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Approximate equations of motion of a charged particle in electric and magnetic fields, obtained by averaging the rapid motion of the particle under the effect of the magnetic field. Drift equations apply if the magnetic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340301.png" /> is changing slowly in space and in time, while the electric field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340302.png" /> is small as compared to the magnetic field:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340304.png" /> is a small parameter, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340305.png" /> is the Larmor frequency, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340306.png" /> is the [[Larmor radius|Larmor radius]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340307.png" /> is the velocity of the particle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340308.png" /> is the velocity component in the direction normal to the magnetic field. Drift equations are obtained from the complete equations of motion by expanding in powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d0340309.png" /> with the aid of the averaging method [[#References|[1]]]. They have the following form:
+
Approximate equations of motion of a charged particle in electric and magnetic fields, obtained by averaging the rapid motion of the particle under the effect of the magnetic field. Drift equations apply if the magnetic field  $  \vec{B} $
 +
is changing slowly in space and in time, while the electric field  $  \vec{E} $
 +
is small as compared to the magnetic field:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
\frac{1}{\omega _ {B} B }
 +
 +
\frac{\partial  R }{\partial  t }
 +
  \sim  \epsilon ,\ \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\frac{\rho _ {B} }{B}
 +
| \nabla B |  \sim  \epsilon ,\ 
 +
\frac{c E }{v B
 +
}
 +
  \sim  \epsilon .
 +
$$
 +
 
 +
Here  $  \epsilon $
 +
is a small parameter,  $  \omega _ {B} = eB / mc $
 +
is the Larmor frequency,  $  \rho _ {B} = v _  \perp  / | \omega _ {B} | $
 +
is the [[Larmor radius|Larmor radius]],  $  v $
 +
is the velocity of the particle, and  $  v _  \perp  $
 +
is the velocity component in the direction normal to the magnetic field. Drift equations are obtained from the complete equations of motion by expanding in powers of  $  \epsilon $
 +
with the aid of the averaging method [[#References|[1]]]. They have the following form:
 +
 
 +
$$ \tag{2 }
 +
 
 +
\frac{d \vec{R} }{dt}
 +
  = V _ {\| } 
 +
\frac{\vec{B} }{B}
 +
+ \vec{V} _ {\textrm{ dr } }  ,
 +
$$
 +
 
 +
$$ \tag{3 }
 +
 
 +
\frac{d}{dt}
 +
\left (
 +
\frac{1}{2}
 +
m ( V _  \perp  ^ {2} + V _ {\| }  ^ {2} )
 +
\right )  = e \vec{E}
 +
\frac{d \vec{R} }{dt}
 +
-  
 +
\frac{m c V _  \perp  ^ {2} }{2 B  ^ {2} }
 +
\vec{B}  \mathop{\rm rot}  \vec{E} ,
 +
$$
 +
 
 +
$$ \tag{4 }
 +
 
 +
\frac{d}{dt}
 +
\left (
 +
\frac{V _  \perp  ^ {2} }{B}
 +
\right ) =  0 ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403013.png" /></td> </tr></table>
+
$$
 +
\vec{V} _ { \mathop{\rm dr}  }  =
 +
\frac{c}{B}
 +
  ^ {2} [ \vec{E} \times \vec{B} ] +
 +
\frac{m c
 +
V _ {\| }  ^ {2} }{c B  ^ {4} }
 +
[ \vec{B} \times ( \vec{B} \nabla ) \vec{B}
 +
] +
 +
\frac{m c V _  \perp  ^ {2} }{2 e B  ^ {3} }
 +
[ \vec{B} \times \nabla B
 +
] .
 +
$$
 +
 
 +
The system (2)–(4), known as the drift system, is written with respect to the auxiliary averaged variables  $  \vec{R} $,
 +
$  V _  \perp  $,
 +
$  V _ {\| }  $,
 +
connected by a certain relation with the initial variables  $  \vec{r} $,
 +
$  \vec{v} $.  
 +
The drift rate  $  \vec{V} _ { \mathop{\rm dr}  } $
 +
in equation (2) describes a slow motion along the averaged trajectory in the direction perpendicular to the magnetic field:
  
The system (2)–(4), known as the drift system, is written with respect to the auxiliary averaged variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403016.png" />, connected by a certain relation with the initial variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403018.png" />. The drift rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403019.png" /> in equation (2) describes a slow motion along the averaged trajectory in the direction perpendicular to the magnetic field:
+
$$
 +
V _ { \mathop{\rm dr}  }  \sim  \epsilon v ,\  \vec{V} _ { \mathop{\rm dr}  } \vec{B}  = 0 .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403020.png" /></td> </tr></table>
+
The equations (3) and (4) have second-order accuracy with respect to  $  \epsilon $
 +
and define the magnitudes  $  V _  \perp  $
 +
and  $  V _ {\| }  $
 +
up to first order terms in the time interval  $  t $
 +
containing many Larmor periods  $  t \sim 1 / \epsilon | \omega _ {B} | $.  
 +
Equation (2) has first-order accuracy with respect to  $  \epsilon $.
  
