  Transport of a particle in a spatially periodic harmonic potential under the
influence of a slowly time-dependent unbiased periodic external force is
studied. The equations of motion are the same as in the problem of a slowly
forced nonlinear pendulum. Using methods of the adiabatic perturbation theory
we show that for a periodic external force of general kind the system
demonstrates directed (ratchet) transport in the chaotic domain on very long
time intervals and obtain a formula for the average velocity of this transport.
Two cases are studied: the case of the external force of small amplitude, and
the case of the external force with amplitude of order one. The obtained
formulas can also be used in case of a non-harmonic periodic potential.
