200B Problem Set I, Solution to No.4
Emily Nardoni
1/21/14
Problem Statement:
2
Let H(q, p, t) = H0 (q, p) + V (q) ddtA
2 , for A(t) periodic with period τ T , and T the period of
the motion governed by H0 .
(a) Derive the mean field (i.e. short time averaged) equations for this system.
Solution:
First, we derive the Lagrangian from the given Hamiltonian, and find the Euler-Lagrange
equations of motion:
d2 A
p2
+ V0 (q) + V (q) 2
2m
dt
∂H
p
L = pq̇ − H, q̇ =
=
⇒ p = mq̇
∂p
m
1
d2 A
L = mq̇ 2 − V0 (q) − V (q) 2
2
dt
H=
By the Euler-Lagrange equations,
d ∂L
∂L
=
∂q
dt ∂ q̇
⇒ −
∂V0 (q) ∂V (q) d2 A
−
= mq̈
∂q
∂q dt2
Now, we separate the motion into a centroid part y(t) and a quiver motion (t):
q(t) = y(t) + (t),
v cos ωt, ω =
2π
τ
We plug this into the Euler-Lagrange equations, expand (assuming is small), and pick
out the slow and fast parts:
−
∂
∂
d2 A(t)
V0 (y + ) −
V (y + )
= mÿ + m¨
∂q
∂q
dt2
2
∂ 2 V0 (y) ∂V (y) ∂ 2 V (y) d A(t)
∂V0 (y) −
−
−
+
= mÿ + m¨
2
2
2
∂y slow
∂y
∂y fast
∂y
dt
fast
slow
slow
fast
1
Rτ
We take the short time average of the previous equation, where <>= τ1 0 dt, and note
that h¨
i = 0, hi = 0: then we are left with the “slow”, or mean field, equation
mÿ = −
∂V0 (y) ∂ 2 V (y)
−
∂y
∂y 2
d2 A
dt2
The “fast” equation is given by
m¨
= −
∂ 2 V0 (y) ∂V d2 A
−
∂q 2
∂y dt2
Note that since v cos ωt,
m¨
v mω 2 ∂ 2 V0 (y)
v mΩ20 ,
∂y 2
ω 2 Ω20
so that we can drop the above term in our fast equation relative to the other terms, and
we write our fast equation as
m¨
=−
∂V d2 A(t)
∂y dt2
The idea is to solve the fast equation and plug into the slow equation. Since V is independent of time, we find
=−
1 ∂V
A
m ∂q
⇒
−
∂ 2 V (y)
∂y 2
d2 A
dt2
∂ 2 V (y) 1 ∂V
d2 A(t)
A(t)
∂y 2
m ∂y
dt2
1 ∂ 2 V (y) ∂V
d2 A(t)
=
A(t)
m ∂y 2 ∂y
dt2
=
Note that we can rewrite
d2 A
A 2
dt
=
d
dt
dA
A
dt
−
*
dA
dt
2 +
=−
*
dA
dt
2 +
since
A v sin ωt →
E
d D
d
(AȦ) v
h(sin ωt cos ωt)i = 0
dt
dt
Thus we plug this result back in to our slow equation, and (finally) find our mean field
equation:
∂V0 (y)
1 ∂ 2 V ∂V
mÿ = −
−
∂y
m ∂y 2 ∂y
2
*
dA
dt
2 +
(b) Show that these mean field equations may be obtained from the effective Hamiltonian
1
K(p, q) = H0 (p, q) +
4m
*
dA
dt
2 + ∂V (q)
∂q
2
where here <> means a short time average, and we may assume H0 =
p2
2m
+ V0 (q).
Solution:
Again, we derive the Lagrangian and Euler-Lagrange equations of motion, but this time
for our Hamiltonian being the given effective Hamiltonian:
p
∂H
=
→ p = mq̇
∂p
m
* + 1
1
dA 2
∂V (q) 2
L = mq̇ 2 − V0 (q) −
2
4m
dt
∂q
* +
d ∂L
∂V0 (q)
1 ∂V ∂ 2 V
dA 2
∂L
= mq̈
=
⇒ −
−
∂q
dt ∂ q̇
∂q
2m ∂q ∂q 2
dt
L = pq̇ − H,
q̇ =
Thus we derive the same mean field equation as before (with m → 2m, which I believe is
a mistake in the problem statement).
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