  We introduce and study extensions of the varying alpha theory of
Bekenstein-Sandvik-Barrow-Magueijo to allow for an arbitrary coupling function
and self-interaction potential term in the theory. We study the full evolution
equations without assuming that variations in alpha have a negligible effect on
the expansion scale factor and the matter density evolution, as was assumed in
earlier studies. The background FRW cosmology of this model in the cases of
zero and non-zero spatial curvature is studied in detail, using dynamical
systems techniques, for a wide class of potentials and coupling functions. All
the asymptotic behaviours are found, together with some new solutions. We study
the cases where the electromagnetic parameter, zeta, is positive and negative,
corresponding to magnetic and electrostatic energy domination in the
non-relativistic matter. In particular, we investigate the cases where the
scalar field driving alpha variations has exponential and power-law
self-interaction potentials and the behaviour of theories where the coupling
constant between matter and alpha variations is no longer a constant.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory