  For an underactuated (simple) Hamiltonian system with two degrees of freedom
and one degree of underactuation, a rather general condition that ensures its
stabilizability, by means of the existence of a (simple) Lyapunov function, was
found in a recent paper by D.E. Chang within the context of the energy shaping
method. Also, in the same paper, some additional assumptions were presented in
order to ensure also asymptotic stabilizability. In this paper we extend these
results by showing that above mentioned condition is not only sufficient, but
also a necessary one. And, more importantly, we show that no additional
assumption is needed to ensure asymptotic stabilizability.
