  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.
