  Rapidly oscillating potentials with a vanishing time average have been used
for a long time to trap charged particles in source-free regions. It has been
argued that the motion of a particle in such a potential can be approximately
described by a time independent effective potential, which does not depend upon
the initial phase of the oscillating potential. However, here we show that the
motion of a particle and its trapping condition significantly depend upon this
initial phase for arbitrarily high frequencies of the potential's oscillation.
We explain this novel phenomenon by showing that the motion of a particle is
determined by the effective potential stated in the literature only if its
initial conditions are transformed according to a transformation which we show
to significantly depend on the potential's initial phase for arbitrarily high
frequencies. We confirm our theoretical findings by numerical simulations.
Further, we demonstrate that the found phenomenon offers new ways to manipulate
the dynamics of particles which are trapped by rapidly oscillating potentials.
Finally, we propose a simple experiment to verify the theoretical findings of
this work.
