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

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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

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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

show all references

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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