In this paper, a method of tri-axial gravity gradient stabilization of satellite in circular orbit is proposed and investigated. In this method, only one actuator is employed. A satellite with varying-length boom is considered consisting of two rigid bodies having the freedom of moving in the boom direction. The only control input is the force between these two bodies to control the varying-length boom. The gravity gradient torque is considered as the only external torque acting on the satellite. The system is under-actuated and has Hamiltonian structure. So, the port-Hamiltonian approach is utilized. The equations of motion of the system are obtained in Hamiltonian formulation. The equilibrium points and their required control inputs are determined. The linearization around the equilibria is carried out and it can be seen that the linear dynamics of pitch-boom and roll-yaw are decoupled. Therefore, the roll-yaw dynamics is linearly uncontrollable. The method of energy shaping and damping injection is used for controller design. The conditions on the energy shaping control law to stabilize the system are determined. Further, the resulting closed-loop system is analyzed. The closed-loop system has center manifolds. Finally, the performance of the closed-loop system, convergence of state trajectory to the center manifold and its non-exponential convergence is shown by simulation.