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 )