Goldstein 928 (3
rd
ed. 9.30)
(a)
Let
H
be a Hamiltonian for some dynamical system. If
A
and
B
are two
constants of the motion that explicitly depend on time, then it must be true
that
∂
A
∂
t
=
−
[
A, H
] = [
H, A
]
,
(1)
and
∂
B
∂
t
=
−
[
B, H
] = [
H, B
]
.
(2)
We want to prove that
[
A, B
]
is also a constant of the motion, or that
∂
C
∂
t
+ [
C, H
] = 0
,
(3)
where
C
= [
A, B
]
.
(4)
For the three functions
A
,
B
, and
H
, we can use Jacobi’s identity to write
[[
A, B
]
, H
] + [[
B, H
]
, A
] + [[
H, A
]
, B
] = 0
,
(5)
which we can rewrite using Eqs.(1) and (2) as
[[
A, B
]
, H
] + [
−
∂
B
∂
t
, A
] + [
∂
A
∂
t
, B
] = 0
.
(6)
This can immediately be rewritten as
[[
A, B
]
, H
] + [
A,
∂
B
∂
t
] + [
∂
A
∂
t
, B
] = 0
,
(7)
and again as
[[
A, B
]
, H
] +
∂
[
A, B
]
∂
t
= 0
,
(8)
by applying the rules of partial di
ff
erentiation.
If we compare Eq.(8) with
Eqs.(3) and (4), we see that the desired result is established.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(b)
If the Hamiltonian
H
and the function
F
are constants of the motion,
then it must be true that
dF
dt
=
∂
F
∂
t
+ [
F, H
] = 0
,
(9)
and
∂
F
∂
t
= [
H, F
]
.
(10)
Since,
H
and
F
are each constants of the motion, we know from part
(a)
that
[
H, F
]
is also a constant of the motion, and from Eq.(10) it is, thus, clear that
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Wilemski
 mechanics, Derivative, Eqs., Eq., partial time

Click to edit the document details