Geography Reference
In-Depth Information
In order to materialize the definition of ( Riemann ) polar coordinates and normal coordinates ,in
particular, Definition 20.4 and formulae ( 20.49 )-( 20.52 ), we have to solve the system of second
order ordinary differential equations ( 20.42 ) of a geodesic in
2
A 1 ,A 2
. We refer to ( 20.42 )asthe
geodesic equations in the Lagrange portrait : they can be derived from a stationary Lagrangean
functional of the arc length ( 20.53 ) with S A and S B and fixed boundaries .
E
S B
δ
dS =0 .
(20.53)
S A =0
Alternatively, the geodesic equations as a system of two second order ordinary differential equa-
tions can be transformed into a system of four first order ordinary differential equations subject
to the Hamilton portrait of a geodesic, for example, following Grafarend and You ( 1995 )asa
sample reference. The four first order ordinary differential equations can be reduced to three in
case of rotational symmetry, for instance, for an ellipsoid-of-revolution in terms of
}
in phase space. Note that for both systems, we here present to you the first solutions in terms
of Legendre series and the second solutions in terms of Hamilton equations, both for the initial
value problem and for the boundary value problem.
{
L, B, α
20-21 Lagrange Portrait of a Geodesic: Legendre Series,
Initial/Boundary Values
A 1 ,A 2
The ellipsoid-of-revolution
E
is an analytic manifold . Therefore, in this context the following
Taylor expansion exists:
n
U A ( S )= U 0 + SU 0 + 1
1
n ! S m U ( m ) A
2! S 2 U 0 + lim n→∞
.
(20.54)
0
m =3
Question: “But how to effectively compute the higher
derivatives of U A ( S ) with respect to an initial point
U 0 and subject to the differential equations which gov-
ern a geodesic , a submanifold in
A 1 ,A 2
?” Answer: “A
proper answer is given by the Legendre recurrence ('Leg-
endre series') of U ( m ) A
E
in terms of U A
summarized in
Box 20.3 .”
 
Search WWH ::




Custom Search