8.5 Lagrangian and Hamiltonian Formalism in Field Theories
233
on the values of the fields
ϕ
α
(
x
,
t
)
and
∂
t
ϕ
α
(
x
,
t
)
at every point in the domain V of
the three-dimensional space.
We say in this case that the Lagrangian is a
functional
of
ϕ
α
(
x
,
t
)
and
∂
t
ϕ
α
(
x
,
t
)
, viewed as functions of
x
.
It will be convenient in the
following to denote by
ϕ
α
(
t
)
the function
ϕ
α
(
x
,
t
)
of the point
x
in space at a given
time
t
, and by
˙
ϕ
α
(
t
)
its time derivative
˙
ϕ
α
(
x
,
t
)
≡
∂
t
ϕ
α
(
x
,
t
)
. We shall presently
explore some property of functionals. Let us consider a functional
F
[
ϕ
]
, and perform
an independent variation of
ϕ(
x
)
, at each space point
x
.
The corresponding variation
of
F
[
ϕ
]
will be:
δ
F
[
ϕ
] ≡
F
[
ϕ
+
δϕ
] −
F
[
ϕ
] =
δ
F
[
ϕ
]
δϕ(
x
)
δϕ(
x
)
d
3
x
,
(8.103)
where
by definition
,
δ
F
[
ϕ
]
δϕ(
x
)
is the
functional derivative
of
F
[
ϕ
]
with respect to
ϕ
at
the point
x
.
Here we have suppressed the possible dependence on time of
ϕ
and of
the functional
F
either explicitly or through
ϕ
:
ϕ
=
ϕ(
x
,
t
),
F
=
F
[
ϕ(
t
),
t
]
.
From its definition it is easy to verify that the functional derivation enjoys the same
properties as the ordinary one, namely it is a linear operator, vanishes on constants
and satisfies the Leibnitz rule.
When the functional depends on more than a single function, its definition can
be extended correspondingly, as for ordinary derivatives. Of particular relevance for
us is the additional dependence of
F
on the time derivative
∂
t
ϕ(
x
,
t
)
of
ϕ(
x
,
t
).
Moreover we may consider a set of fields
ϕ
α
labeled by the index
α
pertaining
to a given representation of a group G. This is the case of the Lagrangian
F
=
L
(ϕ
α
(
t
),
˙
ϕ
α
(
t
),
t
)
, where we recall once again that, in writing
ϕ(
t
),
˙
ϕ(
t
)
among the
arguments of the Lagrangian, we mean that
L
depends on the values
ϕ(
x
,
t
),
˙
ϕ(
x
,
t
)
of these fields in
every point
x
in space at a given time
t
.
Applying the definition
(
8.103
) to the two functions
ϕ
α
(
t
)
and
˙
ϕ
α
(
t
)
we have:
δ
L
(ϕ
α
(
t
),
˙
ϕ
α
(
t
),
t
)
=
d
3
x
δ
L
δϕ
α
(
x
,
t
)
δϕ
α
(
x
,
t
)
+
δ
L
δ
˙
ϕ
α
(
x
,
t
)
δ
˙
ϕ
α
(
x
,
t
)
.
(8.104)
Note that the Lagrangian depends on
t
either through
ϕ
α
and
˙
ϕ
α
or explicitly. The
Lagrangian, as a
functional
with respect to the space-dependence of the fields, can be
thought of as the continuous limit of a function of infinitely many discrete variables:
L
(ϕ
i
(
t
),
˙
ϕ
i
(
t
),
t
)
i
→
x
−→
L
(ϕ(
t
),
˙
ϕ(
t
),
t
).
Here and in the following we shall often omit the index
α
if not essential to our
considerations. Correspondingly, we can show that the
functional derivative
defined
above can be thought of as a suitable continuous limit of the ordinary derivative with
respect to a discrete set of degrees of freedom
q
i
, described by a Lagrangian
L
(
q
i
,
˙
q
i
).