Orbits in a two Dimensional Non-axis-symmetric Galactic Potential

The dynamics of stellar objects, considered as point test particles in a non-axis-symmetric logarithmic galactic potential U(x,y)$\propto \,Ln\left[ {R_c ^2\,+\,x^2\,+\,\frac{y^2}{b^2}} \right]$ are studied numerically and analytically, using a Leapfrog Integrator. This potential is considered as a model of a galaxy in astrophysics since it includes the gravitational effects of both the visible and the dark matter composing it. The ellipticity parameter 0$<$b$<$1~generates a fluctuating torque acting on the moving test particles of the galaxy (actually the stars) and triggers the emergence of a set of non conventional families of orbits. This fluctuating torque provides, in our opinion, a rich field to test concepts like conserved quantities and compare them with the known conventional orbits in mechanics. We provide a simple numerical approach that reproduces most of the orbits associated with this potential. We also analyze the behavior of the torque and angular momentum at the different orbits.