  We investigate classical strings defined by the Nambu-Goto action with the
boundary term added. We demonstrate that the latter term has a significant
bearing on the string dynamics. It is confirmed that new action terms that
depend on higher order derivatives of string coordinates cannot be considered
as continuous perturbations from the starting string functional. In the case
the boundary term reduces to the Gauss-Bonnet term, a stability analysis is
performed on the rotating rigid string solution. We determined the most generic
solution that the fluctuations grow to. Longitudinal string excitations are
found. The Regge trajectories are nonlinear.
