K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses,, Arch. Ration. Mech. Anal., 170 (2001), 293. Google Scholar [3] K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem,, Erg. Thy. Dyn. Sys., 23 (2003), 1691. Google Scholar [4] L. Sbano, Periodic orbits of Hamiltonian systems,, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), (2011), 1212. Google Scholar [5] T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , (). Google Scholar [6] T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , (). Google Scholar [7] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155 (2004), 305. Google Scholar [8] M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits,, Phy. Rev. Lett., 110 (2013). Google Scholar [9] R. Vanderbei, New orbits for the n-body problem,, Ann. N.Y. Acad. Sci., 1017 (2004), 422. Google Scholar [10] L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem,, Celestial Mech. Dynam. Astronom., 108 (2010), 147. Google Scholar [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 (2012), 942. Google Scholar [12] T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the n-body problem,, Rocky Mountain J. Math., 42 (2012), 1601. Google Scholar [13] D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem,, Inter. J. Bifurcation Chaos, 25 (2015). Google Scholar [14] D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., (). Google Scholar
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