Guiding center motion of charged particles in EM field

[Korybut]

TL;DR

Charged particle in EM field

Hello!

I am trying to figure how one can deduce guiding center motion equation according to Hazeltine and Waelbroeck "The Framework of Plasma physics". They suggest the following:
To solve equations
$\frac{d\vec{r}}{dt}=\vec{v},\;\; \frac{d\vec{v}}{dt}=\frac{e}{\epsilon m}(\vec{E}+\vec{v}\times\vec{B})$
Here $\epsilon$ is book-keeping dimensionless variable. The lowest possible order of $\epsilon$ is $\epsilon^{-1}$
they introduce new variables
$\vec{r}(t)=\vec{R}(t)+\epsilon \vec{\rho}(\vec{R},\vec{U},t,\gamma)$
$\vec{v}(t)=\vec{U}(t)+\vec{u}(\vec{R},\vec{U},t,\gamma)$
Here $\gamma$ is new independent variable describing the phase of gyrating particle. Functions $\vec{\rho}$ and $\vec{u}$ has zero mean value, i.e.
$\langle \vec{\rho}\rangle = \int_0^{2\pi} \vec{\rho}(\vec{R},\vec{U},t,\gamma)d\gamma=\langle \vec{u}\rangle= \int_0^{2\pi} \vec{u}(\vec{R},\vec{U},t,\gamma)d\gamma=0$
There is also equation for time evolution of gamma
$\frac{d\gamma}{dt}=\epsilon^{-1}\omega_{-1}(\vec{R},\vec{U},t)+\omega_0(\vec{R},\vec{U},t)+...$
Expansion for other variables looks as follows
$\vec{U}(\vec{R},t)=\vec{U}_0(\vec{R},t)+\epsilon \vec{U}_1(\vec{R},t)+...$
$\vec{\rho}(\vec{R},\vec{U},t,\gamma)=\vec{\rho}_0(\vec{R},\vec{U},t,\gamma)+\epsilon \vec{\rho}_1(\vec{R},\vec{U},t,\gamma)+...$
$\vec{u}(\vec{R},\vec{U},t,\gamma)=\vec{u}_0(\vec{R},\vec{U},t,\gamma)+\epsilon \vec{u}_1(\vec{R},\vec{U},t,\gamma)+...$

Next they plug their new variable in the original equations and taking the average with respect to $\gamma$ obtain the so called solubility conditions. For example equations of motion up to $\epsilon^{-1}$ order looks as
$\omega_{-1} \frac{\partial \vec{u}_0}{\partial \gamma}-\frac{e}{m} \vec{u}_0\times B=\frac{e}{m}(E+U_0\times B)$
taking the $\gamma$ average according to the book one obtains
$E+\vec{U}_0\times B=0$
I don't get how one should rigorously deal with this average for example
$\langle \omega_{-1}\frac{\partial \vec{u}_0}{\partial \gamma} \rangle$
According to the authors it should be zero but I am confused. While differentiating with respect to $t$ I differentiate $\gamma$ however taking avarage $\frac{d\gamma}{dt}$ is supposed to be silent. How this is justified?