Vector Hamiltonian Formalism for Nonlinear Magnetization Dynamics

Vector Hamiltonian formalism (VHF) for the description of a weakly nonlinear magnetization dynamics has been developed. Transformation from the traditional Landau-Lifshitz equation, describing dynamics of a magnetization vector $\vec{m}(\vec{r}, t)$ on a sphere, to a vector Hamiltonian equation, describing dynamics of a \emph{spin excitation vector} $\vec{s}(\vec{r}, t)$ on a plane, is done using the azimuthal Lambert transformation that preserves both the phase-space area and vector structure of dynamical equations, and guarantees that the plane containing vector $\vec{s}(\vec{r}, t)$ is at each value of the coordinate $\vec{r}$ perpendicular to the a stationary vector $\vec{m}_0(\vec{r})$ describing the magnetization ground state of the system. By expanding vector $\vec{s}(\vec{r}, t)$ in a complete set of linear magnetic vector eigemodes $\vec{s}_\nu(\vec{r})$ of the studied system, and using a weakly nonlinear approximation $|\vec{s}(\vec{r}, t)| \ll 1$, it is possible to express the Hamiltonian function of the system in the form of integrals over the vector eigenmode profiles $\vec{s}_\nu(\vec{r})$, and calculate all the coefficients of this Hamiltonian. The developed approach allows one to describe weakly nonlinear dynamics in micro- and nano-scale magnetic systems with complicated geometries and spatially non-uniform ground states by numerically calculating linear spectrum and eigenmode profiles, and semi-analytically evaluating amplitudes of multi-mode nonlinear interactions. Examples of applications of the developed formalism to the magnetic systems having spatially nonuniform ground state of magnetization are presented.