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Poincaré invariants are generally of little practical interest
unless the curve closely corresponds to the trajectories of
actual particles. Now, for the motion of magnetized particles
it is clear from Eqs. (60) and (73) that points having the same
guiding centre at a certain time will continue to have approximately
the same guiding centre at a later time. An approximate Poincaré
invariant may thus be obtained by choosing the curve to be a circle of points
corresponding to a gyrophase period. In other words,

(106) 
Here, is an adiabatic invariant.
To evaluate for a magnetized plasma recall that the canonical momentum
for charged particles is

(107) 
where is the vector potential. We express in terms
of its Taylor series about the guiding centre position:

(108) 
The element of length along the curve is [see Eq. (74)]

(109) 
The adiabatic invariant is thus

(110) 
which reduces to

(111) 
The final term on the righthand side is written [see Eq. (76)]

(112) 
It follows that

(113) 
Thus, to lowest order the adiabatic invariant is proportional to the magnetic moment .
Next: Magnetic Mirrors
Up: Charged Particle Motion
Previous: Poincaré Invariants
Richard Fitzpatrick
20110331