  We investigate stability issues for steady states of the spherically
symmetric Einstein-Vlasov system numerically in Schwarzschild, maximal areal,
and Eddington-Finkelstein coordinates. Across all coordinate systems we confirm
the conjecture that the first binding energy maximum along a one-parameter
family of steady states signals the onset of instability. Beyond this maximum
perturbed solutions either collapse to a black hole, form heteroclinic orbits,
or eventually fully disperse. Contrary to earlier research, we find that a
negative binding energy does not necessarily correspond to fully dispersing
solutions. We also comment on the so-called turning point principle from the
viewpoint of our numerical results. The physical reliability of the latter is
strengthened by obtaining consistent results in the three different coordinate
systems and by the systematic use of dynamically accessible perturbations.
