  We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if
the soliton is orbitally stable, then it is also asymptotically stable. The
main assumptions are transversal nondegeneracy of the manifold of the ground
states, linear dispersion (in the form of Strichartz estimates) and nonlinear
Fermi Golden Rule. We allow the linearization of the equation at the soliton to
have an arbitrary number of eigenvalues. The theory is tailor made for the
application to the translational invariant NLS in space dimension 3. The proof
is based on the extension of some tools of the theory of Hamiltonian systems
(reduction theory, Darboux theorem, normal form) to the case of systems
invariant under a symmetry group with unbounded generators.
