  By applying a variational principle on a magnetic system within the framework
of extended irreversible thermodynamics, we find that the presence of a
temperature gradient in a ferromagnet leads to a generalisation of the
Landau-Lifshitz equation with an additional magnetic induction field
proportional to the temperature gradient. This field modulates the damping of
the magnetic excitation. It can increase or decrease the damping, depending on
the orientation of the magnetisation wave-vector with respect to the
temperature gradient. This variational approach confirms the existence of the
Magnetic Seebeck effect which was derived from thermodynamics and provides a
quantitative estimate of the strength of this effect.
