  We discuss the modelling of grain boundary dynamics within an amplitude
equations description, which is derived from classical density functional
theory or the phase field crystal model. The relation between the conditions
for periodicity of the system and coincidence site lattices at grain boundaries
is investigated. Within the amplitude equations framework we recover
predictions of the geometrical model by Cahn and Taylor for coupled grain
boundary motion, and find both $\langle100\rangle$ and $\langle110\rangle$
coupling. No spontaneous transition between these modes occurs due to
restrictions related to the rotational invariance of the amplitude equations.
Grain rotation due to coupled motion is also in agreement with theoretical
predictions. Whereas linear elasticity is correctly captured by the amplitude
equations model, open questions remain for the case of nonlinear deformations.
