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.   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

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.   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

[1]

Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925
[2]

Hsin-Yuan Huang. Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1763-1774. doi: 10.3934/dcds.2012.32.1763
[3]

Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909
[4]

Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
[5]

Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143
[6]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009
[7]

Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
[8]

Ernesto A. Lacomba, Mario Medina. Oscillatory motions in the rectangular four body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 557-587. doi: 10.3934/dcdss.2008.1.557
[9]

Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158
[10]

Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85
[11]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062
[12]

Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323
[13]

Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569
[14]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523
[15]

Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745
[16]

Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003
[17]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229
[18]

Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379
[19]

Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169
[20]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045

 Impact Factor: 

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences