  We describe how a simple class of out of equilibrium, rotating and
asymmetrical mass distributions evolve under their self-gravity to produce a
quasi-planar spiral structure surrounding a virialized core, qualitatively
resembling a spiral galaxy. The spiral structure is transient, but can survive
tens of dynamical times, and further reproduces qualitatively noted features of
spiral galaxies as the predominance of trailing two-armed spirals and large
pitch angles. As our models are highly idealized, a detailed comparison with
observations is not appropriate, but generic features of the velocity
distributions can be identified to be potential observational signatures of
such a mechanism. Indeed, the mechanism leads generically to a characteristic
transition from predominantly rotational motion, in a region outside the core,
to radial ballistic motion in the outermost parts. Such radial motions are
excluded in our Galaxy up to 15 kpc, but could be detected at larger scales in
the future by GAIA. We explore the apparent motions seen by external observers
of the velocity distributions of our toy galaxies, and find that it is
difficult to distinguish them from those of a rotating disc with sub-dominant
radial motions at levels typically inferred from observations. These simple
models illustrate the possibility that the observed apparent motions of spiral
galaxies might be explained by non-trivial non-stationary mass and velocity
distributions without invoking a dark matter halo or modification of Newtonian
gravity. In this scenario the observed phenomenological relation between the
centripetal and gravitational acceleration of the visible baryonic mass could
have a simple explanation.
