Geography Reference
In-Depth Information
e 2 )cos 2 r 1 ( q 1 ) 2 +( q 2 ) 2 1+sin r 1 ( q 1 ) 2 +( q 2 ) 2 1
A 2 (1
p μ =
2 c 2 n 1
e 2 sin 2 r 1 ( q 1 ) 2 +( q 2 ) 2 1
e sin r 1 ( q 1 ) 2 +( q 2 ) 2 2 ×
2
r 1 ( q 1 ) 2 +( q 2 ) 2
1+ e sin r 1 ( q 1 ) 2 +( q 2 ) 2
2 n−
1
e
× tan π
e sin r 1 ( q 1 ) 2 +( q 2 ) 2
4
2
1
e 2 )cos r 1 ( q 1 ) 2 +( q 2 ) 2 sin r 1 ( q 1 ) 2 +( q 2 ) 2
c 2 n 2 1
A 2 (1
×
(E.110)
e 2 sin 2 r 1 ( q 1 ) 2 +( q 2 ) 2 2
r 1 ( q 1 ) 2 +( q 2 ) 2
tan π
×
4
2
2 n
×
2
1+ e sin r 1 ( q 1 ) 2 +( q 2 ) 2
1 − e sin r 1 ( q 1 ) 2 +( q 2 ) 2
e
∂r 1 ( q 1 ) 2 +( q 2 ) 2
∂q μ
×
.
E-35 Maupertuis Gauged Geodesics (Normal Coordinates, Local
Tangent Plane)
The unit tangent vector (Darboux one-leg) d at a point q of a geodesic can be represented in the
local base { g 1 , g 2 } , the local tangent vectors (Gauss two-leg), which spans the tangent space T q M
2
3 ij } .
Note that from now on, we assume { g 1 , g 2 } to be an orthogonal, but not normalized Gauss
two-leg which can be materialized by orthogonal coordinates
at a point q ,namelyby( E.111 ), where x ( q 1 ,q 2 ) denotes the immersion { M
2 ,g μν }→{ R
{
q 1 ,q 2
}
.
d = x ( q 1 ,q 2 )= x
∂q μ q μ = g 1 q 1 + g 2 q 2
(E.111)
Alternatively, we can represent d by polar coordinates as follows: by means of the scalar products
(inner products) ( E.112 ), we introduce the azimuth α with respect to the first Gauss two-leg
g 1 . Inserting ( E.111 )into( E.112 ) and differentiating ( E.112 ), we obtain ( E.113 ) or, in terms of
conformal coordinates, we obtain ( E.114 ). The parameter change s
t ,( E.112 ) implemented,
 
Search WWH ::




Custom Search