Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 37J45, 70F16; Secondary: 37N05.

 Citation:

•  [1] R. Devaney, Triple collision in the planar isosceles three-body problem, Inventiones Math., 60 (1980), 249-267.doi: 10.1007/BF01390017. [2] R. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.doi: 10.1090/S0002-9947-1976-0402815-3. [3] R. McGehee, Triple collision in the collinear three-body problem, Inventiones Math., 27 (1974), 191-227.doi: 10.1007/BF01390175. [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. [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. [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. [7] R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones, SIAM J. Math. Anal., 26 (1995), 978-998. [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. [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. [10] C. Simó and J. Llibre, Characterization of transversal homothetic solutions in the n-body problem, Arch. Ration Mech. Anal., 77 (1981), 189-198. [11] C. Simó and R. Martínez, Qualitative study of the planar isosceles three-body problem, Celestial Mechanics, 41 (1987/88), 179-251. doi: 10.1007/BF01238762. [12] J. Schubart, Numerische Ausfsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.doi: 10.1002/asna.19562830105. [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. [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.