Non-integrability of some Hamiltonian systems in polar coordinates

In this paper we first reformulate a non-integrability criterion obtained by Yoshida for Hamiltonian systems with two degrees of freedom in order to make it easier to handle those problems whose natural formulation is given in polar coordinates, as occurs with those that have harmonic potential. Among other applications, we prove the non-integrability of the satellite problem under McCullagh's approximation of the potential, i.e. truncated at the r(-3) term that, in most cases, is the main problem of the satellite of a triaxial primary body, hence its importance.