  When continuous symmetry is spontaneously broken, there appear
Nambu-Goldstone modes (NGMs) with linear and quadratic dispersion relations,
which are called type-I and type-II, respectively. We propose a framework to
count these modes including the coefficients of the dispersion relations with
applying the standard Gross-Pitaevskii-Bogoliubov theory. Our method is mainly
based on (i) zero-mode solutions of the Bogoliubov equation originated from
spontaneous symmetry breaking and (ii) their generalized orthogonal relations,
which naturally arises from well-known Bogoliubov transformations and is
referred to as "$\sigma$-orthogonality" in this paper. Unlike previous works,
our framework is applicable without any modification to the cases where there
are additional zero modes, which do not have a symmetry origin, such as
quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the
presence of a topological soliton or a vortex. As a by-product of the
formulation, we also give a compact summary for mathematics of bosonic
Bogoliubov equations and Bogoliubov transformations, which becomes a foundation
for any problem of Bogoliubov quasiparticles. The general results are
illustrated by various examples in spinor Bose-Einstein condensates (BECs). In
particular, the result on the spin-3 BECs includes new findings such as a
type-I-type-II transition and an increase of the type-II dispersion coefficient
caused by a linearly-independent pair of zero modes.
