Landau-Lifshitz magnetodynamics as a Hamilton model: Magnons in an instanton background

To take full advantage of the well-developed field-theoretic methods, Magnonics needs a yet-existing Lagrangian formulation. Here, we show that Landau-Lifshitz magnetodynamics is a member of the covariant-Schrödinger- equation family of Hamilton models and apply the covariant background method arriving at the Ginzburg-Landau Lagrangian formalism for magnons in an instanton background. Magnons appear to be nonrelativistic spinless bosons, which feel instantons as a gauge field and as a Bose condensate. Among the examples of the usefulness of the proposition is the recognition of the instanton-induced phase shifts in magnons as the Berry phase and the interpretation of the spin-transfer-torque generation as a ferromagnetic counterpart of the Josephson supercurrent.