|
Hamiltonian systems of perturbed
oscillators in 1:1:1:1 resonance are studied, which include several
models for
perturbed Keplerian systems.
Assuming symmetries in those systems we
make use of regular and singular reductions techniques. Normalization
by
Lie-transforms is approached both in Poisson and symplectic formalisms.
The
first procedure relies on the quadratic invariants, and is based on
previous
work of the authors (see CASA 09-27 and 09-31). Hinging on the
maximally
superintegrable character of the isotropic oscillator, the symplectic
reduction
is carried out a la Delaunay using a generalization of those
variables
to 4-DOF.
As an application the 4D
generalized Van der Waals family of oscillators in 1:1:1:1 resonance is
studied, which includes several models for perturbed Keplerian systems.
Explicit expressions of the Delaunay normalization up to the second
order are
presented, showing that they may be extended to higher orders. We focus
on the
search of the relative equilibria and the bifurcations with both
treatments.
Due to the symmetries of the system there are isolated as well as
circles of
stationary points. This feature manifest rather differently in the
analysis of
the normalized flow in the formulations considered, due to the
constraints
among the invariants versus the singularities associated to the
Delaunay chart.
The pros and cons will be presented in detail.
|
