Approximate First Integrals with the Method of Lie Transform Normalization

Abstract

We investigate the phase-space structure of perturbed resonant harmonic oscillators with the method of Lie Transform Normalization. The perturbation is a quartic polynomial which can be used as a model for the central part of an elliptical galaxy and is analyzed computing approximate integrals of motion in the form of truncated series expansions. We compute surfaces of sections and compare the results with numerical integrations verifying that a global agreement is achieved at the first order which incorporates the resonance.

Stability of axial orbits in galactic potentials

“…A Lie transform normalization truncated to the second order gives the following expression of the first-order normal form (cfr. Belmonte et al 2006)…”

“…Its expression is quite involved (cfr. Belmonte et al 2006), but we can exploit the change of variables to action-angle coordinates to see that the normal form has the structure K = J 1 +2J 2 +ε 2 (2δJ 1 +P 2 (J 1 , J 2 ))+ε 4 (P 3 (J 1 , J 2 )+kJ 2 1 J 2 cos(4θ 1 −2θ 2 )), (74) where Eq. (29) has been used, that in the present instance reads…”

“…Its expression is quite involved (cfr. Belmonte et al 2006), but we can exploit the change of variables to action-angle coordinates to see that the normal form has the structure…”