August  2014, 19(6): 1523-1548. doi: 10.3934/dcdsb.2014.19.1523

Symmetric periodic orbits in three sub-problems of the $N$-body problem

1. 

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

Received  November 2013 Revised  March 2014 Published  June 2014

We study three sub-problems of the $N$-body problem that have two degrees of freedom, namely the $n-$pyramidal problem, the planar double-polygon problem, and the spatial double-polygon problem. We prove the existence of several families of symmetric periodic orbits, including ``Schubart-like" orbits and brake orbits, by using topological shooting arguments.
Citation: 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
References:
[1]

R. Broucke, On the isosceles triangle configuration in the planar general three body problem,, Astron. Astrophys., 73 (1979), 303.   Google Scholar

[2]

N. C. Chen, Periodic brake orbits in the planar isosceles three-body problem,, Nonlinearity, 26 (2013), 2875.  doi: 10.1088/0951-7715/26/10/2875.  Google Scholar

[3]

J. Delgado and C. Vidal, The tetrahedral $4$-body problem,, J. Dynam. Differential Equations, 11 (1999), 735.  doi: 10.1023/A:1022667613764.  Google Scholar

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D. Ferrario and A. Portaluri, On the dihedral $n$-body problem,, Nonlinearity, 21 (2008), 1307.  doi: 10.1088/0951-7715/21/6/009.  Google Scholar

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R. Martínez, On the existence of doubly symmetric "Schubart-like" periodic orbits,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943.  doi: 10.3934/dcdsb.2012.17.943.  Google Scholar

[6]

R. Martínez, Families of double symmetric ‘Schubart-like' periodic orbits,, Celest. Mech. Dyn. Astr., 117 (2013), 217.  doi: 10.1007/s10569-013-9509-4.  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, R. Montgomery and A. Venturelli, From brake to syzygy,, Arch. Ration. Mech. Anal., 204 (2012), 1009.  doi: 10.1007/s00205-012-0502-y.  Google Scholar

[9]

R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones,, SIAM J. Math. Anal., 26 (1995), 978.  doi: 10.1137/S0036141093248414.  Google Scholar

[10]

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

[11]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astron. Nachr., 283 (1956), 17.  doi: 10.1002/asna.19562830105.  Google Scholar

[12]

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

[13]

C. Simó, Analysis of triple collision in the isosceles problem,, in Classical Mechanics and Dynamical Systems, (1981), 203.   Google Scholar

[14]

C. Simó and R. Martínez, Qualitative study of the planar isosceles three-body problem,, Celest. Mech. Dyn. Astr., 41 (): 179.  doi: 10.1007/BF01238762.  Google Scholar

[15]

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

show all references

References:
[1]

R. Broucke, On the isosceles triangle configuration in the planar general three body problem,, Astron. Astrophys., 73 (1979), 303.   Google Scholar

[2]

N. C. Chen, Periodic brake orbits in the planar isosceles three-body problem,, Nonlinearity, 26 (2013), 2875.  doi: 10.1088/0951-7715/26/10/2875.  Google Scholar

[3]

J. Delgado and C. Vidal, The tetrahedral $4$-body problem,, J. Dynam. Differential Equations, 11 (1999), 735.  doi: 10.1023/A:1022667613764.  Google Scholar

[4]

D. Ferrario and A. Portaluri, On the dihedral $n$-body problem,, Nonlinearity, 21 (2008), 1307.  doi: 10.1088/0951-7715/21/6/009.  Google Scholar

[5]

R. Martínez, On the existence of doubly symmetric "Schubart-like" periodic orbits,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943.  doi: 10.3934/dcdsb.2012.17.943.  Google Scholar

[6]

R. Martínez, Families of double symmetric ‘Schubart-like' periodic orbits,, Celest. Mech. Dyn. Astr., 117 (2013), 217.  doi: 10.1007/s10569-013-9509-4.  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, R. Montgomery and A. Venturelli, From brake to syzygy,, Arch. Ration. Mech. Anal., 204 (2012), 1009.  doi: 10.1007/s00205-012-0502-y.  Google Scholar

[9]

R. Moeckel and C. Simó, Bifurcation of spatial central configurations from planar ones,, SIAM J. Math. Anal., 26 (1995), 978.  doi: 10.1137/S0036141093248414.  Google Scholar

[10]

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

[11]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem,, Astron. Nachr., 283 (1956), 17.  doi: 10.1002/asna.19562830105.  Google Scholar

[12]

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

[13]

C. Simó, Analysis of triple collision in the isosceles problem,, in Classical Mechanics and Dynamical Systems, (1981), 203.   Google Scholar

[14]

C. Simó and R. Martínez, Qualitative study of the planar isosceles three-body problem,, Celest. Mech. Dyn. Astr., 41 (): 179.  doi: 10.1007/BF01238762.  Google Scholar

[15]

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

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