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For periodic motions the theory of mechanics shows that quantities called actions can remain invariant
for slow changes in the system. An action J can be defined in terms of generalized coordinates
qgen and conjugate momenta pgen by
![\begin{displaymath}J = \oint p_{gen} d q_{gen} \ ,
\end{displaymath}](img81.gif) |
(2.31) |
where the integral is over one period of the motion. A particle's gyromotion is one example of a
periodic motion amenable to the construction of an approximate constant of the motion (or invariant).
Defining qgen to be the gyrophase
,
then the angular momentum
is the conjugate momentum. Inserting these variables into (2.31) and integrating one finds
![\begin{displaymath}J = 2\pi \frac{m}{q} \frac{v_{\perp}^{2}}{B} = \frac{4\pi m}{q} \mu
\end{displaymath}](img84.gif) |
(2.32) |
for slowly varying B. The quantity
![\begin{displaymath}\mu = \frac{v_{\perp}^{2}}{2 B}
\end{displaymath}](img85.gif) |
(2.33) |
is known as the first adiabatic invariant of a plasma particle. This implies that a
particle's perpendicular energy
is proportional to
B if
is constant.
Another, perhaps more obvious derivation of the first adiabatic invariant
is as follows. Assume that the particle sees a small change in
during a
gyroperiod, whether due to temporal or spatial variations in
.
I.E.,
![\begin{displaymath}\frac{1}{\Omega_{c}} \vert \frac{\partial B}{\partial t} B^{-1} \vert \ll 1 \ .
\end{displaymath}](img88.gif) |
(2.34) |
The change in
in one gyroperiod is
Assuming the orbit size changes very little in one gyroperiod then
![\begin{displaymath}\Delta W_{\perp} \approx q \pi r_{L}^{2} \frac{1}{c} \frac{\partial B}{\partial t} .
\end{displaymath}](img93.gif) |
(2.37) |
Since the change in B in one gyroperiod is
![\begin{displaymath}\Delta B = \frac{2\pi}{\Omega_{c}} \frac{\partial B}{\partial t}
\end{displaymath}](img94.gif) |
(2.38) |
then
or
![\begin{displaymath}\Delta \left( \frac{W}{B} \right) = 0 = \Delta( \mu )\ .
\end{displaymath}](img96.gif) |
(2.39) |
That is,
is a constant.
Other adiabatic invariants also exist. The second or longitudinal
adiabatic invariant is associated with the periodic bouncing of particles in
magnetic flux tubes and magnetic bottles:
![\begin{displaymath}J_{L} = m \oint v_{\parallel} d z
\end{displaymath}](img97.gif) |
(2.40) |
A third adiabatic invariant can be associated with the periodic drift of a particle
(due to
and curvature drifts) around a dipole magnetic field. It is
useful for studying particle motions in Earth's magnetosphere but is not
addressed further here.
Next: Magnetic Mirrors: the effects
Up: Single Particle Motions: ``Orbit
Previous: Applications
Iver Cairns
1999-08-04