The equations (3) and (4) have second-order accuracy with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403021.png" /> and define the magnitudes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403023.png" /> up to first order terms in the time interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403024.png" /> containing many Larmor periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403025.png" />. Equation (2) has first-order accuracy with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403026.png" />.
+
The magnitude  $  \mu = V _  \perp  ^ {2} / B $,
 +
which is the integral of the drift system (2)(4), is an approximate integral of the true motion. It is known as the [[Adiabatic invariant|adiabatic invariant]]. In the static case, when  $  \mathop{\rm rot}  \vec{E} = 0 $
 +
and $  \vec{E} = - \nabla \phi $,
 +
equation (3) admits the energy integral
  
The magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403027.png" />, which is the integral of the drift system (2)–(4), is an approximate integral of the true motion. It is known as the [[Adiabatic invariant|adiabatic invariant]]. In the static case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403029.png" />, equation (3) admits the energy integral
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034030/d03403030.png" /></td> </tr></table>
+
\frac{1}{2}
 +
m ( V _  \perp  ^ {2} + V _ {\| }  ^ {2} ) + e \phi  = \
 +
\textrm{ const }
 +
$$
  
 
for the averaged motion.
 
for the averaged motion.

Latest revision as of 19:36, 5 June 2020


Approximate equations of motion of a charged particle in electric and magnetic fields, obtained by averaging the rapid motion of the particle under the effect of the magnetic field. Drift equations apply if the magnetic field $ \vec{B} $ is changing slowly in space and in time, while the electric field $ \vec{E} $ is small as compared to the magnetic field:

$$ \tag{1 } \frac{1}{\omega _ {B} B } \frac{\partial R }{\partial t } \sim \epsilon ,\ \ \frac{\rho _ {B} }{B} | \nabla B | \sim \epsilon ,\ \frac{c E }{v B } \sim \epsilon . $$

Here $ \epsilon $ is a small parameter, $ \omega _ {B} = eB / mc $ is the Larmor frequency, $ \rho _ {B} = v _ \perp / | \omega _ {B} | $ is the Larmor radius, $ v $ is the velocity of the particle, and $ v _ \perp $ is the velocity component in the direction normal to the magnetic field. Drift equations are obtained from the complete equations of motion by expanding in powers of $ \epsilon $ with the aid of the averaging method [1]. They have the following form:

$$ \tag{2 } \frac{d \vec{R} }{dt} = V _ {\| } \frac{\vec{B} }{B} + \vec{V} _ {\textrm{ dr } } , $$

$$ \tag{3 } \frac{d}{dt} \left ( \frac{1}{2} m ( V _ \perp ^ {2} + V _ {\| } ^ {2} ) \right ) = e \vec{E} \frac{d \vec{R} }{dt} - \frac{m c V _ \perp ^ {2} }{2 B ^ {2} } \vec{B} \mathop{\rm rot} \vec{E} , $$

$$ \tag{4 } \frac{d}{dt} \left ( \frac{V _ \perp ^ {2} }{B} \right ) = 0 , $$

where

$$ \vec{V} _ { \mathop{\rm dr} } = \frac{c}{B} ^ {2} [ \vec{E} \times \vec{B} ] + \frac{m c V _ {\| } ^ {2} }{c B ^ {4} } [ \vec{B} \times ( \vec{B} \nabla ) \vec{B} ] + \frac{m c V _ \perp ^ {2} }{2 e B ^ {3} } [ \vec{B} \times \nabla B ] . $$

The system (2)–(4), known as the drift system, is written with respect to the auxiliary averaged variables $ \vec{R} $, $ V _ \perp $, $ V _ {\| } $, connected by a certain relation with the initial variables $ \vec{r} $, $ \vec{v} $. The drift rate $ \vec{V} _ { \mathop{\rm dr} } $ in equation (2) describes a slow motion along the averaged trajectory in the direction perpendicular to the magnetic field:

$$ V _ { \mathop{\rm dr} } \sim \epsilon v ,\ \vec{V} _ { \mathop{\rm dr} } \vec{B} = 0 . $$

The equations (3) and (4) have second-order accuracy with respect to $ \epsilon $ and define the magnitudes $ V _ \perp $ and $ V _ {\| } $ up to first order terms in the time interval $ t $ containing many Larmor periods $ t \sim 1 / \epsilon | \omega _ {B} | $. Equation (2) has first-order accuracy with respect to $ \epsilon $.

The magnitude $ \mu = V _ \perp ^ {2} / B $, which is the integral of the drift system (2)–(4), is an approximate integral of the true motion. It is known as the adiabatic invariant. In the static case, when $ \mathop{\rm rot} \vec{E} = 0 $ and $ \vec{E} = - \nabla \phi $, equation (3) admits the energy integral

$$ \frac{1}{2} m ( V _ \perp ^ {2} + V _ {\| } ^ {2} ) + e \phi = \ \textrm{ const } $$

for the averaged motion.

The drift system may be generalized to include the relativistic case [2], [3].

References

[1] N.N. Bogolyubov, Yu.A. Mitropol'skii, "Asymptotic methods in the theory of non-linear oscillations" , Hindushtan Publ. Comp. , Delhi (1961) (Translated from Russian)
[2] D.V. Sivukhin, , Problems in the theory of plasma , 1 , Moscow (1963) pp. 7–97 (In Russian)
[3] A.I. Morozov, L.S. Solov'ev, , Problems in the theory of plasma , 2 , Moscow (1963) pp. 177–261 (In Russian)
How to Cite This Entry:
Drift equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Drift_equations&oldid=46776
This article was adapted from an original article by D.P. Kostomarov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article