# American Institute of Mathematical Sciences

May  2012, 32(5): 1763-1774. doi: 10.3934/dcds.2012.32.1763

## Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof

 1 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States

Received  December 2010 Revised  May 2011 Published  January 2012

The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.
Citation: 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
##### References:
 [1] 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 Mechanics and Dynamical Astronomy, 108 (2010), 147.  doi: 10.1007/s10569-010-9298-y.  Google Scholar [2] K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses,, Arch. Ration. Mech. Anal., 158 (2001), 293.  doi: 10.1007/s002050100146.  Google Scholar [3] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math. (2), 152 (2000), 881.  doi: 10.2307/2661357.  Google Scholar [4] R. Easton, Regularization of vector fields by surgery,, J. Differential Equations, 10 (1971), 92.  doi: 10.1016/0022-0396(71)90098-2.  Google Scholar [5] M. ElBialy, Simultaneous binary collisions in the collinear $N$-body problem,, J. Differential Equations, 102 (1993), 209.  doi: 10.1006/jdeq.1993.1028.  Google Scholar [6] R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems,, Nonlinearity, 13 (2000), 2107.  doi: 10.1088/0951-7715/13/6/312.  Google Scholar [7] R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.  doi: 10.1007/BF01390175.  Google Scholar [8] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609.  doi: 10.3934/dcdsb.2008.10.609.  Google Scholar [9] F. R. Moulton, The straight line solutions of the problem of $n$ bodies,, Ann. of Math. (2), 12 (1910), 1.  doi: 10.2307/2007159.  Google Scholar [10] T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem,, Celestial Mech. Dynam. Astronom., 109 (2011), 229.   Google Scholar [11] G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,, Ergodic Theory Dynam. Systems, 27 (2007), 1947.  doi: 10.1017/S0143385707000284.  Google Scholar [12] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astr. Nachr., 283 (1956), 17.  doi: 10.1002/asna.19562830105.  Google Scholar [13] M. Sekiguchi and K. Tanikawa, On the symmetric collinear four-body problem,, Publication of the Astronomical Society of Japan, 56 (2004), 235.   Google Scholar [14] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem,, Arch. Ration. Mech. Anal., 199 (2011), 821.  doi: 10.1007/s00205-010-0334-6.  Google Scholar [15] C. Simó and E. Lacomba, Regularization of simultaneous binary collisions in the $n$-body problem,, J. Differential Equations, 98 (1992), 241.  doi: 10.1016/0022-0396(92)90092-2.  Google Scholar [16] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34,, Springer-Verlag, (2008).   Google Scholar [17] W. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation,, The Restless Universe (Blair Atholl, 82 (2002), 179.  doi: 10.1023/A:1014599918133.  Google Scholar [18] W. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem,, Celestial Mech. Dynam. Astronom., 94 (2006), 37.  doi: 10.1007/s10569-005-2289-8.  Google Scholar [19] A. Venturelli, "Application de la Minimisation de l'Action au Problḿe des $n$ Corps dans le Plan et dans l'Espace,", Thése de Doctorat, (2002).   Google Scholar [20] A. Venturelli, A variational proof of the existence of von Schubart's orbit,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699.  doi: 10.3934/dcdsb.2008.10.699.  Google Scholar [21] A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, (1941).   Google Scholar

show all references

##### References:
 [1] 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 Mechanics and Dynamical Astronomy, 108 (2010), 147.  doi: 10.1007/s10569-010-9298-y.  Google Scholar [2] K.-C. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses,, Arch. Ration. Mech. Anal., 158 (2001), 293.  doi: 10.1007/s002050100146.  Google Scholar [3] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math. (2), 152 (2000), 881.  doi: 10.2307/2661357.  Google Scholar [4] R. Easton, Regularization of vector fields by surgery,, J. Differential Equations, 10 (1971), 92.  doi: 10.1016/0022-0396(71)90098-2.  Google Scholar [5] M. ElBialy, Simultaneous binary collisions in the collinear $N$-body problem,, J. Differential Equations, 102 (1993), 209.  doi: 10.1006/jdeq.1993.1028.  Google Scholar [6] R. Martínez and C. Simó, The degree of differentiability of the regularization of simultaneous binary collisions in some $N$-body problems,, Nonlinearity, 13 (2000), 2107.  doi: 10.1088/0951-7715/13/6/312.  Google Scholar [7] R. McGehee, Triple collision in the collinear three-body problem,, Invent. Math., 27 (1974), 191.  doi: 10.1007/BF01390175.  Google Scholar [8] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609.  doi: 10.3934/dcdsb.2008.10.609.  Google Scholar [9] F. R. Moulton, The straight line solutions of the problem of $n$ bodies,, Ann. of Math. (2), 12 (1910), 1.  doi: 10.2307/2007159.  Google Scholar [10] T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem,, Celestial Mech. Dynam. Astronom., 109 (2011), 229.   Google Scholar [11] G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,, Ergodic Theory Dynam. Systems, 27 (2007), 1947.  doi: 10.1017/S0143385707000284.  Google Scholar [12] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astr. Nachr., 283 (1956), 17.  doi: 10.1002/asna.19562830105.  Google Scholar [13] M. Sekiguchi and K. Tanikawa, On the symmetric collinear four-body problem,, Publication of the Astronomical Society of Japan, 56 (2004), 235.   Google Scholar [14] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem,, Arch. Ration. Mech. Anal., 199 (2011), 821.  doi: 10.1007/s00205-010-0334-6.  Google Scholar [15] C. Simó and E. Lacomba, Regularization of simultaneous binary collisions in the $n$-body problem,, J. Differential Equations, 98 (1992), 241.  doi: 10.1016/0022-0396(92)90092-2.  Google Scholar [16] M. Struwe, "Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems," Fourth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34,, Springer-Verlag, (2008).   Google Scholar [17] W. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation,, The Restless Universe (Blair Atholl, 82 (2002), 179.  doi: 10.1023/A:1014599918133.  Google Scholar [18] W. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem,, Celestial Mech. Dynam. Astronom., 94 (2006), 37.  doi: 10.1007/s10569-005-2289-8.  Google Scholar [19] A. Venturelli, "Application de la Minimisation de l'Action au Problḿe des $n$ Corps dans le Plan et dans l'Espace,", Thése de Doctorat, (2002).   Google Scholar [20] A. Venturelli, A variational proof of the existence of von Schubart's orbit,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699.  doi: 10.3934/dcdsb.2008.10.699.  Google Scholar [21] A. Wintner, "The Analytical Foundations of Celestial Mechanics,", Princeton Mathematical Series, (1941).   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] 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 [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] 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 [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] 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 [7] Shiqing Zhang, Qing Zhou. Nonplanar and noncollision periodic solutions for $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 679-685. doi: 10.3934/dcds.2004.10.679 [8] A.V. Borisov, I.S. Mamaev, A.A. Kilin. New periodic solutions for three or four identical vortices on a plane and a sphere. Conference Publications, 2005, 2005 (Special) : 110-120. doi: 10.3934/proc.2005.2005.110 [9] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [10] Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] Qunyao Yin, Shiqing Zhang. New periodic solutions for the circular restricted 3-body and 4-body problems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 249-260. doi: 10.3934/cpaa.2010.9.249 [18] 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 [19] Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623 [20] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090

2019 Impact Factor: 1.338