8.9 Hamiltonian Formalism in Field Theory
259
˙
F
(
t
)
=
∂
F
∂
t
+
V
δ
F
δϕ(
x
)
˙
ϕ(
x
)
+
δ
F
δπ(
x
)
˙
π(
x
)
d
3
x
=
∂
F
∂
t
+
V
δ
F
δϕ(
x
)
δ
H
δπ(
x
)
−
δ
F
δπ(
x
)
δ
H
δϕ(
x
)
d
3
x
=
∂
F
∂
t
+ {
F
,
H
}
,
(8.215)
where we have used (
8.212
)–(
8.213
).
In particular, if
F
does not have an explicit dependence on time:
˙
F
(
t
)
= {
F
,
H
}
.
In this case the dynamic variable
F
is conserved
if and only if its Poisson bracket
with the Hamiltonian vanishes.
Writing:
ϕ(
x
,
t
)
=
V
δ
3
(
x
−
x
)ϕ(
x
,
t
)
d
3
x
,
π(
x
,
t
)
=
V
δ
3
(
x
−
x
)π(
x
,
t
)
d
3
x
,
from the definition of
functional derivative
we have:
δϕ(
x
,
t
)
δϕ(
x
,
t
)
=
δπ(
x
,
t
)
δπ(
x
,
t
)
=
δ
3
(
x
−
x
).
(8.216)
Applying this relations we find:
{
ϕ(
x
,
t
),
H
} =
δ
H
δπ(
x
,
t
)
,
(8.217)
{
π(
x
,
t
),
H
} = −
δ
H
δϕ(
x
,
t
)
,
(8.218)
and using the Hamilton (
8.212
)–(
8.213
):
˙
ϕ(
x
,
t
)
= {
ϕ(
x
,
t
),
H
}
,
(8.219)
˙
π(
x
,
t
)
= {
π(
x
,
t
),
H
}
.
(8.220)
From (
8.216
) we also derive the fundamental relations:
{
ϕ(
x
,
t
), π(
x
,
t
)
} =
δ
3
(
x
−
x
),
(8.221)
{
ϕ(
x
,
t
), ϕ(
x
,
t
)
} = {
π(
x
,
t
), π(
x
,
t
)
} =
0
.
(8.222)