Fundamentals of Plasma Physics

3.3 Adiabatic invariant of a pendulum 67

The assumption of slowness is thus at least self-consistent, for ifω(t)is indeed slowly
changing, Eq.(3.24) shows thatA(t)will also be slowly changing and the dropping of the
last term in Eq.(3.22) is justified. The slowness requirement can be quantified by assuming
that the frequency has an exponential dependence

ω(t) =ω 0 eαt. (3.25)

Thus

α=

1

ω

dω

dt

(3.26)

is a measure of how fast the frequency is changing compared to the frequency itself. Hence
dropping the last term in Eq.(3.22) is legitimate if

α << 4 ω 0 (3.27)

or
1
ω

dω

dt

<< 4 ω. (3.28)

In other words, if Eq.(3.28) is satisfied, then the fractional change of thependulum period
per period is small.
Equation (3.24) indicates that whenωis time-dependent, the pendulum amplitude is not
constant and so the pendulum energy isnotconserved. It turns out that whatisconserved
is theaction integral

S=

∮

vdx (3.29)

where the integration is over one period of oscillation. This integral can also be written in
terms of time as

S=

∫t 0 +τ

t 0

v

dx

dt

dt (3.30)

wheret 0 is a time whenxis at an instantaneous maximum andτ,the period of a complete
cycle, is defined as the interval between two successive times whendx/dt= 0andd^2 x/dt^2
has the same sign (e.g., for a pendulum,t 0 would be a time when the pendulum has swung
all the way to the right and so is reversing its velocity whileτis the time one has to wait
for this to happen again). To show that action is conserved, Eq. (3.29) can be integrated by
parts as

S =

∫t 0 +τ

t 0

[

d

dt

(

x

dx

dt

)

−x

d^2 x

dt^2

]

dt

=

[

x

dx

dt

]t 0 +τ

t 0

−

∫t 0 +τ

t 0

x

d^2 x

dt^2

dt

=

∫t 0 +τ

t 0

ω^2 x^2 dt (3.31)

where (i) the integrated term has vanished by virtue of the definitions oft 0 andτ,and (ii)
Eq.(3.17) has been used to substitute ford^2 x/dt^2. Equations (3.19) and (3.24) can be
combined to give

x(t) =x(t 0 )

√

ω(t 0 )

ω(t)

cos

(∫t

t 0

ω(t′)dt′

)

(3.32)