Models of Elliptical Galaxies in 1-1-1 Resonances with Axial and Discrete Symmetries.

(1) Grupo de Mecanica Espacial, Univ. de Zaragoza, 50009 Zaragoza, Spain,
(2) Departamento de Matematica e Informatica, Univ. Publica de Navarra, 31006 Pamplona, Spain
(3) Departamento de Matematicas y Computacion, Univ. de La Rioja, 26004 Logrono, Spain,
(4) Departamento de Matematica e Informatica, Univ. Publica de Navarra, 31006 Pamplona, Spain

The composition of the Whittaker transformation with a Lissajous transformation in the instantaneous trajectory plane, opens the possibility to normalize 3 dimensions perturbed isotropic harmonic oscillators in a 1-1-1 resonance. In this way we manage to achieve an extension of the studies involving solutions in the equatorial and meridian planes, by now considering orbits at any inclination. A main feature of this transformation is the expression of the unperturbed Hamiltonian as proportional to one of the momentum coordinates; thus, when the perturbation is periodic, the solution of the equation that has to be solved in each step of the normalization process is simply an average with respect to the elliptic anomaly . Recognizing the fact that, from the Lissajous normalization and the axial symmetry, the doubly reduced orbit space is S², we introduce coordinates which allow to look for relative equilibria and their bifurcations over the domain of parameters.

As a first application we deal with 2-DOF cubic and quartic polynomial potentials with axial and discrete symmetries proposed by de Zeeuw. In particular we deal with the relation between degenerate cases of the averaged system with the integrable cases of the full problem. As far as we know only first order studies have been done; we have extended the normalization up to higher orders. As an example, dealing solely with the cubic perturbation, we extend the study done by Miller for the meridian case. As a second application we apply the concept of the elimination of the parallax and radial simplifications to perturbed isotropic oscillators.

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