72 Chapter 3. Motion of a single plasma particle
From Eq.(3.16) it is seen thatP/mis not, in general, the velocity and so the velocity
dQ/dtis not, in general, proportional toP.Thus, the turning points are not necessarily
at the locations wherePvanishes, and in factPneed not change sign during a period.
However,Sstill corresponds to the area of phase-space enclosed by one period of the
phase-space trajectory.
We can now calculate
dS
dt=
d
dt∮
PdQ=d
dt∫Qtp(t+τ)Qtp(t)P(E(t),Q,λ(t))dQ=
[
P
dQ
dt]Qtp(t+τ)Qtp(t)+
∫Qtp(t+τ)Qtp(t)(
∂P
∂t)
QdQ=
∫Qtp(t+τ)Qtp(t)[(
∂P
∂E
)
Q,λdE
dt+
(
∂P
∂λ)
Q,Edλ
dt]
dQ.(3.50)
BecausedQ/dt= 0at the turning point, the integrated term vanishes and so there is no
contribution from motion of the turning point. From Eq.(3.48) we have
1 =
∂H
∂P
(
∂P
∂E
)
Q,λ(3.51)
and
0 =∂H
∂P
(
∂P
∂λ)
Q,E+
∂H
∂λ(3.52)
so that Eq.(3.50) becomes
dS
dt=
∮ (
∂H
∂P
)− 1 [
dE
dt−
∂H
∂λdλ
dt]
dQ. (3.53)From Eq.(3.48) we have
dE
dt=
∂H
∂P
dP
dt+
∂H
∂Q
dQ
dt+
∂H
∂λdλ
dt=
∂H
∂λdλ
dt(3.54)
since the first two terms cancelled due to Hamilton’s equations. Substitution of Eq.(3.54)
into Eq.(3.53) givesdS/dt= 0,completing the proof of adiabatic invariance. No assump-
tion has been made here thatP,Qare close to the values associated with an extremum of
H.
This proof seems too neat, because it has established adiabatic invariance simply by
careful use of the chain rule, and by taking partial derivatives. However, this observation
reveals the underlying essence of adiabaticity, namely it is the differentiability ofH,P
with respect toλfrom one period to the next and the Hamilton nature of the system which
together provide the conditions for the adiabatic invariant to exist. Ifthe motion had been
such that after one cycle the motion had changed so drastically that taking a derivative ofH
orPwith respect toλwould not make sense, then the adiabatic invariant would not exist.