Fundamentals of Plasma Physics

68 Chapter 3. Motion of a single plasma particle

so Eq.(3.31) becomes

S =

∫t 0 +τ

t 0

ω(t′)^2

{

x(t 0 )

√

ω(t 0 )

ω(t′)

cos

(∫

t′

t 0 ω(t

′′

)dt

′′)

} 2

dt′

= [x(t 0 )]^2 ω(t 0 )

∫t 0 +τ

t 0

ω(t′)cos^2 (

∫t′

t 0 ω(t

′′

)dt

′′

)dt′

= [x(t 0 )]^2 ω(t 0 )

∫ 2 π

0

dξcos^2 ξ=π[x(t 0 )]^2 ω(t 0 ) =const.

(3.33)

whereξ=

∫t′

t 0 ω(t

′′
)dt
′′
anddξ=ω(t′)dt′.Equation (3.29) shows thatSis the area in
phase-space enclosed by the trajectory{x(t),v(t)}and Eq.(3.33) shows that for a slowly
changing pendulum frequency,this area is a constant of the motion. Since the average
energy of the pendulum scales as∼[ω(t)x(t)]^2 ,we see from Eq.(3.24) that the ratio

energy

frequency

∼ω(t)x^2 (t)∼S∼const. (3.34)

The ratio in Eq.(3.34) is the classical equivalent of the quantum numberNof a simple
harmonic oscillator because in quantum mechanics the energyEof a simple harmonic
oscillator is related to the frequency by the relationE/hω=N+ 1/ 2.
This analysis clearly applies toanydynamical system having an equation of motion of
the form of Eq.(3.17). Hence, if the dynamics of plasma particles happens to be of this
form, thenScan be added to our repertoire of constants of the motion.

3.4 Extension of WKB method to general adiabatic invariant

Action has the dimensions of (canonical momentum)×(canonical coordinate) so we may
anticipate that for general Hamiltonian systems, the action integralgiven in Eq.(3.29) is not
an invariant becausevisnot, in general, proportional toP. We postulate that the general
form for the action integral is

S=

∮

PdQ (3.35)

where the integral is over one period of the periodic motion andP,Qare the relevant
canonical momentum-coordinate conjugate pair. The proof of adiabatic invariance used for
Eq.(3.29) does not work directly for Eq.(3.35);we now present a slightly more involved
proof to show that Eq.(3.35) is indeed the more general form of adiabatic invariant.
Let us define the radius vector in theQ−P plane to beR= (Q,P)and define unit
vectors in theQandP directions byQˆandPˆ;these definitions are shown in Fig.3.1. Fur-
thermore we define thezdirection as being normal to theQ−Pplane;thus the unit vector
ˆzis ‘out of the paper’, i.e.,ˆz=Qˆ×P.ˆ Hamilton’s equations [i.e.,P ̇=−∂H/∂Q, Q ̇=
∂H/∂P] may be written in vector form as

dR

dt

=−zˆ×∇H (3.36)