Phase-plane analysis of the timelike geodesics around a spherically symmetric static dilaton black hole

In this note we take a dynamical systems approach to the equations of motion of a free test particle moving around a spherically symmetric static dilaton black hole, written in the Einstein frame. The equations of motion are obtained using the Euler-Lagrange formalism. Using the first integrals of motion, we reach the conclusion that the free test particles are moving in a plane, named \emph{plane of motion}. In it we analyze the existence and nature of the equilibrium points and compare the behavior of free test particles near the equilibrium points using the dynamics systems approach. The study revealed that in the exact phase-plane exist distinct regions of motion, separated through a curve named \emph{separatrix}. In the end we obtained a relation between the parameters describing the black hole and the free test particle that holds on a parabolic separatrix.