Environmental Engineering Reference
In-Depth Information
way we can express the variation in the length
L
of a curve:
B
δL
=
δ(
d
s).
(A.3)
A
The variation in the element d
s
is given by
δ((
d
s)
2
)
=
d
x
i
d
x
j
δg
ij
+
2
g
ij
d
x
i
δ(
d
x
j
)
1
2
˙
δ(
d
s)
=
x
i
d
x
j
δg
ij
+
g
ij
˙
x
i
d
(δx
j
)
(A.4)
and we use the dot notation to indicate differentiation with respect to
s
. Writing
δg
ij
=
=
δx
k
g
ij,k
(this comma notation is a common shorthand for
differentiation with respect to
x
)gives
δx
k
(∂g
ij
/∂x
k
)
1
2
˙
d
s.
x
i
d
(δx
j
)
d
s
δ(
d
s)
=
x
i
˙
x
j
δx
k
g
ij,k
+
g
ij
˙
(A.5)
Under the integration we can re-arrange the second term on the RHS of this equation
using integration by parts, i.e.
B
B
=
δx
j
g
ij
˙
x
i
B
x
i
d
(δx
j
)
d
s
d
(g
ij
˙
x
i
)
d
s
g
ij
˙
d
s
A
−
δx
j
d
s.
(A.6)
A
A
The first term on the RHS vanishes since we require the variation to vanish at the
end points (i.e. the curve must pass through these two points and hence
δx
i
=
0
there). Thus the variation in the length is
B
1
2
˙
δx
k
d
s.
d
(g
ik
˙
x
i
)
δL
=
x
i
x
j
g
ij,k
˙
−
(A.7)
d
s
A
Now since each element
δx
i
can be varied independently it follows that the term
in parenthesis must vanish identically for the extremal path, i.e.
d
(g
ik
˙
1
2
˙
x
i
)
˙
−
=
x
i
x
j
g
ij,k
0
.
(A.8)
d
s
We are almost done now. Differentiating the product gives
1
2
˙
x
i
x
j
g
ij,k
˙
−
g
ik,l
x
l
˙
x
i
˙
−
g
ik
x
i
¨
=
0
(A.9)
which can be re-arranged (utilizing the symmetry of the metric to give the final
result a more symmetric appearance and re-naming some of the indices) to give
the final answer:
2
g
ij,k
g
jk,i
˙
1
g
ij
¨
x
j
+
+
g
ik,j
−
x
j
x
k
˙
=
0
.
(A.10)