Figure 1.
A sketch of the first sixth of the orbit
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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
Universidade Federal Rural de Pernambuco, Departamento de Matemática, Rua Dom Manoel de Medeiros, s/n, Recife, PE 52171-900, Brasil |
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.
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 and Continuous Dynamical Systems,
2017, 37
(4)
: 1903-1922.
doi: 10.3934/dcds.2017080
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H. Brezis,
Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1983. |
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K. -C. Chen,
Variational Methods and Periodic Solutions of Newtonian N-Body Problems Ph. D thesis, University of Minnesota, 2001. |
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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. |
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Z. Coti-Zelati,
Periodic solution for $N$-body problems, Ann. Inst. Henri Poincaré, Anal Non Linéaire, 7 (1990), 477-492.
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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.
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W. B. Gordon,
A minimizing property of Kleperian orbits, American Journal of Math, 99 (1977), 961-971.
doi: 10.2307/2373993. |
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T. Levi-Civita,
Sur la Régularization du probléme des trois corps, Acta. Math, 42 (1920), 99-144.
doi: 10.1007/BF02404404. |
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R. Palais,
The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
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H. Poincaré,
Sur Les Solutions Périodiques et le Principe de Moindre Action, C. R. A. S. , 1896. |
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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. |
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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. |
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J. Schubart,
Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astronom. Nachr., 283 (1956), 17-22.
doi: 10.1002/asna.19562830105. |
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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. |
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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. |
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L. C. Young,
Lectures on the Calculus of Variations and Optimal Control Theory W. B. Saunders Company, 1969. |
show all references
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. |
Figure 2.
Orbit in the configuration space
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