  We study impact of inertia on directed transport of a Brownian particle under
non-equilibrium conditions: the particle moves in a one-dimensional periodic
and symmetric potential, is driven by both an unbiased time-periodic force and
a constant force, and is coupled to a thermostat of temperature T. Within
selected parameter regimes this system exhibits negative mobility, which means
that the particle moves in the direction opposite to the direction of the
constant force. It is known that in such a setup the inertial term is essential
for the emergence of negative mobility and it cannot be detected in the
limiting case of overdamped dynamics. We analyse inertial effects and show that
negative mobility can be observed even in the strong damping regime. We
determine the optimal dimensionless mass for the presence of negative mobility
and reveal three mechanisms standing behind this anomaly: deterministic
chaotic, thermal noise induced and deterministic non-chaotic. The last origin
has never been reported. It may provide guidance to the possibility of
observation of negative mobility for strongly damped dynamics which is of
fundamental importance from the point of view of biological systems, all of
which in situ operate in fluctuating environments.
