Fundamentals of Plasma Physics

15.2 Manley-Rowe relations 431

where it is assumed that
A ̇j
Aj

<<ωjandδ ̇j<<ωj. (15.5)

Each oscillator is considered as a linear mode of the system and, from now on,the term
mode will be used interchangeably with oscillator. The energy associated with each mode
is

Wj=

1

2 m

Pj^2 +

1

2

κjQ^2 j=

m

2

Q ̇^2 j+^1

2

κjQ^2 j=

m

2

ω^2 jA^2 j (15.6)

and the associated action is

S =

∮

PjdQj=m

∮

Q ̇jdQj=−mωjA^2 j

∫ 2 π

0

sinψjdcosψj

= mωjA^2 j

∫ 2 π

0

sin^2 ψjdψj=

m

2

ωjA^2 j (15.7)

whereψj=ωjt+δj(t)is the phase of the mode.

SinceA ̈jand ̈δjare small compared toωjA ̇jandωjδ ̇jrespectively, the former terms
may be dropped when calculating the second derivative ofQjwhich is therefore

Q ̈j = −ω^2 jAjcos(ωjt+δj)− 2 ωjA ̇jsin(ωjt+δj)− 2 ωjAjδ ̇j cos(ωjt+δj).(15.8)

When the above expression is inserted into Eqs.(15.3) the terms involvingω^2 jcancel and
what remains is
[
2 ωjA ̇jsin(ωjt+δj)
+2ωjAjδ ̇jcos(ωjt+δj)

]

=

λ

m

AkAlcos(ωkt+δk)cos(ωlt+δl)

=

λ

2 m

AkAl

[

cos((ωk+ωl)t+δk+δl)

+cos((ωk−ωl)t+δk−δl)

]

.

(15.9)

It is necessary to write out the three coupled equations explicitly becausethe coupling
terms on the right hand side are not fully symmetric. To identify resonant interactions it is
assumed that

ω 3 = ω 1 +ω 2 (15.10)

θ(t) = δ 1 +δ 2 −δ 3 (15.11)

in which case the coupled mode equations can be written

[

A ̇ 1 sin(ω 1 t+δ 1 )

+A 1 ̇δ 1 cos(ω 1 t+δ 1 )

]

=

λA 2 A 3

4 mω 1





cos((ω 2 +ω 3 )t+δ 2 +δ 3 )

+cos((ω 2 −ω 3 )t+δ 2 −δ 3 )

︸ ︷︷ ︸

resonant at−ω 1





[ ̇

A 2 sin(ω 2 t+δ 2 )

+A 2 ̇δ 2 cos(ω 2 t+δ 2 )

]

=

λA 1 A 3

4 mω 2





cos((ω 1 +ω 3 )t+δ 1 +δ 3 )

+cos((ω 1 −ω 3 )t+δ 1 −δ 3 )

︸ ︷︷ ︸

resonant at−ω 2





[

A ̇ 3 sin(ω 3 t+δ 3 )

+A 3 ̇δ 3 cos(ω 3 t+δ 3 )

]

=

λA 1 A 2

4 mω 3





resonant at+ω 3

︷ ︸︸ ︷

cos((ω 1 +ω 2 )t+δ 1 +δ 2 )

+cos((ω 1 −ω 2 )t+δ 1 −δ 2 )



.

(15.12)