  Using the in-in formalism, we generalize the recently constructed
magnetoelastic EFT arXiv:2112.13873 [hep-th] to describe the damping dynamics
of ferromagnetic systems at long wavelengths. We find that the standard Gilbert
damping term naturally arises as the simplest leading-order symmetry-consistent
non-conservative contribution within the in-in framework. The EFT is easily
generalized to scenarios with anisotropy and inhomogeneity. In particular, we
find the classic Landau-Lifshitz damping term emerges when isotropy is broken
by a constant external background field. This provides a first principle
explanation for distinguishing the two types of damping dynamics that were
originally constructed phenomenologically. Furthermore, the EFT framework could
also incorporate intrinsic anisotropy of the material in a straightforward way
using the spurion method. For systems with inhomogeneity such as nontrivial
spin textures, we find that the leading order derivative correction yields the
generalized Gilbert damping equations that were found in condensed matter
literature. This shows that the EFT approach enables us to derive the form of
higher-derivative-order corrections in a systematic way. Lastly, using the
phonon-magnon coupling deduced in the magnetoelastic EFT, we are able to make a
prediction for the generic form of the phononic contribution to the damping
equation.
