  We present classical and quantum mechanical descriptions of lattice dynamics,
from the atomic to the continuum scale, using atomic scale symmetry modes and
their constraint equations. This approach is demonstrated for a one-dimensional
chain and a two-dimensional square lattice with a monatomic basis. For the
classical description, we find that rigid modes, in addition to the
distortional modes found before, are necessary to describe the kinetic energy.
The long wavelength limit of the kinetic energy terms expressed in terms of
atomic scale modes is shown to be consistent with the continuum theory, and the
leading order corrections are obtained. For the quantum mechanical description,
we find conjugate momenta for the atomic scale symmetry modes. In direct space,
graphical rules for their commutation relations are obtained. Commutation
relations in the reciprocal space are also calculated. As an example, phonon
modes are analyzed in terms of symmetry modes. We emphasize that the approach
based on atomic scale symmetry modes could be useful, for example, for the
description of multiscale lattice dynamics and the dynamics near structural
phase transition.
