Geography Reference
In-Depth Information
1
q
μ
=
λ
2
(
q
1
,q
2
)
q
μ
,
(E.36)
λ
2
q
μ
=
λ
−
2
q
μ
λ
−
4
(
∂
ν
λ
2
)
q
ν
q
μ
.
−
Implementing (
E.36
)into(
E.21
) or directly deriving from (
E.32
), we are led to Corollary
E.1
.
Obviously, thanks to the Maupertuis gauge of a geodesic, in particular, by introducing dynamic
time
t
, we have found the Newton form of a geodesic, an extremely elegant form we shall apply
furtheron.
Corollary E.1 (Newton portrait of a Maupertuis gauged geodesic).
∂
μ
λ
2
2
q
μ
−
=0
.
(E.37)
End of Corollary.
E-322 Hamilton Portrait
It should not be too surprising that by introducing the Maupertuis gauge, in particular dynamic
time
t
, we are led to familiar Hamiltonian equations. The Hamiltonian
H
2
(
q
(
τ
)
,p
(
τ
)) is trans-
formed into the Hamiltonian
H
2
(
q
(
t
)
,p
(
t
)) subject to the metric
g
μν
=
λ
2
(
q
1
,q
2
)
δ
μν
and
λ
2
=(
q
1
)
2
+(
q
2
)
2
.
H
2
(
q
(
t
)
,p
(
t
)) =
(E.38)
=
1
1
2
δ
μν
p
μ
p
ν
−
2
λ
2
(
q
1
,q
2
)
.
Equation (
E.38
) used as the input to minimize the functional leads to Corollary
E.2
.
Corollary E.2 (Newton portrait of a Maupertuis gauged geodesic in phase space).
d
q
μ
d
t
=
∂H
2
∂p
μ
=
δ
μν
p
ν
,
(E.39)
Search WWH ::
Custom Search