Lagrangian approach and dissipative magnetic systems

A Lagrangian is introduced which includes the coupling between magnetic moments m and the degrees of freedom σ of a reservoir. In case the system–reservoir coupling breaks the time reversal symmetry the magnetic moments perform a damped precession around an effective field which is self-organized by the mutual interaction of the moments. The resulting evolution equation has the form of the Landau–Lifshitz–Gilbert equation. In case the bath variables are constant vector fields the moments m fulfill the reversible Landau–Lifshitz equation. Applying Noetherʼs theorem we find conserved quantities under rotation in space and within the configuration space of the moments.

► We propose a new approach for describing magnetic systems with dissipation on a mesoscopic scale.
► The Lagrangian consists of an active magnetic system and a bath.
► The coupling between both subsystems breaks the time reversal symmetry.
► The suggested Lagrangian leads to the Landau–Lifshitz equation with damping.
► We consider symmetry operations by means of Noetherʼs theorem.