American Institute of Mathematical Sciences

Advanced
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

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), 408-421.

[2]

K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293-318.

[3]

K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem, Erg. Thy. Dyn. Sys., 23 (2003), 1691-1715.

[4]

L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 1212-1236.

[5]

T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , (). 

[6]

T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , (). 

[7]

D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362.

[8]

M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301.

[9]

R. Vanderbei, New orbits for the n-body problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422-433.

[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), 147-164.

[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), 942-951.

[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), 1601-1614.

[13]

D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Inter. J. Bifurcation Chaos, 25 (2015), 1550116.

[14]

D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., (). 

show all references

References:
[1]

R. Broucke, Classification of periodic orbits in the four- and five-body problems, Ann. N.Y. Acad. Sci., 1017 (2004), 408-421.

[2]

K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293-318.

[3]

K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem, Erg. Thy. Dyn. Sys., 23 (2003), 1691-1715.

[4]

L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 1212-1236.

[5]

T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , (). 

[6]

T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , (). 

[7]

D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362.

[8]

M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301.

[9]

R. Vanderbei, New orbits for the n-body problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422-433.

[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), 147-164.

[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), 942-951.

[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), 1601-1614.

[13]

D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Inter. J. Bifurcation Chaos, 25 (2015), 1550116.

[14]

D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., (). 

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