  In this paper we study dynamics of solitons in the generalized nonlinear
Schr\"odinger equation (NLS) with an external potential in all dimensions
except for 2. For a certain class of nonlinearities such an equation has
solutions which are periodic in time and exponentially decaying in space,
centered near different critical points of the potential. We call those
solutions which are centered near the minima of the potential and which
minimize energy restricted to $\mathcal{L}^2-$unit sphere, trapped solitons or
just solitons.
  In this paper we prove, under certain conditions on the potentials and
initial conditions, that trapped solitons are asymptotically stable. Moreover,
if an initial condition is close to a trapped soliton then the solution looks
like a moving soliton relaxing to its equilibrium position. The dynamical law
of motion of the soliton (i.e. effective equations of motion for the soliton's
center and momentum) is close to Newton's equation but with a dissipative term
due to radiation of the energy to infinity.
