I Guiding center motion of charged particles in EM field

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The discussion centers on deriving the guiding center motion equation for charged particles in an electromagnetic field, referencing Hazeltine and Waelbroeck's work. The equations of motion involve new variables that account for the particle's gyration, with specific expansions for velocity and position. The authors introduce averaging techniques over the gyration phase, leading to solubility conditions that simplify the equations. A key point of confusion arises regarding the treatment of averages, particularly how certain terms are considered zero when averaged over the gyration phase. Clarification is sought on the justification for differentiating with respect to time while maintaining the average of the gyration phase as silent.
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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?
 
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