  Bondi (1952) and Parker (1958} derived a steady-state solution for
Bernouilli's equation in spherical symmetry around a point mass for two cases,
respectively, an inward accretion flow and an outward wind. Left unanswered
were the stability of the steady-state solution, the solution itself of
time-dependent flows, whether the time-dependent flows would evolve to the
steady-state, and under what conditions a transonic flow would develop. In a
Hamiltonian description, we find that the steady state solution is equivalent
to the Lagrangian implying that time-dependent flows evolve to the steady
state. We find that the second variation is definite in sign for isothermal and
adiabatic flows, implying at least linear stability. We solve the partial
differential equation for the time-dependent flow as an initial-value problem
and find that a transonic flow develops under a wide range of realistic initial
conditions. We present some examples of time-dependent solutions.
