3.7 Non-adiabatic motion in symmetric geometry 101defined by Eq.(3.161). From Eq.(3.157) the angular velocity isθ ̇=^1
mr^2(
Pθ−
qψ
2 π)
. (3.162)
The sign ofθ ̇ reverses periodically as the particle bounces back and forth in theχ
potential well. This corresponds to localized gyromotion as shown in Fig.3.15.non -axis-encircling axis-encircling
Figure 3.15: Localized gyro motion associated with particle bouncing in effective potential
well.
- qψ/Pθ isnegative. In this caseχcan never vanish, becausePθ−qψ/ 2 πnever
vanishes. Nevertheless, it is still possible forχ to have a minimum and hence a
potential well. This possibility can be seen by setting∂χ/∂r= 0 which occurs
when (
Pθ−
q
2 π
ψ)∂
∂r(
Pθ−qψ/ 2 π
r)
= 0. (3.163)
Equation (3.163) can be satisfied by having∂
∂r
Pθ−q
2 πψ
r
= 0 (3.164)
which implies
Pθ=−qr^2
2 π∂
∂r(
ψ
r)
. (3.165)
Recall thatψhad a maximum, thatψ ∼r^2 nearr= 0,and also thatψ → 0 as
r→∞.Thusψ/r ∼rfor smallrandψ/r→ 0 forr→∞so thatψ/ralso
has a maximum;this maximum is located at anrsomewhat inside of the maximum
ofψ.Thus Eq.(3.165) can only be valid at points inside of this maximum;other-
wise the assumption of opposite signs forPθandψwould be incorrect. Furthermore