  We discuss the equations of motion of test particles for a version of
Kaluza-Klein theory where the cylinder condition is not imposed. The metric
tensor of the five-dimensional manifold is allowed to depend on the fifth
coordinate. This is the usual working scenario in brane-world, induced-matter
theory and other Kaluza-Klein theories with large extra dimensions. We present
a new version for the fully covariant splitting of the 5D equations. We show
how to change the usual definition of various physical quantities in order to
make physics in 4D invariant under transformations in 5D. These include the
redefinition of the electromagnetic tensor, force and Christoffel symbols. With
our definitions, each of the force terms in the equation of motion is gauge
invariant and orthogonal to the four-velocity of the particle. The "hidden"
parameter associated with the rate of motion along the extra dimension is
identified with the electric charge, regardless of whether there is an
electromagnetic field or not. In addition, for charged particles, the
charge-to-mass ratio should vary. Therefore, the motion of a charged particle
should differ from the motion of a neutral particle, with the same initial mass
and energy, even in the absence of electromagnetic field. These predictions
have important implications and could in principle be experimentally detected.
