conservation laws for energy, linear momentum and angular momentum.
Let us apply the same argument of
invariance
to the Hamiltonian of a dynamic
system: A canonical transformation is an
invariance
of the system if it leaves the
Hamilton equations of motion invariant in form, and this is the case if the functional
dependence of the Hamiltonian on the old and new canonical variables is the same
13
H
(
p
,
q
,
t
)
=
H
(
p
,
q
,
t
).
(8.99)
If we consider infinitesimal canonical transformations (
8.89
), (
8.90
), (
8.91
),
p
,
q
differ from
p
,
q
by infinitesimals
δ
p
, δ
q
, so that (
8.99
) amounts to requiring:
δ
H
=
H
(
p
,
q
)
−
H
(
p
,
q
)
= −
δθ
r
{
H
,
G
r
} = −
δθ
r
∂
G
r
∂
t
,
(8.100)
where we have used (
8.94
) and (
8.95
). From Eqs. (
8.69
) and (
8.100
), being
δθ
r
arbitrary, we conclude that
dG
r
dt
=
∂
G
r
∂
t
+ {
G
r
,
H
} =
0
,
namely that the
G
r
are constants of motion. In particular we see that
the infinitesi-
mal generators G
r
of the canonical transformations in the Hamiltonian formalism
correspond to the Noether charges Q
r
of the Lagrangian formalism.
As an example, we want to retrieve once again the three conservation laws of
linear momentum, angular momentum and energy, in the Hamiltonian formalism.
Let us start implementing the condition of invariance of a system of
n
particles under
space translations. The (Cartesian) coordinates and momenta of the particles are
denoted, as usual, by
x
(
k
)
and
p
(
k
)
, (
k
=
1
, . . . ,
n
)
, respectively. Let us perform the
infinitesimal translations
x
(
k
)
→
x
(
k
)
=
x
(
k
)
−
;
| |
1
p
(
k
)
→
p
(
k
)
=
p
(
k
)
,
which are supposed to leave the action invariant. To compute the infinitesimal gen-
erator of a translation on the
k
th particle we use (
8.92
) and (
8.93
):
δ
x
i
(
k
)
= −
j
∂
G
j
∂
p
i
(
k
)
= −
i
;
δ
p
i
(
k
)
=
j
∂
G
j
∂
x
i
(
k
)
=
0
.
From the above equations it follows that:
∂
G
j
∂
x
i
(
k
)
=
0
;
∂
G
j
∂
p
i
(
k
)
=
δ
i
j
⇒
G
j
(
x
(
k
)
,
p
(
k
)
)
=
k
p
j
(
k
)
=
P
j
(
tot
)
,
13
In this subsection we use the notations
p
i
,
q
i
instead of
P
i
,
Q
i
.