  Recent astrophysical observations of quasar absorption systems indicate that
the fine structure constant $\alpha$ and the proton-electron mass ratio $\mu$
may have evolved through the history of the universe. Motivated by these
observations, we consider the cosmological evolution of a quintessence-like
scalar field $\phi$ coupled to gauge fields and matter which leads to effective
modifications of the coupling constants and particle masses over time. We show
that a class of models where the scalar field potential $V(\phi)$ and the
couplings to matter $B(\phi)$ admit common extremum in $\phi$ naturally
explains constraints on variations of both the fine structure constant and the
proton-electron mass ratio.
