Investigation of the nonlinear stability of a pendulum with variable length in elliptic orbit

The dynamics of a simple pendulum attached to the center of mass of a satellite in an elliptic orbit is studied. We restrict the study to the case where the pendulum is in the orbital plane, with the length of the pendulum rod varying proportionally to the radius vector of the orbit. Considering the satellite approximation (namely, the size of the pendulum rod is smaller than the radius vector of the orbit) and using the Hamiltonian function of the problem, we analyze, taking into account the parameter of the eccentricity of the orbit, the conditions to obtain the linear and nonlinear stability of four equilibrium positions found.