apply the transformation by the substitutional function (3).
The result is:
\begin{equation}
\tilde H({\bf p, r}, t) = \eps_0({\bf k}) + \lambda_{ik}({\bf k})u_{ik}
+\left({\bf p} -m \f{\p \eps}{\p {\bf k}}\right)\dot {\bf u}.
\ee
\par
This procedure has been used in the most of
recent works. However it cannot be well-grounded due to its internal
inconsistency. In fact, if we consider the expression (5)
 as energy, and take a constant velocity ${\bf {\dot u}}$
then we come to the wrong conclusion that the energy in an inertial frame
could depend on the frame velocity. The same confusion follows
for the energy density which  must strictly obey
 the Galilean principle.
Note, that we may not consider (3) as a Hamiltonian, because the Hamilton
  function in $C$-system coincides with the dispersion law in accordance with
the Hamilton  equation
$ \dx
\dot {\bf r'} = {\bf v} = \frac{\p \eps}{\p {\bf k}}
$.
\par
These shortcomings have been removed in our previous works \cite{Push,AP,DSc}
 (see also
\cite{Singapore,Nauka})  where
transformations to replace the Galilean ones have been deduced and the
quasiparticle mechanics equations in Hamiltonian form have been presented.
This enabled us to derive  a Boltzmann-like kinetic equation valid in the whole
Brillouin zone. We 