# American Institute of Mathematical Sciences

May  2012, 17(3): 943-975. doi: 10.3934/dcdsb.2012.17.943

## On the existence of doubly symmetric "Schubart-like" periodic orbits

 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain

Received  July 2010 Revised  October 2011 Published  January 2012

We give sufficient conditions to ensure the existence of symmetrical periodic orbits for a class of Hamiltonian systems having some singularities. The results are applied to different subproblems of the gravitational $n$-body problem where singularities appear due to collisions.
Citation: Regina Martínez. On the existence of doubly symmetric "Schubart-like" periodic orbits. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 943-975. doi: 10.3934/dcdsb.2012.17.943
##### References:
 [1] R. Devaney, Triple collision in the planar isosceles three-body problem, Inventiones Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar [2] R. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [3] R. McGehee, Triple collision in the collinear three-body problem, Inventiones Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.  Google Scholar [4] R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, Journal of Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6.  Google Scholar [5] R. Moeckel, Orbits of the three-body problem which pass infinitely close to triple collision, Amer. Journ. of Math., 103 (1981), 1323-1341. doi: 10.2307/2374233.  Google Scholar [6] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Dis. Con. Dyn. Syst. Series B, 10 (2008), 609-620. doi: 10.3934/dcdsb.2008.10.609.  Google Scholar [7] R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones, SIAM J. Math. Anal., 26 (1995), 978-998.  Google Scholar [8] T. Ouyang, S. C. Simmons and D. Yan, Periodic solutions with singularities in two dimensions in the $n$-body problem, preprint, arXiv:0811.0227v3, 2008. Google Scholar [9] C. Simó, Analysis of triple collision in the isosceles problem, in "Classical Mechanics and Dynamical Systems" (ed. R. Devaney and Z. Nitecki) (Medford, Mass., 1979), Lecture Notes in Pure and Appl. Math., 70, Dekker, New York, (1981), 203-224.  Google Scholar [10] C. Simó and J. Llibre, Characterization of transversal homothetic solutions in the n-body problem, Arch. Ration Mech. Anal., 77 (1981), 189-198.  Google Scholar [11] C. Simó and R. Martínez, Qualitative study of the planar isosceles three-body problem,, Celestial Mechanics, 41 (): 179.  doi: 10.1007/BF01238762.  Google Scholar [12] J. Schubart, Numerische Ausfsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.  Google Scholar [13] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n-$body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.  Google Scholar [14] K. Tanikawa and H. Umehara, Oscillatory orbits in the planar three-body problem with equal masses, Celest. Mech. Dyn. Astr., 70 (1998), 167-180. doi: 10.1023/A:1008301405839.  Google Scholar

show all references

##### References:
 [1] R. Devaney, Triple collision in the planar isosceles three-body problem, Inventiones Math., 60 (1980), 249-267. doi: 10.1007/BF01390017.  Google Scholar [2] R. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113. doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [3] R. McGehee, Triple collision in the collinear three-body problem, Inventiones Math., 27 (1974), 191-227. doi: 10.1007/BF01390175.  Google Scholar [4] R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, Journal of Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6.  Google Scholar [5] R. Moeckel, Orbits of the three-body problem which pass infinitely close to triple collision, Amer. Journ. of Math., 103 (1981), 1323-1341. doi: 10.2307/2374233.  Google Scholar [6] R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Dis. Con. Dyn. Syst. Series B, 10 (2008), 609-620. doi: 10.3934/dcdsb.2008.10.609.  Google Scholar [7] R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones, SIAM J. Math. Anal., 26 (1995), 978-998.  Google Scholar [8] T. Ouyang, S. C. Simmons and D. Yan, Periodic solutions with singularities in two dimensions in the $n$-body problem, preprint, arXiv:0811.0227v3, 2008. Google Scholar [9] C. Simó, Analysis of triple collision in the isosceles problem, in "Classical Mechanics and Dynamical Systems" (ed. R. Devaney and Z. Nitecki) (Medford, Mass., 1979), Lecture Notes in Pure and Appl. Math., 70, Dekker, New York, (1981), 203-224.  Google Scholar [10] C. Simó and J. Llibre, Characterization of transversal homothetic solutions in the n-body problem, Arch. Ration Mech. Anal., 77 (1981), 189-198.  Google Scholar [11] C. Simó and R. Martínez, Qualitative study of the planar isosceles three-body problem,, Celestial Mechanics, 41 (): 179.  doi: 10.1007/BF01238762.  Google Scholar [12] J. Schubart, Numerische Ausfsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22. doi: 10.1002/asna.19562830105.  Google Scholar [13] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n-$body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841. doi: 10.1007/s00205-010-0334-6.  Google Scholar [14] K. Tanikawa and H. Umehara, Oscillatory orbits in the planar three-body problem with equal masses, Celest. Mech. Dyn. Astr., 70 (1998), 167-180. doi: 10.1023/A:1008301405839.  Google Scholar
 [1] 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 [2] Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229 [3] 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 [4] Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745 [5] 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 [6] Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080 [7] 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 [8] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 [9] Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057 [10] Abimael Bengochea, Manuel Falconi, Ernesto Pérez-Chavela. Horseshoe periodic orbits with one symmetry in the general planar three-body problem. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 987-1008. doi: 10.3934/dcds.2013.33.987 [11] 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, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137 [12] Shiqing Zhang, Qing Zhou. Nonplanar and noncollision periodic solutions for $N$-body problems. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 679-685. doi: 10.3934/dcds.2004.10.679 [13] Carlota Rebelo, Alexandre Simões. Periodic linear motions with multiple collisions in a forced Kepler type problem. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3955-3975. doi: 10.3934/dcds.2018172 [14] Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909 [15] Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609 [16] Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009 [17] Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629 [18] Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589 [19] Marshall Hampton, Anders Nedergaard Jensen. Finiteness of relative equilibria in the planar generalized $N$-body problem with fixed subconfigurations. Journal of Geometric Mechanics, 2015, 7 (1) : 35-42. doi: 10.3934/jgm.2015.7.35 [20] Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495

2020 Impact Factor: 1.327