A Fluctuating Torque

The existence of a fluctuating torque generates a wide variety of possible orbits. This situation contrasts with those examples where the torque vanishes and the angular momentum remains constant. Here we study a two dimensional example with a logarithmic effective potential $ V(x,y)= \frac{1}{2}\,m\,V^2_o\,\ln[ x^2 + (y/b)^2]$, with a small deviation from the axis symmetry given by the constant b with $b < 1$. Briefly, the effective potential models the gravitational force exerted by the {\bf N} point particles on a test object. This potential is used to learn about the dynamics of galaxies and among other features, generates a fluctuating torque which is our main interest here. There is not an analytical solution for these two equations of motion. A simple numerical approach (provided) is required. Also, a change on the initial conditions may generate a different shape for the orbit. This apparently simple potential, represents a challenge for the students. We propose it as a good pedagogical tool for reviewing the main concepts of newtonian dynamics.