  A family of effective actions in Hamiltonian form is derived for a
self-gravitating sphere of isotropic homogeneous dust. Starting from the
Einstein-Hilbert action for barotropic perfect fluids and making use of the
symmetry and equation of state of the matter distribution we obtain reduced
actions for two canonical variables, namely the radius of the sphere and its
ADM energy, the latter being conserved along trajectories of the former. These
actions differ by the value of the (conserved) geodesic energy of the radius of
the sphere which defines (disconnected) classes of solutions in correspondence
to the inner geometry and proper volume of the sphere. Each class is thus
treated as one constrained dynamical system and the union of all classes covers
the full phase space of the model. Generalization to the (inhomogeneous) Tolman
model is shown to be straightforward. Quantization is also discussed.
