- #1
Korybut
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- TL;DR Summary
- 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?
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?