  We present explicitly a relation between the Nambu-Goldstone boson and the
order parameter in non-relativistic systems with spontaneous symmetry breaking.
We show that the Nambu-Goldstone bosons are characterized by transformation
property of the order parameter under symmetry transformation of a system. We
give an explicit formula for the Nambu-Goldstone boson for a general Lie group
$G$, and then the number of the Nambu-Goldstone boson is derived
straightforwardly from the symmetry of the order parameter, i.e. the type of
symmetry breaking. We show that the Ward-Takahashi identity is modified in the
presence of the Nambu-Goldstone boson, where the generalized Ward-Takahashi
identity includes the coupling (the vertex function) between fermions and
Nambu-Goldstone bosons. The closed equation for the Green's functions of
Nambu-Goldstone bosons is derived by introducing the fermion-Nambu-Goldstone
boson vertex function. Examples are given for $G=SU(2)$ (ferromagnetic), $U(1)$
(superconductor) and $SU(3)$ symmetry breaking.
