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. doi: 10.1007/BF01390017. Google Scholar

[2]

R. Devaney, Reversible diffeomorphisms and flows,, Trans. Amer. Math. Soc., 218 (1976), 89. 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. 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. 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. 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. 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. Google Scholar

[8]

T. Ouyang, S. C. Simmons and D. Yan, Periodic solutions with singularities in two dimensions in the $n$-body problem,, preprint, (2008). Google Scholar

[9]

C. Simó, Analysis of triple collision in the isosceles problem,, in, 70 (1981), 203. 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. 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. 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. 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. 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. doi: 10.1007/BF01390017. Google Scholar

[2]

R. Devaney, Reversible diffeomorphisms and flows,, Trans. Amer. Math. Soc., 218 (1976), 89. 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. 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. 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. 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. 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. Google Scholar

[8]

T. Ouyang, S. C. Simmons and D. Yan, Periodic solutions with singularities in two dimensions in the $n$-body problem,, preprint, (2008). Google Scholar

[9]

C. Simó, Analysis of triple collision in the isosceles problem,, in, 70 (1981), 203. 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. 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. 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. 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. doi: 10.1023/A:1008301405839. Google Scholar

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