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

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

[1]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

[2]

Yuika Kajihara, Misturu Shibayama. Variational proof of the existence of brake orbits in the planar 2-center problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5785-5797. doi: 10.3934/dcds.2019254

[3]

Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733

[4]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[5]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971

[6]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123

[7]

David Jerison, Nikola Kamburov. Free boundaries subject to topological constraints. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7213-7248. doi: 10.3934/dcds.2019301

[8]

Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial & Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045

[9]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018

[10]

Giuseppe Buttazzo, Luigi De Pascale, Ilaria Fragalà. Topological equivalence of some variational problems involving distances. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 247-258. doi: 10.3934/dcds.2001.7.247

[11]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133

[12]

Sergio Grillo, Marcela Zuccalli. Variational reduction of Lagrangian systems with general constraints. Journal of Geometric Mechanics, 2012, 4 (1) : 49-88. doi: 10.3934/jgm.2012.4.49

[13]

Gang Qian, Deren Han, Lingling Xu, Hai Yang. Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities. Journal of Industrial & Management Optimization, 2013, 9 (1) : 255-274. doi: 10.3934/jimo.2013.9.255

[14]

Rafael Ortega, Andrés Rivera. Global bifurcations from the center of mass in the Sitnikov problem. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 719-732. doi: 10.3934/dcdsb.2010.14.719

[15]

Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577

[16]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967

[17]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201

[18]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423

[19]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621

[20]

Mohammad Safdari. The regularity of some vector-valued variational inequalities with gradient constraints. Communications on Pure & Applied Analysis, 2018, 17 (2) : 413-428. doi: 10.3934/cpaa.2018023

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]