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On the Periodic Structure of the Anisotropic Manev Problem

  • Juan Luis García GuiraoEmail author
  • José Luis Roca
  • Juan Antonio Vera López
Article
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Abstract

The aim of the present work is to provide sufficient conditions for the existence of periodic orbits of the first and second kind in the sense of Poincaré for the Anisotropic Manev problem. Moreover, we are also able to provide information on the stability and bifurcations of the orbits obtained. The main tool that we use is the averaging theory of dynamical systems.

Keywords

Celestial mechanics Anisotropic Manev problem Averaging theory of dynamical systems Periodic orbits 

Mathematics Subject Classification

Primary 70F05 70F15 

Notes

References

  1. 1.
    Abouelmagd, E.I., Alhothuali, M.S., Guirao, J.L.G., Malaikah, H.M.: The effect of zonal harmonic coefficients in the framework of the restricted three-body problem. Adv. Space Res. 55, 1660–1672 (2015)CrossRefGoogle Scholar
  2. 2.
    Abouelmagd, E.I.: Existence and stability of triangular points in the restricted three-body problem with numerical applications. Astrophys. Space Sci. 342, 45–53 (2012)CrossRefGoogle Scholar
  3. 3.
    Abouelmagd, E.I., Guirao, J.L.G.: On the perturbed restricted three-body problem. Appl. Math. Nonlinear Sci. 1(1), 123–144 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Abouelmagd, E.I., Guirao, J.L.G., Mostafa, A.: Numerical integration of the restricted three-body problem with Lie series. Astrophys. Space Sci. 354, 369–378 (2014)CrossRefGoogle Scholar
  5. 5.
    Ammar, M.K., Amin, M.R., Hassan, M.H.M.: Calculation of line of site periods between two artificial satellites under the action air drag. Appl. Math. Nonlinear Sci. 3(2), 339–352 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Alberti, A., Vidal, C.: First kind symmetric periodic solutions of the generalized van der Waals Hamiltonian. J. Math. Phys. 57, 072902 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Casasayas, J., Llibre, J.: Qualitative Analysis of the Anisotropic Kepler Problem, vol. 52, No. 312. Memoirs of the American Mathematical Society, Providence (1984)zbMATHGoogle Scholar
  8. 8.
    Craig, S., Diacu, F., Lacomba, E.A., Perez, E.: On the anisotropic Manev problem. J. Math. Phys. 40, 1359–1375 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Contopoulos, G., Harsoula, M.: Stability and instability in the anisotropic Kepler problem. J. Phys. A 38, 8897–8920 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cordani, B.: The Kepler Problem, Progress in Mathematical Physics 29. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cors, J.M., Hall, G.R.: Coorbital periodic orbits in the three body problem. SIAM J. Appl. Dyn. Syst. 2(2), 219–237 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Diacu, F., Santoprete, M.: On the global dynamics of the anisotropic Manev problem. Phys. D: Nonlinear Phenom. 194(1–2), 75–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Farrelly, D., Uzer, T.: Normalization and detection of the integrability: the generalized van der waals potential. Celest. Mech. Dyn. Astron. 61, 71–95 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Greuel, G.M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer Monographs in Mathematics. Springer, Berlin (2007)zbMATHGoogle Scholar
  15. 15.
    Gutzwiller, M.C.: The anisotropic Kepler problem in two dimensions. J. Math. Phys. 14, 139 (1973)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gutzwiller, M., Martin, C.: The quantization of a classically ergodic system. Classical quantum models and arithmetic problems. Lecture Notes in Pure and Applied Mathematics, vol. 92, pp. 287–351. Dekker, New York (1984)Google Scholar
  17. 17.
    Meyer, K.R., Hall, G.R., Offin, D.: Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Applied Mathematical Sciences 90. Springer, New York (2009)zbMATHGoogle Scholar
  18. 18.
    Meyer, K.R., Palacián, J.F., Yanguas, P., Dumas, H.S.: Periodic solutions in Hamiltonian systems, averaging, and the Lunar problem. SIAM J. Appl. Dyn. Syst. 7, 311–340 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meyer, K.R., Palacián, J.F., Yanguas, P.: Geometric averaging of Hamiltonian systems: periodic solutions, stability, and KAM tori. SIAM J. Appl. Dyn. Syst. 10, 817–856 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Robinson, C.: An Introduction to Dynamical Systems—Continuous and Discrete. Pure and Applied Undergraduate Texts, vol. 19, Second edn. American Mathematical Society, Providence (2012)zbMATHGoogle Scholar
  21. 21.
    Santoprete, M.: Symmetric periodic solutions of the anisotropic Manev problem. J. Math. Phys. 43, 3207 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tsetkova, K., Mioc, V.: Manev’s field problem in contemporary science. AIP Conf. Proc. 1043, 137–141 (2008)CrossRefGoogle Scholar
  23. 23.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin (1991)zbMATHGoogle Scholar
  24. 24.
    Vidal, C.: Periodic solutions for any planar symmetric perturbation of the Kepler problem. Celest. Mech. Dyn. Astron. 80, 119–132 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de Cartagena, Hospital de MarinaCartagenaSpain
  2. 2.Centro Universitario de la Defensa, Academia General del AireUniversidad Politécnica de CartagenaSantiago de la RiberaSpain

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