Fundamentals of Plasma Physics

70 Chapter 3. Motion of a single plasma particle

whereα(t)andβ(t)have the same sign. The termHextremumin Eq.(3.39) has been
dropped because it is just an additive constant to the energy and does not affect Hamilton’s
equations. From Eq. (3.36) the direction of rotation ofRis seen to be counterclockwise if
the extremum ofHis a hill, and clockwise if a valley.
Hamilton’s equations operating on Eq.(3.40) give

dP

dt

=−βQ,

dQ

dt

=αP. (3.41)

These equations do not directly generate the simple harmonic oscillator equation because
of the time dependence ofα,β.However, if we define the auxiliary variable

τ=

∫t

β(t′)dt′ (3.42)

then
d
dt

=

dτ

dt

d

dτ

=β

d
dτ
so Eq.(3.41) becomes
dP
dτ

=−Q,

dQ

dτ

=

α

β

P. (3.43)

Taking theτderivative of the left equation above, and substituting the right hand equation
gives
d^2 P
dτ^2

+

α

β

P= 0 (3.44)

which now is a simple harmonic oscillator withω^2 (τ) =α(τ)/β(τ).The action integral
may be rewritten as

S=

∫

P

dQ

dτ

dτ (3.45)

where the integral is over one period of the motion. Using Eqs.(3.43) and following the
same procedure as was used with Eqs.(3.32) and Eq.(3.33), this becomes

S=

∫

P^2

α

β

dτ=λ^2

∫[(

α(τ′)

β(τ′)

) 1 / 2

cos^2

(∫

τ′

(α/β)^1 /^2 dτ

′′

)]

dτ′ (3.46)

where∫ λis a constant dependent on initial conditions. By introducing the orbit phaseφ=
τ
(α/β)^1 /^2 dτ,Eq.(3.46) becomes

S=λ^2

∫ 2 π

0

dφcos^2 φ=const. (3.47)

Thus, the general action integral is indeed an adiabatic invariant. This proof is of course
only valid in the vicinity of an extremum ofH,i.e., only whereHcan be adequately
represented by Eq.(3.40).

3.4.1 General Proof for the General Adiabatic Invariant

We now develop a proof for the general adiabatic invariant. This proof is not restricted
to small oscillations (i.e., being near an extremum ofH) as was the previous discussion.