  In contrast to the Euler-Poincar{\'e} reduction of geodesic flows of left- or
right-invariant metrics on Lie groups to the corresponding Lie algebra (or its
dual), one can consider the reduction of the geodesic flows to the group
itself. The reduced vector field has a remarkable hydrodynamic interpretation:
it is a velocity field for a stationary flow of an ideal fluid. Right- or
left-invariant symmetry fields of the reduced field define vortex manifolds for
such flows.
  Consider now a mechanical system, whose configuration space is a Lie group
and whose Lagrangian is invariant to left translations on that group, and
assume that the mass geometry of the system may change under the action of
internal control forces. Such system can also be reduced to the Lie group. With
no controls, this mechanical system describes a geodesic flow of the
left-invariant metric, given by the Lagrangian, and thus its reduced flow is a
stationary ideal fluid flow on the Lie group. The standard control problem for
such system is to find the conditions, under which the system can be brought
from any initial position in the configuration space to another preassigned
position by changing its mass geometry. We show that under these conditions, by
changing the mass geometry, one can also bring one vortex manifold to any other
preassigned vortex manifold.
