  This work is mainly based on postgraduate lectures at Tongji University since
2020 spring. We firstly revisit the elastic spin and orbital angular momentum
[Proc. Natl. Acad. Sci. USA 115, 9951 (2018)] but more general for anisotropic
systems by applying Noether's theorem to the elastic Lagrangian and by applying
the symmetry argument in the field theory. Then, fundamental relations between
elastic energy flux and elastic spin are uncovered. In particular cases, the
wave spin is closely related to the vorticity of energy flux and momentum.
Secondly, we move forward from the elastic spin to revisit the phonon spin
[Fizika Tverdogo Tela 3, 2160 (1961)] by applying the second quantization to
elastic fields. We show that the uncovered phonon spin, a polarized
elastic-vibration quanta, is generally not restricted to transverse phonon
modes, but applying to general phonon modes, such as the longitudinal phonon
modes, surface phonon modes, and hybridized phonon modes, regarded as a
consequence of mode interferences. The elastic spin and phonon spin originate
from the local rotating of the field polarization in time domain, not the local
circulation (vorticity) of displacement or velocity in space domain. It is
hopeful that the present results could advance the fundamental understanding of
phonon spin and elastic spin, and promote the spin phononics for hybrid quantum
sensing and technology with multiple degrees of freedom.
