  Using a new scheme of the derivation of the non-linear $\sigma$-model we
consider the electron motion in a random magnetic field (RMF) in two
dimensions. The derivation is based on writing quasiclassical equations and
representing their solutions in terms of a functional integral over
supermatrices $Q$ with the constraint $Q^2=1$. Contrary to the standard scheme,
neither singling out slow modes nor saddle-point approximation are used. The
$\sigma$-model obtained is applicable at the length scale down to the electron
wavelength. We show that this model differs from the model with a random
potential (RP).However, after averaging over fluctuations in the Lyapunov
region the standard $\sigma$-model is obtained leading to the conventional
localization behavior.
