ABSTRACT
This chapter considers magnets and their characteristic collective modes: spin waves. It considers the magnetic energy density, which obeys a conservation law and will therefore lead to a hydrodynamic mode. There is a propagating hydrodynamic mode in the system, called a spin wave, with the a bit unusual characteristic that its group velocity increases with decreasing wavelength. Apart from the emergence of spin waves in the transverse variables, the most noteworthy thing that happens in the ordered ferromagnetic state is that the energy density can now couple to the longitudinal component of the magnetization. The chapter deals with a minor subtlety. It considers the former whose fluctuations are by far the more interesting ones. The chapter then provides a straightforward application of the formalism.
