# American Institute of Mathematical Sciences

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 [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

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
 [1] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [2] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [3] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [4] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [5] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [6] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031 [7] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 [8] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [9] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [10] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [11] Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052 [12] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [13] Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $A_n$-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118 [14] Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240 [15] Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $(n, m)$-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117 [16] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349 [17] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [18] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [19] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [20] Lei Liu, Li Wu. Multiplicity of closed characteristics on $P$-symmetric compact convex hypersurfaces in $\mathbb{R}^{2n}$. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.27