# American Institute of Mathematical Sciences

April  2017, 37(4): 1903-1922. doi: 10.3934/dcds.2017080

## An existence proof of a symmetric periodic orbit in the octahedral six-body problem

 Universidade Federal Rural de Pernambuco, Departamento de Matemática, Rua Dom Manoel de Medeiros, s/n, Recife, PE 52171-900, Brasil

Received  May 2016 Revised  November 2016 Published  December 2016

Fund Project: The first author is supported by CAPES grants.

We present a proof of the existence of a periodic orbit for the Newtonian six-body problem with equal masses. This orbit has three double collisions each period and no multiple collisions. Our proof is based on the minimization of the lagrangian action functional on a well chosen class of symmetric loops.

Citation: Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
##### References:
 [1] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. [2] K. -C. Chen, Variational Methods and Periodic Solutions of Newtonian N-Body Problems Ph. D thesis, University of Minnesota, 2001. [3] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal mass, Annals of Mathematics, 152 (2000), 881-901.  doi: 10.2307/2661357. [4] Z. Coti-Zelati, Periodic solution for $N$-body problems, Ann. Inst. Henri Poincaré, Anal Non Linéaire, 7 (1990), 477-492. [5] M. Degiovanni, F. Gianonni and A. Marino, Periodic solutions of dynamical systems with newtonian type potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 467-494. [6] W. B. Gordon, A minimizing property of Kleperian orbits, American Journal of Math, 99 (1977), 961-971.  doi: 10.2307/2373993. [7] T. Levi-Civita, Sur la Régularization du probléme des trois corps, Acta. Math, 42 (1920), 99-144.  doi: 10.1007/BF02404404. [8] R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322. [9] H. Poincaré, Sur Les Solutions Périodiques et le Principe de Moindre Action, C. R. A. S. , 1896. [10] E. Serra and S. Terracini, Collisionless periodic solutions to some three-body problems, Arch. Rational Mech. Anal., 120 (1992), 305-325.  doi: 10.1007/BF00380317. [11] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the n-body problem, Arch. Rational Mech. Anal., 199 (2011), 821-841.  doi: 10.1007/s00205-010-0334-6. [12] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astronom. Nachr., 283 (1956), 17-22.  doi: 10.1002/asna.19562830105. [13] A. Venturelli, A Variational proof of the existence of Von Schubart's Orbits, Discrete and Continuous Dynamical Systems B, 10 (2008), 699-717.  doi: 10.3934/dcdsb.2008.10.699. [14] A. Venturelli, Application de la Minimisation De L'action au Probléme des N Corps Dans le Plan et Dans L'espace Ph. D. thesis, Université Denis Diderot in Paris, 2002. [15] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders Company, 1969.

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##### References:
 [1] H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. [2] K. -C. Chen, Variational Methods and Periodic Solutions of Newtonian N-Body Problems Ph. D thesis, University of Minnesota, 2001. [3] A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal mass, Annals of Mathematics, 152 (2000), 881-901.  doi: 10.2307/2661357. [4] Z. Coti-Zelati, Periodic solution for $N$-body problems, Ann. Inst. Henri Poincaré, Anal Non Linéaire, 7 (1990), 477-492. [5] M. Degiovanni, F. Gianonni and A. Marino, Periodic solutions of dynamical systems with newtonian type potentials, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 15 (1988), 467-494. [6] W. B. Gordon, A minimizing property of Kleperian orbits, American Journal of Math, 99 (1977), 961-971.  doi: 10.2307/2373993. [7] T. Levi-Civita, Sur la Régularization du probléme des trois corps, Acta. Math, 42 (1920), 99-144.  doi: 10.1007/BF02404404. [8] R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322. [9] H. Poincaré, Sur Les Solutions Périodiques et le Principe de Moindre Action, C. R. A. S. , 1896. [10] E. Serra and S. Terracini, Collisionless periodic solutions to some three-body problems, Arch. Rational Mech. Anal., 120 (1992), 305-325.  doi: 10.1007/BF00380317. [11] M. Shibayama, Minimizing periodic orbits with regularizable collisions in the n-body problem, Arch. Rational Mech. Anal., 199 (2011), 821-841.  doi: 10.1007/s00205-010-0334-6. [12] J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astronom. Nachr., 283 (1956), 17-22.  doi: 10.1002/asna.19562830105. [13] A. Venturelli, A Variational proof of the existence of Von Schubart's Orbits, Discrete and Continuous Dynamical Systems B, 10 (2008), 699-717.  doi: 10.3934/dcdsb.2008.10.699. [14] A. Venturelli, Application de la Minimisation De L'action au Probléme des N Corps Dans le Plan et Dans L'espace Ph. D. thesis, Université Denis Diderot in Paris, 2002. [15] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders Company, 1969.
A sketch of the first sixth of the orbit
Orbit in the configuration space
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