Hostname: page-component-6bf8c574d5-r8w4l Total loading time: 0 Render date: 2025-03-07T16:48:45.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Models of Elliptical Galaxies in 1–1–1 Resonance and their Normalization: The 3D Hénon and Heiles System

Published online by Cambridge University Press:  12 April 2016

S. Ferrer ,
A. Viartola ,
J. Palacián ,
P. Yanguas  and
J.F. San Juan
S. Ferrer
Affiliation:
Universidad de Zaragoza, Zaragoza, Spain
A. Viartola
Affiliation:
Universidad de Zaragoza, Zaragoza, Spain
J. Palacián
Affiliation:
Universidad Pública de Navarra, Pamplona, Spain
P. Yanguas
Affiliation:
Universidad Pública de Navarra, Pamplona, Spain
J.F. San Juan
Affiliation:
Universidad de la Rioja, Logroño, Spain

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this Note we study perturbed isotropic harmonic oscillators in 1–1–1 resonance, which is one of the typical cases of galactic potential models. We focus on cubic and quartic axial symmetric potentials giving explicitly their normal form. The 3D normalized Hénon and Heiles case, which requires to reach fourth order, is studied showing its relative equilibria and bifurcations.

Type
Theory of Motion
Information
International Astronomical Union Colloquium , Volume 165: Dynamics and Astrometry of Natural and Artificial Celestial Bodies , 1997 , pp. 425 - 430
Copyright
Copyright © Kluwer 1997

References

Abad, A. and San Juan, J.F.: 1993, “PSPC: A Poisson Series Processor Coded in C, in: Dynamics and Astrometry of Natural and Artificial Celestial Bodies, (Kurzynska, K., Barlier, F., Seidelmann, P.K., Wytrzyszczak, I., eds), Astronomical Observatory, Poznań, 383–388.Google Scholar
Churchill, R.C., Kummer, M., and Rod, D.L.: 1983, “On averaging, reduction, and symmetry in Hamiltonian systems”, J. Diff. Equat. 49, 359–414.CrossRefGoogle Scholar
de Zeeuw, T.: 1985, “Motion in the core of a triaxial potential”, Mon. Not. R. Astron. Soc. 215, 731–760.Google Scholar
Deprit, A.: 1969, “Canonical transformations depending on a small parameter”, Ce-lest. Mech. 1, 12–30.CrossRefGoogle Scholar
Deprit, A.: 1991, “The Lissajous transformation. I: Basics”, Celest. Mech. & Dyn. Astron. 51, 201–226.Google Scholar
Ferrer, S. and Gárate, J.: 1996, “On perturbed 3D elliptic oscillators: A case of critical inclination in galactic dynamics”, in: New Trends in Hamiltonian Systems and Celestial Mechanics, (Lacomba, E. Llibre, J., eds), ASND 8 World Scientific, Singapore, 179–197.Google Scholar
Ferrer, S., Lara, M., Palacián, J., and Yanguas, P.: 1996 a, “On perturbed oscillators in 1-1-1 Resonance: Critical inclination in the 3D Hénon and Heiles potential”, to appear in: Hamiltonian Systems with Three or More Degrees of Freedom, (Delshams, A., ed.), Kluwer, Dordrecht.Google Scholar
Ferrer, S., Lara, M., Palacián, J., San Juan, J.F., Viartola, A., and Yanguas, P.: 1996 b, “The Hénon and Heiles system in three dimensions: The atypical cubic oscillator in 1-1-1 resonance, Physica D, submitted.Google Scholar
Hénon, M. and Heiles, C.: 1964, “The applicability of the third integral of motion: Some numerical experiments”, Astron. J. 69, 73–79.Google Scholar
Robe, H.: 1985, “Periodic orbits in a triaxial galaxy”, Astron. & Astrophys. 142, 351–354 and 1987, ibid 182, 202-206.Google Scholar
Verhulst, F.: 1979, “Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies”, Phil. Trans. Roy. Soc. London A 290, 435–465.Google Scholar
Yanguas, P.: 1996, Ph.D. Thesis, in preparation.Google Scholar