96 Chapter 3. Motion of a single plasma particle
Lagrange’s equationP ̇j=−∂L/∂Qjhas no limitations on the rate at which changes can
occur. In effect, being geometrically symmetric trumps being non-analytic. The absolute
invariance ofPjwhen∂L/∂Qj= 0reduces the number of equations and allows a partial
or sometimes even a complete solution of the motion. Solutions to symmetricproblems
give valuable insight regarding the more general situation of being both non-adiabatic and
asymmetric.
Two closed related examples of non-adiabatic particle motion will now be analyzed: (i)
sudden temporal and (ii) sudden spatial reversal of the polarity of an azimuthally symmetric
magnetic field having no azimuthal component. The most general form of such a fieldcan
be written in cylindrical coordinates(r,θ,z)as
B=
1
2 π
∇ψ(r,z,t)×∇θ; (3.146)a field of this form is called poloidal. Rather than usingˆθexplicitly, the form∇θhas been
used because∇θis better suited for use with the various identities of vector calculus (e.g.,
∇×∇θ= 0) and leads to greater algebraic clarity. The relationship between∇θandˆθis
seen by simply taking the gradient:
∇θ=(
ˆr∂
∂r+
ˆθ
r∂
∂θ+ ˆz∂
∂z)
θ=ˆθ
r. (3.147)
Equation(3.146) automatically satisfies∇·B= 0[by virtue of the vector identity
∇·(G×H) =H·∇×G−G·∇×H],has noθcomponent, and is otherwise arbitrary
sinceψis arbitrary. As shown in Fig.3.12, the magneticflux linking a circle of radiusr
with center at axial positionzis
∫
B·ds =∫r02 πrdrzˆ·[
1
2 π∇ψ(r,z,t)×∇θ]
=
∫r0dr∂ψ(r,z,t)
∂r
=ψ(r,z,t)−ψ(0,z,t).(3.148)
However,
Br(r,z,t) =−1
2 πr∂ψ
∂z(3.149)
and since∇·B=0, Brmust vanish atr= 0,and so∂ψ/∂z= 0on the symmetry axis
r= 0.