Fundamentals of Plasma Physics

3.4 Extension of WKB method to general adiabatic invariant 71

Let the Hamiltonian depend on time via a slowly changing parameterλ(t),so thatH=

H(P,Q,λ(t)).From Eq.(3.16) the energy is given by

E(t) =H(P,Q,λ(t)) (3.48)

and, in principle, this relation can be inverted to giveP =P(E(t),Q,λ(t)).Suppose a

particle is executing nearly periodic motion in theQ−Pplane. We define the turning

pointQtp as a position wheredQ/dt= 0.SinceQis oscillating there will be a turning

point associated withQhaving its maximum value and a turning point associated withQ

having its minimum value. From now on let us only consider turning points whereQhas

its maximum value, that is we only consider the turning points on the right hand side of the

nearly periodic trajectories in theQ−Pplane shown in Fig.3.2.

Q

P

Qtp
t

Figure 3.2: Nearly periodic phase space trajectory for slowly changing Hamiltonian. The
turningQtp(t)point is whereQis at its maximum.

If the motion is periodic, then the turning point for theN+1thperiod will be the same

as the turning point for theNthperiod, but if the motion is only nearly periodic, there will

be a slight difference as shown in Fig.3.2. This difference can be characterized by making

the turning point a function of time soQtp=Qtp(t).This function is only defined for the

times whendQ/dt= 0.When the motion is not exactly periodic, this turning point is such

thatQtp(t+τ)=Qtp(t)whereτis the time interval required for the particle to go from

the first turning point to the next turning point. The action integral is over one entire period

of oscillation starting from a right hand turning point and then going to the next right hand

turning point (cf. Fig. 3.2) and so can be written as

S =

∮

PdQ

=

∫Qtp(t+τ)

Qtp(t)

PdQ. (3.49)