3.4 Extension of WKB method to general adiabatic invariant 69where
∇=Qˆ∂
∂Q
+Pˆ
∂
∂P
(3.37)
is the gradient operator in theQ−Pplane. Equation (3.36) shows that the phase-space
‘velocity’dR/dtisorthogonalto∇H.Hence,Rstays on a level contour ofH.IfHis
constant, then, in order for the motion to be periodic, the path along this level contour must
circle around and join itself, like a road ofconstant elevationaround the rim of a mountain
(or a crater). IfHis not constant, but slowly changing in time, the contour will circle
around andnearlyjoin itself.
P
Q
RQ,P
constant H
contourP̂
Q̂
Figure 3.1: Q-P planeEquation (3.36) can be inverted by crossing it withzˆto give∇H= ˆz×dR
dt. (3.38)
For periodic and near-periodic motions,dR/dtis always in the same sense (always clock-
wise or always counterclockwise). Thus, Eq. (3.38) shows that an “observer” following
the path would always see thatHis increasing on the left hand side of the path and de-
creasing on the right hand side (or vice versa). For clarity, the origin oftheQ−P plane
is re-defined to be at a local maximum or minimum ofH. Hence, near the extremumH
must have the Taylor expansion
H(P,Q) =Hextremum+P^2
2
[
∂^2 H
∂P^2
]
P=0,Q=0+
Q^2
2
[
∂^2 H
∂Q^2
]
P=0,Q=0(3.39)
where
[
∂^2 H/∂P^2
]
P=0,Q=0and[
∂^2 H/∂Q^2
]
P=0,Q=0are either both positive (valley) or
both negative (hill). SinceHis assumed to have a slow dependence on time, these second
derivatives will be time-dependent so that Eq.(3.39) has the form
H=α(t)P^2
2
+β(t)