$\dx  \frac{\d}{\d t} = \frac{\partial }{\partial t} + (\dot{\bf
u}\nabla)$ and all quantities are differentiated with respect to the coordinates and the time at constant [$\kappa$]{}.

The term $m \dx \frac{\d\dot{\bf u}}{\d t}$ takes into account noninertial properties of the local frame. This is the term which is responsible for the Stewart-Tolman effect in metals.

The term $$m\frac{\partial\eps}{\partial \kappa_{\beta}}
{\bf a}_{\beta}\times  \curl \dot{\bf u}$$ is of [*essentially new kind*]{} and cannot be obtained in linear theories. It is proportional to the bare mass $\,m\,$ and, hence, is also responsible for noninertial effects.

The Lorentz force in our notation has the form $${\bf  F}  =  -e{\bf  E}  -  \frac{e}{c}\,
\frac{\partial\eps}{\partial
\kappa_{\alpha}}{\bf a}_{\alpha}\times {\bf B} - \frac{e}{c}
\dot{\bf u} \times {\bf B}$$ where [**E**]{} and [**B**]{} are the strengths of the electric field and magnetic induction, respectively. Substituting (40) into (38) yields the kinetic equation for charged quasiparticles with a charge $-e$ ( $e>0$): $$\frac{\d f}{\d t}  +  {\bf a}_{\alpha}\left(\frac{\partial    f}{\partial
{\bf r}}\right)_{\mbox {\boldmath $\kappa$}}
\frac{\partial\eps}{\partial
\kappa_{\alpha}} -
{\bf a}_\alpha
\frac{\partial f}{\partial \kappa_{\alpha}}\left\{ \left(
\frac{\partial\eps}{\partial{\bf r}}\right)_{\mbox {\boldmath $\kappa$}} +
e\tilde {\bf E}+ \frac{e}{c}\, \frac{\partial\eps}{\partial
\kappa_{\beta}}
 {\bf a}_{\beta}\times \tilde {\bf B} \right\} = \hat{I}f ,$$ where $$e \tilde{ \bf E} = e{\bf E} 