  A collective-variable approach for the study of non-linear dynamics of
magnetic textures in planar nano-magnets is proposed. The variables are just
arbitrary parameters (complex or real) in the specified analytical function of
a complex variable, describing the texture in motion. Starting with such a
function, a formal procedure is outlined, allowing a (non-linear) system of
differential equations of motion to be obtained for the variables. The
resulting equations are equivalent to Landau-Lifshitz-Gilbert dynamics as far
as the definition of collective variables allows it. Apart from the
collective-variable specification, the procedure does not involve any
additional assumptions (such as translational invariance or steady-state
motion). As an example, the equations of weakly non-linear motion of a magnetic
vortex are derived and solved analytically. A simple formula for the dependence
of the vortex precession frequency on its amplitude is derived. The results are
verified against special cases from the literature and agree quantitatively
with experiments and simulations.
