# American Institute of Mathematical Sciences

September  2016, 36(9): 5131-5162. doi: 10.3934/dcds.2016023

## Periodic solutions of the planar N-center problem with topological constraints

 1 Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, Ontario, M5S 2E4, Canada

Received  July 2015 Revised  November 2015 Published  May 2016

In the planar $N$-center problem, given a non-trivial free homotopy class of the configuration space satisfying certain conditions, we show that there is at least one collision-free $T$-periodic solution for any positive $T.$ The direct method of calculus of variations is used and the main difficulty is to show that minimizers under certain topological constraints are free of collision.
Citation: Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
##### References:
 [1] A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems,, Progress in Nonlinear Differential Equations and their Applications, 10 (1993).  doi: 10.1007/978-1-4612-0319-3.  Google Scholar [2] A. Bahri and P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials,, J. Funct. Anal., 82 (1989), 412.  doi: 10.1016/0022-1236(89)90078-5.  Google Scholar [3] V. Barutello, D. L. Ferrario and S. Terracini, Symmetry groups of the planar three-body problem and action-minimizing trajectories,, Arch. Ration. Mech. Anal., 190 (2008), 189.  doi: 10.1007/s00205-008-0131-7.  Google Scholar [4] V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem,, Arch. Ration. Mech. Anal., 207 (2013), 583.  doi: 10.1007/s00205-012-0565-9.  Google Scholar [5] V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions,, Calc. Var. Partial Differential Equations, 49 (2014), 391.  doi: 10.1007/s00526-012-0587-z.  Google Scholar [6] K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions,, Arch. Ration. Mech. Anal., 170 (2003), 247.  doi: 10.1007/s00205-003-0277-2.  Google Scholar [7] K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses,, Ann. of Math. (2), 167 (2008), 325.  doi: 10.4007/annals.2008.167.325.  Google Scholar [8] A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry,, in Proceedings of the International Congress of Mathematicians, (2002), 279.   Google Scholar [9] A. Chenciner, J. Gerver, R. Montgomery and C. Simó, Simple choreographic motions of N bodies: A preliminary study,, in Geometry, (2002), 287.  doi: 10.1007/0-387-21791-6_9.  Google Scholar [10] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math. (2), 152 (2000), 881.  doi: 10.2307/2661357.  Google Scholar [11] A. Chenciner and A. Venturelli., Minima de l'intégrale d'action du probléme newtonien de 4 corps de masses égales dans $\mathbbR^3$: orbites "hip-hop'',, Celestial Mech. Dynam. Astronom., 77 (2000), 139.  doi: 10.1023/A:1008381001328.  Google Scholar [12] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar [13] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155 (2004), 305.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [14] G. Fusco, G. F. Gronchi and P. Negrini, Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem,, Invent. Math., 185 (2011), 283.  doi: 10.1007/s00222-010-0306-3.  Google Scholar [15] W. B. Gordon, Conservative dynamical systems involving strong forces,, Trans. Amer. Math. Soc., 204 (1975), 113.  doi: 10.1090/S0002-9947-1975-0377983-1.  Google Scholar [16] W. B. Gordon, A minimizing property of Keplerian orbits,, Amer. J. Math., 99 (1977), 961.  doi: 10.2307/2373993.  Google Scholar [17] J. Hass and P. Scott, Intersections of curves on surfaces,, Israel J. Math., 51 (1985), 90.  doi: 10.1007/BF02772960.  Google Scholar [18] M. W. Hirsch, Differential Topology,, Texts in Mathematics, (1994).   Google Scholar [19] R. Montgomery, The N-body problem, the braid group, and action-minimizing periodic solutions,, Nonlinearity, 11 (1998), 363.  doi: 10.1088/0951-7715/11/2/011.  Google Scholar [20] M. Shibayama, Variational proof of the existence of the super-eight orbit in the four-body problem,, Arch. Ration. Mech. Anal., 214 (2014), 77.  doi: 10.1007/s00205-014-0753-x.  Google Scholar [21] N. Soave, Symbolic dynamics: From the N-centre to the (N +1)-body problem, a preliminary study,, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 371.  doi: 10.1007/s00030-013-0251-0.  Google Scholar [22] N. Soave and S. Terracini, Symbolic dynamics for the N-centre problem at negative energies,, Discrete Contin. Dyn. Syst. - A, 32 (2012), 3245.  doi: 10.3934/dcds.2012.32.3245.  Google Scholar [23] S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses,, Arch. Ration. Mech. Anal., 184 (2007), 465.  doi: 10.1007/s00205-006-0030-8.  Google Scholar [24] A. Venturelli, Application de la Minimisation de L'action au Problme des N Corps Dans Leplan et Dans L'espace,, Ph.D thesis, (2002).   Google Scholar [25] G. Yu, Simple choreography solutions of newtonian n-body problem,, , (2015).   Google Scholar [26] G. Yu, Shape space figure 8 solutions of three body problem with two equalmasses,, , (2015).   Google Scholar [27] G. Yu, Periodic solutions of the rotating N-center and restricted N + 1-body problems with topological constraints,, work in progress., ().   Google Scholar

