8.2 Conservation Laws
219
Summarizing we have seen that a transformation of the kind (
8.36
) is a symmetry
of the system if it leaves the action
invariant
, that is if the actions
S
and
S
exhibit
the
same functional dependence
on the paths described by
q
i
and
q
i
:
S
[
q
i
;
t
1
,
t
2
] =
S
[
q
i
;
t
1
,
t
2
]
,
(8.44)
or, equivalently, if
δ
S
≡
t
2
t
1
dt L
(
q
,
˙
q
,
t
)
−
t
2
t
1
dtL
(
q
,
˙
q
,
t
)
=
0
.
(8.45)
After these preliminaries we may state the
Noether theorem
as follows:
If the action of a dynamic system is invariant under a continuous group of (non
singular) transformations of the generalized coordinates and time, of the form
q
i
=
q
i
(
q
,
t
),
t
=
t
(
t
)
,
and if the equations of motion are satisfied, then the
quantity:
Q
≡
i
∂
L
∂
˙
q
i
δ
q
i
+
L
δ
t
,
(8.46)
is conserved.
We stress that the variations
δ
q
i
=
q
i
(
t
)
−
q
i
(
t
)
corresponding to infinitesi-
mal
local
transformations of the form (
8.36
),
are not arbitrary
as those used in
the discussion of the Hamilton action principle, but correspond to the subclass of
transformations leaving the action invariant.
Let us now start from the invariance property (
8.45
) to derive the conserved
quantities.
We set
δ
S
=
t
2
t
1
dt L
(
q
(
t
),
˙
q
(
t
),
t
)
−
t
2
t
1
dtL
(
q
(
t
),
˙
q
(
t
),
t
)
=
t
2
t
1
dtL
(
q
(
t
),
˙
q
(
t
),
t
)
−
t
2
t
1
dtL
(
q
(
t
),
˙
q
(
t
),
t
),
(8.47)
where, on the right hand side of the above equation, we have used the fact that
t
is
an integration variable. We now decompose the integration over
(
t
2
,
t
1
)
as follows:
Setting
t
1
=
t
1
+
δ
t
1
,
t
2
=
t
2
+
δ
t
2
, we can write
t
2
t
1
=
t
2
t
1
+
t
2
+
δ
t
2
t
2
−
t
1
+
δ
t
1
t
1
,
(8.48)