  In this second article of a series we propose to base criteria of stability
on the hamiltonian functional that is provided by the variational principle, to
replace the reliance that has often been placed on {\it ad hoc} definitions of
the "energy". We introduce a new virial principle that is formulated entirely
within the Eulerian description of hydrodynamics, which allows a simpler
derivation of a well known stability criterion for polytropic stellar
configurations. Boundary conditions are based entirely on mass conservation.
