On the Lyapunov stability of stationary points around a central body

It is well known that for a satellite about the rotating Earth, in a frame fixed in the planet, there are four stationary solutions, two of which are stable in the linear sense and the other two-unstable. This result is proved by computing the eigenvalues of the linearized equations of motion. Determining the orbital stability (or Lyapunov stability) is more complicated because it requires firstly the computing of the normal form around the equilibrium and secondly the expression of this normal form in action-and-angle variables in order to apply the so-called Arnold's theorem of stability. Higher-order normal forms are necessary for some specific values. However, Arnold's theorem is useless in the presence of resonances, in which case a new technique (again related to normal forms) must be applied in order to determine the orbital stability. In this paper, we proceed symbolically, taking the harmonic coefficients as parameters. We find the Lyapunov stability diagram on the parametric plane for the stationary points. We also study resonances 2:1 and 3:1, as well as the case in which a higher-order normalization is needed. Therefore, whatever the values of the harmonic coefficients of the potential expansion are the Lyapunov stability of the stationary solutions is determined. The advantage of a symbolic analysis is that the replacement of the actual values of a planet or celestial body is sufficient to obtain the orbital stability of the stationary points. For Earth-like planets, these points are indeed stable.