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CCCC – p. 11
Low energy effective action
To second adiabatic order, as second order equation of motion:
( )
as it results from
Γ eff [q(t)] =
m+
U ′′′ (q) 2
32m 2 ω 5 (1+ U′′ (q)
mω 2 ) 5 2
+ ˙q2 (
4mω 2 U ′′′ (q)U ′′′′ (q)
+mω 2 q +U ′ (q)+
(
¨q
128m 3 ω 7 (1+ U′′ (q)
mω 2 ) 7 2
) )
1+ U′′ (q)
mω 2 −5U ′′′ (q) 3
U ′′′ (q)
4mω(1+ U′′ (q)
mω 2 ) 1 2
= 0.
∫ ( (
)
1 U ′′′ (q) 2
dt m+
2 2 5 m 2 (ω 2 +m −1 U ′′ (q)) 5 2
− 1 2 mω2 q 2 −U(q)− ω ( )
)1
1+ U′′ (q)
2
2 mω 2
˙q 2