Periodic orbits of the three dimensional logarithm galactic potential

Abstract

We apply the averaging theory of first order to study analytically families of periodic orbits for a three dimensional logarithmic galactic potential $H=\dfrac12 (p_x^2+p_y^2+p_z^2) + \dfrac{ v_0^2}{2} \ln(x^2 - \lambda x^3 + \alpha y^2+b z^2 + c_b^2)$, that is relevant in the study of elliptic galactic dynamics. We first introduce a scale transformation in the coordinates and momenta with a parameter $\varepsilon$ and we find, using averaging theory of first order in $\varepsilon$, the existence up to three periodic orbits if $\alpha,\beta$ are irrational, and one periodic orbit if either $\alpha$ is irrational and $\beta$ is rational, or $\beta$ is irrational and $\alpha$ is rational, for $\varepsilon$ sufficiently small.