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
f
erentiation.
If we compare Eq.(8) with
Eqs.(3) and (4), we see that the desired result is established.
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