  The following principle of minimum energy may be a powerful substitute to the
dynamical perturbation method, when the latter is hard to apply. Fluid elements
of self-gravitating barotropic flows, whose vortex lines extend to the boundary
of the fluid, are labelled in such a way that any change of trial
configurations automatically preserves mass and circulation. The velocity field
is given by a mass conserving Clebsch representation. With three independent
Lagrangian functions, the total energy is stationary for all small variations
about a flow with fixed linear and angular momenta provided Euler's equations
for steady motion are satisfied. Thus, steady flows are stable if their energy
is minimum. Since energy is here minimized subject to having local and global
contants of the motion fixed, stability limits obtained that way are expected
to be close to limits given by dynamical perturbation methods. Moreover, the
stability limits are with respect to arbitrary, not necessary small,
perturbations. A weaker form of the energy principle is also given which may be
easier to apply. The Lagrangian functional, with the same three Lagrange
variables is stationary for the fully time dependent Euler equations. It
follows that the principle of minimum energy gives stability conditions that
are both necessary and sufficient if terms linear in time derivatives
(gyroscopic terms) are absent from the Lagrangian. The gyroscopic term for
small deviations around steady flows is given explicitly. Key words: Energy
variational principle; Self-gravitating systems; Stability of fluids.
