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.