  We examine the ADM reformulation of the 5-D KK model: the dimensional
reduction is provided to commute with the ADM splitting and we show how the
time component of the gauge vector is given by combination of the Lagrangian
multipliers for the 5-D gravitational field. We consider 5D particles motion
and after dimensional reduction the definition of charge is recovered within
electrodynamic coupling. A time-varying fine structure constant is recognized
because an extra scalar field is present in the 4-D theory.
