  Starting from a travelling wave ansatz we show successively that the shape of
a nonlinear excitation generally depends also on the 1st, 2nd, ... time
derivative of the position X of the excitation. From the Hamilton equations we
derive a hierarchy of equations of motion for X. The type of the excitation
determines on which levels the hierarchy can be truncated consistently:
"Gyrotropic" excitations are governed by odd-order equations, non-gyrotropic
ones by even-order equations. Examples for the latter case are kinks in
1-dimensional models and planar vortices of the 2D anisotropic (easy-plane)
Heisenberg model. The non-planar vortices of this model are the simplest
gyrotropic example. For this case we solve the Hamilton equations for a finite
system with one vortex and free boundary conditions and calculate the
parameters of the 3rd-order equation of motion. This equation yields
trajectories which are a superposition of two cycloids with different
frequencies, which is in full agreement with computer simulations of the full
many-spin model. Finally we demonstrate that the additional effects from the
5th-order equation are negligible.
