  The generalized elastic model encompasses several physical systems such as
polymers, membranes, single file systems, fluctuating surfaces and rough
interfaces. We consider the case of an applied localized potential, namely an
external force acting only on a single (tagged) probe, leaving the rest of the
system unaffected. We derive the fractional Langevin equation for the tagged
probe, as well as for a generic (untagged) probe, where the force is not
directly applied. Within the framework of the fluctuation-dissipation
relations, we discuss the unexpected physical scenarios arising when the force
is constant and time periodic, whether or not the hydrodynamic interactions are
included in the model. For short times, in case of the constant force, we show
that the average drift is linear in time for long range hydrodynamic
interactions and behaves ballistically or exponentially for local hydrodynamic
interactions. Moreover, it can be opposite to the direction of external
disturbance for some values of the model's parameters. When the force is time
periodic, the effects are macroscopic: the system splits into two distinct
spatial regions whose size is proportional to the value of the applied
frequency. These two regions are characterized by different amplitudes and
phase shifts in the response dynamics.
