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

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

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

Duokui Yan ^1, , Tiancheng Ouyang ^2, and Zhifu Xie ^3,

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

Abstract
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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.

Keywords: Four-body problem, periodic orbit, variational method..

Mathematics Subject Classification: Primary: 70F10, 70F15; Secondary: 49M1.

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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