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##### References:
 [1] A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems,, Progress in Nonlinear Differential Equations and their Applications, 10 (1993).  doi: 10.1007/978-1-4612-0319-3.  Google Scholar [2] A. Bahri and P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials,, J. Funct. Anal., 82 (1989), 412.  doi: 10.1016/0022-1236(89)90078-5.  Google Scholar [3] V. Barutello, D. L. Ferrario and S. Terracini, Symmetry groups of the planar three-body problem and action-minimizing trajectories,, Arch. Ration. Mech. Anal., 190 (2008), 189.  doi: 10.1007/s00205-008-0131-7.  Google Scholar [4] V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem,, Arch. Ration. Mech. Anal., 207 (2013), 583.  doi: 10.1007/s00205-012-0565-9.  Google Scholar [5] V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions,, Calc. Var. Partial Differential Equations, 49 (2014), 391.  doi: 10.1007/s00526-012-0587-z.  Google Scholar [6] K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions,, Arch. Ration. Mech. Anal., 170 (2003), 247.  doi: 10.1007/s00205-003-0277-2.  Google Scholar [7] K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problem with various choices of masses,, Ann. of Math. (2), 167 (2008), 325.  doi: 10.4007/annals.2008.167.325.  Google Scholar [8] A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry,, in Proceedings of the International Congress of Mathematicians, (2002), 279.   Google Scholar [9] A. Chenciner, J. Gerver, R. Montgomery and C. Simó, Simple choreographic motions of N bodies: A preliminary study,, in Geometry, (2002), 287.  doi: 10.1007/0-387-21791-6_9.  Google Scholar [10] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses,, Ann. of Math. (2), 152 (2000), 881.  doi: 10.2307/2661357.  Google Scholar [11] A. Chenciner and A. Venturelli., Minima de l'intégrale d'action du probléme newtonien de 4 corps de masses égales dans $\mathbbR^3$: orbites "hip-hop'',, Celestial Mech. Dynam. Astronom., 77 (2000), 139.  doi: 10.1023/A:1008381001328.  Google Scholar [12] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).   Google Scholar [13] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem,, Invent. Math., 155 (2004), 305.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [14] G. Fusco, G. F. Gronchi and P. Negrini, Platonic polyhedra, topological constraints and periodic solutions of the classical N-body problem,, Invent. Math., 185 (2011), 283.  doi: 10.1007/s00222-010-0306-3.  Google Scholar [15] W. B. Gordon, Conservative dynamical systems involving strong forces,, Trans. Amer. Math. Soc., 204 (1975), 113.  doi: 10.1090/S0002-9947-1975-0377983-1.  Google Scholar [16] W. B. Gordon, A minimizing property of Keplerian orbits,, Amer. J. Math., 99 (1977), 961.  doi: 10.2307/2373993.  Google Scholar [17] J. Hass and P. Scott, Intersections of curves on surfaces,, Israel J. Math., 51 (1985), 90.  doi: 10.1007/BF02772960.  Google Scholar [18] M. W. Hirsch, Differential Topology,, Texts in Mathematics, (1994).   Google Scholar [19] R. Montgomery, The N-body problem, the braid group, and action-minimizing periodic solutions,, Nonlinearity, 11 (1998), 363.  doi: 10.1088/0951-7715/11/2/011.  Google Scholar [20] M. Shibayama, Variational proof of the existence of the super-eight orbit in the four-body problem,, Arch. Ration. Mech. Anal., 214 (2014), 77.  doi: 10.1007/s00205-014-0753-x.  Google Scholar [21] N. Soave, Symbolic dynamics: From the N-centre to the (N +1)-body problem, a preliminary study,, NoDEA Nonlinear Differential Equations Appl., 21 (2014), 371.  doi: 10.1007/s00030-013-0251-0.  Google Scholar [22] N. Soave and S. Terracini, Symbolic dynamics for the N-centre problem at negative energies,, Discrete Contin. Dyn. Syst. - A, 32 (2012), 3245.  doi: 10.3934/dcds.2012.32.3245.  Google Scholar [23] S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses,, Arch. Ration. Mech. Anal., 184 (2007), 465.  doi: 10.1007/s00205-006-0030-8.  Google Scholar [24] A. Venturelli, Application de la Minimisation de L'action au Problme des N Corps Dans Leplan et Dans L'espace,, Ph.D thesis, (2002).   Google Scholar [25] G. Yu, Simple choreography solutions of newtonian n-body problem,, , (2015).   Google Scholar [26] G. Yu, Shape space figure 8 solutions of three body problem with two equalmasses,, , (2015).   Google Scholar [27] G. Yu, Periodic solutions of the rotating N-center and restricted N + 1-body problems with topological constraints,, work in progress., ().   Google Scholar
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