Constants of motion are important because they may reduce the number of independent variables in an equation characterizing some dynamic property of particles. In the absence of other force fields, the total energy E0 of a charged particle in a magnetic field is made up solely of the kinetic energy, since the only acting force, i. e. the Lorentz force, does not possess a potential energy; hence

Eo = imvj +|mVj

is a constant so that

where, as before, Vjj and v± refer to velocity components parallel and perpendicular to the applied В-field direction.

Consider now the motion of a charged particle moving in a non-uniform B — field as described in the preceding section. Then, the parallel component of the force on this particle was seen to be given by the grad-B force, Eq. (5.72), as

Evidently, this relation holds only if

dt (5.80)

as the individual isolated charged particle moves in the spatially varying magnetic field. That is, the magnetic moment does not vary with time.

Note that the grad-B force of Eq. (5.72), used here for establishing Eq. (5.80), had been derived by approximating the radial magnetic field component according to Eq. (5.65). Hence, for the case considered, the magnetic moment |_t is conserved only in this approximation. From the definition of (I, Eq. (5.76), it is evident that (I is an invariant if В is constant. For magnetic fields slowly varying in space and/or in time, that is, relative changes in В are very small over a distance equal to the radius of gyration,

fv«i.

and/or within a period of gyration,

the magnetic moment is found to remain invariant in this first-order approximation. Consequently, (I is only approximately conserved and therefore called an adiabatic invariant.

The practical consequence of this is as it relates to the so-called magnetic mirror effect. As В increases toward the "throat" of the mirror region, vx must increase and, in view of Eq.(5.74), vM must decrease; thus, ions tend to decelerate along the axial direction as they move into the higher magnetic field of the mirror throat where vx can even become zero, if В is high enough. Note that Гц, which is given in Eq.(5.72) and which points opposite to VB, is still acting on the particles thus causing a reflection.