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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

Anete S. Cavalcanti ,

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.

Keywords: Celestial mechanics, octahedral six body problem, periodic solutions, symmetric orbits.
Mathematics Subject Classification: Primary:70F15, 70F16;Secondary:37J50.
Citation: Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080
References:
[1]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.  Google Scholar

[2]

K. -C. Chen, Variational Methods and Periodic Solutions of Newtonian N-Body Problems Ph. D thesis, University of Minnesota, 2001.  Google Scholar

[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.  Google Scholar

[4]

Z. Coti-Zelati, Periodic solution for $N$-body problems, Ann. Inst. Henri Poincaré, Anal Non Linéaire, 7 (1990), 477-492.   Google Scholar

[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.   Google Scholar

[6]

W. B. Gordon, A minimizing property of Kleperian orbits, American Journal of Math, 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[7]

T. Levi-Civita, Sur la Régularization du probléme des trois corps, Acta. Math, 42 (1920), 99-144.  doi: 10.1007/BF02404404.  Google Scholar

[8]

R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[9]

H. Poincaré, Sur Les Solutions Périodiques et le Principe de Moindre Action, C. R. A. S. , 1896. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astronom. Nachr., 283 (1956), 17-22.  doi: 10.1002/asna.19562830105.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders Company, 1969.  Google Scholar

show all references

References:
[1]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983.  Google Scholar

[2]

K. -C. Chen, Variational Methods and Periodic Solutions of Newtonian N-Body Problems Ph. D thesis, University of Minnesota, 2001.  Google Scholar

[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.  Google Scholar

[4]

Z. Coti-Zelati, Periodic solution for $N$-body problems, Ann. Inst. Henri Poincaré, Anal Non Linéaire, 7 (1990), 477-492.   Google Scholar

[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.   Google Scholar

[6]

W. B. Gordon, A minimizing property of Kleperian orbits, American Journal of Math, 99 (1977), 961-971.  doi: 10.2307/2373993.  Google Scholar

[7]

T. Levi-Civita, Sur la Régularization du probléme des trois corps, Acta. Math, 42 (1920), 99-144.  doi: 10.1007/BF02404404.  Google Scholar

[8]

R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar

[9]

H. Poincaré, Sur Les Solutions Périodiques et le Principe de Moindre Action, C. R. A. S. , 1896. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astronom. Nachr., 283 (1956), 17-22.  doi: 10.1002/asna.19562830105.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders Company, 1969.  Google Scholar

Figure 1.  A sketch of the first sixth of the orbit
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Figure 2.  Orbit in the configuration space
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