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

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

Guowei Yu 1,

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.
Keywords: topological constraints., variational method, N-center problem.
Mathematics Subject Classification: Primary: 70F10, 37N0.
Citation: Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete and Continuous Dynamical Systems, 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, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.

[2]

A. Bahri and P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., 82 (1989), 412-428. doi: 10.1016/0022-1236(89)90078-5.

[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), 189-226. doi: 10.1007/s00205-008-0131-7.

[4]

V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, Arch. Ration. Mech. Anal., 207 (2013), 583-609. doi: 10.1007/s00205-012-0565-9.

[5]

V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, Calc. Var. Partial Differential Equations, 49 (2014), 391-429. doi: 10.1007/s00526-012-0587-z.

[6]

K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2.

[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), 325-348. doi: 10.4007/annals.2008.167.325.

[8]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry, in Proceedings of the International Congress of Mathematicians, Higher Ed. Press, Beijing, (2002), 279-294.

[9]

A. Chenciner, J. Gerver, R. Montgomery and C. Simó, Simple choreographic motions of N bodies: A preliminary study, in Geometry, mechanics, and dynamics, (2002), 287-308. doi: 10.1007/0-387-21791-6_9.

[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), 881-901. doi: 10.2307/2661357.

[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), 139-152. doi: 10.1023/A:1008381001328.

[12]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[13]

D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.

[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), 283-332. doi: 10.1007/s00222-010-0306-3.

[15]

W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975), 113-135. doi: 10.1090/S0002-9947-1975-0377983-1.

[16]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.

[17]

J. Hass and P. Scott, Intersections of curves on surfaces, Israel J. Math., 51 (1985), 90-120. doi: 10.1007/BF02772960.

[18]

M. W. Hirsch, Differential Topology, Texts in Mathematics, 33. Springer-Verlag, New York, 1994.

[19]

R. Montgomery, The N-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity, 11 (1998), 363-376. doi: 10.1088/0951-7715/11/2/011.

[20]

M. Shibayama, Variational proof of the existence of the super-eight orbit in the four-body problem, Arch. Ration. Mech. Anal., 214 (2014), 77-98. doi: 10.1007/s00205-014-0753-x.

[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), 371-413. doi: 10.1007/s00030-013-0251-0.

[22]

N. Soave and S. Terracini, Symbolic dynamics for the N-centre problem at negative energies, Discrete Contin. Dyn. Syst. - A, 32 (2012), 3245-3301. doi: 10.3934/dcds.2012.32.3245.

[23]

S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Ration. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.

[24]

A. Venturelli, Application de la Minimisation de L'action au Problme des N Corps Dans Leplan et Dans L'espace, Ph.D thesis, Universit Denis Diderot in Paris, 2002.

[25]

G. Yu, Simple choreography solutions of newtonian n-body problem, arXiv:1509.04999, 2015.

[26]

G. Yu, Shape space figure 8 solutions of three body problem with two equalmasses, arXiv:0707.0078, 2015.

[27]

G. Yu, Periodic solutions of the rotating N-center and restricted N + 1-body problems with topological constraints,, work in progress., (). 

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, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0319-3.

[2]

A. Bahri and P. H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., 82 (1989), 412-428. doi: 10.1016/0022-1236(89)90078-5.

[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), 189-226. doi: 10.1007/s00205-008-0131-7.

[4]

V. Barutello, S. Terracini and G. Verzini, Entire minimal parabolic trajectories: The planar anisotropic Kepler problem, Arch. Ration. Mech. Anal., 207 (2013), 583-609. doi: 10.1007/s00205-012-0565-9.

[5]

V. Barutello, S. Terracini and G. Verzini, Entire parabolic trajectories as minimal phase transitions, Calc. Var. Partial Differential Equations, 49 (2014), 391-429. doi: 10.1007/s00526-012-0587-z.

[6]

K.-C. Chen, Binary decompositions for planar N-body problems and symmetric periodic solutions, Arch. Ration. Mech. Anal., 170 (2003), 247-276. doi: 10.1007/s00205-003-0277-2.

[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), 325-348. doi: 10.4007/annals.2008.167.325.

[8]

A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: From homology to symmetry, in Proceedings of the International Congress of Mathematicians, Higher Ed. Press, Beijing, (2002), 279-294.

[9]

A. Chenciner, J. Gerver, R. Montgomery and C. Simó, Simple choreographic motions of N bodies: A preliminary study, in Geometry, mechanics, and dynamics, (2002), 287-308. doi: 10.1007/0-387-21791-6_9.

[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), 881-901. doi: 10.2307/2661357.

[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), 139-152. doi: 10.1023/A:1008381001328.

[12]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[13]

D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.

[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), 283-332. doi: 10.1007/s00222-010-0306-3.

[15]

W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975), 113-135. doi: 10.1090/S0002-9947-1975-0377983-1.

[16]

W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1977), 961-971. doi: 10.2307/2373993.

[17]

J. Hass and P. Scott, Intersections of curves on surfaces, Israel J. Math., 51 (1985), 90-120. doi: 10.1007/BF02772960.

[18]

M. W. Hirsch, Differential Topology, Texts in Mathematics, 33. Springer-Verlag, New York, 1994.

[19]

R. Montgomery, The N-body problem, the braid group, and action-minimizing periodic solutions, Nonlinearity, 11 (1998), 363-376. doi: 10.1088/0951-7715/11/2/011.

[20]

M. Shibayama, Variational proof of the existence of the super-eight orbit in the four-body problem, Arch. Ration. Mech. Anal., 214 (2014), 77-98. doi: 10.1007/s00205-014-0753-x.

[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), 371-413. doi: 10.1007/s00030-013-0251-0.

[22]

N. Soave and S. Terracini, Symbolic dynamics for the N-centre problem at negative energies, Discrete Contin. Dyn. Syst. - A, 32 (2012), 3245-3301. doi: 10.3934/dcds.2012.32.3245.

[23]

S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Ration. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.

[24]

A. Venturelli, Application de la Minimisation de L'action au Problme des N Corps Dans Leplan et Dans L'espace, Ph.D thesis, Universit Denis Diderot in Paris, 2002.

[25]

G. Yu, Simple choreography solutions of newtonian n-body problem, arXiv:1509.04999, 2015.

[26]

G. Yu, Shape space figure 8 solutions of three body problem with two equalmasses, arXiv:0707.0078, 2015.

[27]

G. Yu, Periodic solutions of the rotating N-center and restricted N + 1-body problems with topological constraints,, work in progress., (). 

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