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)
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