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Stability of axial orbits in galactic potentials

  • Cinzia Belmonte
  • Dino Boccaletti
  • Giuseppe Pucacco
Conference paper

Abstract

We investigate the dynamics in a galactic potential with two reflection symmetries. The phase-space structure of the real system is approximated with a resonant detuned normal form constructed with the method based on the Lie transform. Attention is focused on the stability properties of the axial periodic orbits that play an important role in galactic models. Using energy and ellipticity as parameters, we find analytical expressions of bifurcations and compare them with numerical results available in the literature.

Keywords

Normal forms of Hamiltonian systems Stability of periodic orbits Galactic potentials 

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References

  1. Belmonte, C., Boccaletti, D., Pucacco, G.: Approximate first integrals for a model of galactic potential with the method of Lie transform normalization. In: Perez-Chavela, E., Xia, J. (eds.) Submitted to the Proceedings of the Saarifest (2005)Google Scholar
  2. Birkhoff, G.D.: Dynamical systems, Amer. Math. Soc. Coll. Publ., vol. 9, New York, USA (1927)Google Scholar
  3. Binney, J., Tremaine, S.: Galactic Dynamics, Princeton University Press (1987)Google Scholar
  4. Boccaletti, D., Pucacco, G.: Theory of Orbits, vol. 2, Springer-Verlag, Berlin (1999)zbMATHGoogle Scholar
  5. Contopoulos, G.: Higher order resonances in dynamical systems. Cel. Mech. 18, 195–204 (1978)CrossRefADSMathSciNetzbMATHGoogle Scholar
  6. Contopoulos, G.: Order and Chaos in Dynamical Astronomy. Springer-Verlag, Berlin (2002)zbMATHGoogle Scholar
  7. Contopoulos, G., Efthymiopoulos, C., Giorgilli, A.: Nonconvergence of formal integrals of motion. J. Phys. A: Math. Gen. 36, 8639–8660 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  8. de Zeeuw, T., Merritt, D.: Stellar orbits in a triaxial galaxy. I. Orbits in the plane of rotation. Astrophys. J. 267, 571–595 (1983)CrossRefADSMathSciNetGoogle Scholar
  9. Dragt, A., Finn, J.M.: Lie series and invariant functions for analytic symplectic maps. J. Mat. Phys. 17, 2215–2227 (1976)CrossRefADSMathSciNetzbMATHGoogle Scholar
  10. Efthymiopoulos, C., Giorgilli, A., Contopoulos, G.: Nonconvergence of formal integrals: II. Improved estimates for the optimal order of truncation. J. Phys. A: Math. Gen. 37, 10831–10858 (2004)CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. Finn, J.M.: Lie series: a perspective. Local and global methods of nonlinear dynamics. Lecture Notes in Physics, vol. 252, pp. 63–86 (1984)ADSMathSciNetCrossRefGoogle Scholar
  12. Fridman, T., Merritt, D.: Periodic orbits in triaxial galaxies with weak cusps. Astron. J. 114, 1479–1487 (1997)CrossRefADSGoogle Scholar
  13. Gustavson, F.: On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71, 670–686 (1966)CrossRefADSGoogle Scholar
  14. Koseleff, P.V.: Comparison between Deprit and Dragt-Finn perturbation methods. Cel. Mech. & Dynam. Astron. 58, 17–36 (1994)CrossRefADSMathSciNetGoogle Scholar
  15. Kummer, M.: On resonant Hamiltonians with two degrees of freedom near an equilibrium point. Lecture Notes in Physics, vol. 93, pp. 57–75 (1977)ADSMathSciNetCrossRefGoogle Scholar
  16. Miralda-Escudé, J., Schwarzschild, M.: On the orbit structure of the logarithmic potential. Astrophys. J. 339, 752–762 (1989)CrossRefADSGoogle Scholar
  17. Moser, J.: Lectures on Hamiltonian systems. Mem. Am. Math. Soc. 81, 1–60 (1968)Google Scholar
  18. Sanders, J.A., Verhulst, F.: Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York (1985)zbMATHGoogle Scholar
  19. Schwarzschild, M.: A numerical model for a triaxial stellar system in dynamical equilibrium. Astrophys. J. 232, 236–247 (1979)CrossRefADSGoogle Scholar
  20. Scuflaire, R.: Stability of axial orbits in analytic galactic potentials. Cel. Mech. & Dynam. Astron. 61, 261–285 (1995)CrossRefADSMathSciNetzbMATHGoogle Scholar
  21. Tuwankotta, J.M., Verhulst, F.: Symmetry and resonance in Hamiltonian systems. SIAM J. Appl. Math. 61, 1369–1385 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  22. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, Berlin (1996)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Cinzia Belmonte
    • 1
  • Dino Boccaletti
    • 2
  • Giuseppe Pucacco
    • 3
  1. 1.Dipartimento di FisicaUniversità di Roma “la Sapienza”RomeItaly
  2. 2.Dipartimento di MathematicaUniversità di Roma “la Sapienza”RomeItaly
  3. 3.Dipartimento di FisicaUniversità di Roma “Tor Vergata” and INFN-Sezione Tor VergataRomeItaly

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