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Let us now demonstrate that the magnetic moment
is indeed a constant of the motion, at least to lowest order. The scalar
product of the equation of motion (59) with the velocity yields

(95) 
This equation governs the evolution of the particle energy during its
motion. Let us make the substitution
,
as before, and then average the above equation over gyrophase. To lowest order, we obtain

(96) 
Here, use has been made of the result

(97) 
which is valid for any . The final term on the righthand side of
Eq. (96) can be written

(98) 
Thus, Eq. (96) reduces to

(99) 
Here, is the guiding centre velocity, evaluated to first order, and

(100) 
is the kinetic energy of the particle. Evidently, the kinetic energy can
change in one of two ways. Either by motion of the guiding centre along the
direction of the electric field, or by the acceleration of the gyration due
to the electromotive force generated around the Larmor orbit by a
changing magnetic field.
Equations (70), (85), and (86) can be used to eliminate
and from Eq. (99). The final result is

(101) 
Thus, the magnetic moment is a constant of the motion to lowest order.
Kruskal^{} has shown that
is the lowest order
approximation to a quantity which is a constant of the motion to
all orders in the perturbation expansion. Such a quantity
is called an adiabatic invariant.
Next: Poincaré Invariants
Up: Charged Particle Motion
Previous: Magnetic Drifts
Richard Fitzpatrick
20110331