546 E Geodesics
E-3 The Newton form of a geodesic in Maupertuis gauge (sphere,
ellipsoid-of-revolution)
Geodesics, in particular minimal geodesics, are of focal geodetic interest. In terms of Riemann normal
coordinates, they are used in map projections to establish azimuthal maps – maps on a local tangential
plane T p M – which are geodetic with respect to the point p of evaluation. Straight lines in a Riemann
map (plane chart) are the shortest on the surface, for example the Earth, with respect to the point
p of evaluation. In geodetic navigation – aerial navigation, space navigation – minimal geodesies
are applied to connect points on the Earth surface or in space. In both applications, initial value
problems, boundary value problems as well as their mixed forms play the dominant role in solving the
differential equations of a geodesic. (See E. Lichtenegger (1987) for a review of the four fundamental
geodesic problems.)
Section E-31.
The differential equations of a geodesic can be written either as a system of two differential equations
of second order (Lagrange portrait) or as a system of four differential equations of first order (Hamilton
portrait) as long as we refer to two-dimensional surfaces. The Lagrange portrait and the Hamilton
portrait of a geodesic is presented in the first section.
Section E-32.
Recently, H. Goenner, E. Grafarend and R. J. You (1994) have shown that the Newton law balancing
inertial forces, in particular accelerations, and acting forces, in particular, those forces which are being
derived from a potential, can be interpreted as a set of three geodesics in a three-dimensional Riemann manifold. In this case, the three-dimensional Riemann manifold is parameterized by conformal
coordinates (isometric coordinates), in short, the three-dimensional Riemann manifold is said to be
conformally flat. In addition, the factor of conformality represents as a Maupertuis gauge the potential
of conservative forces. In turn, we try in the second section to express geodesics firstly by conformal
coordinates (isometric coordinates) and secondly by Maupertuis gauge, in particular, aiming at a
representation of geodesics in their Newton form (Newton portrait).
Section E-33, Section E-34.
The program to express the minimal geodesies as the Newton law is realized for geodesics in the
sphere S
2
R in the third section and in the ellipsoid-of-revolution E
2
A 1 ,A 2
in the fourth section. Extensive
numerical examples in the form of computer graphics are given.
Section E-35, Section E-36.
We refer to Section E-35 for a review of Maupertuis gauged geodesics parameterized in normal coordinates with respect to a local tangent plane. We refer to Section E-36 for a review of Lie series,
Maupertuis gauged geodesics, and the Hamilton portrait.
E-31 The Lagrange portrait and the Hamilton portrait of a geodesic
Let there be given a two-dimensional Riemann manifold {M
2 , g µν } with the standard metric G given
by G = {g µν } ∈ R
2×2 , symmetric and positive-definite, shortly a surface. For geodetic applications,
we shall assume the following important properties.
E-3 The Newton form of a geodesic in Maupertuis gauge (sphere,
ellipsoid-of-revolution)
Geodesics, in particular minimal geodesics, are of focal geodetic interest. In terms of Riemann normal
coordinates, they are used in map projections to establish azimuthal maps – maps on a local tangential
plane T p M – which are geodetic with respect to the point p of evaluation. Straight lines in a Riemann
map (plane chart) are the shortest on the surface, for example the Earth, with respect to the point
p of evaluation. In geodetic navigation – aerial navigation, space navigation – minimal geodesies
are applied to connect points on the Earth surface or in space. In both applications, initial value
problems, boundary value problems as well as their mixed forms play the dominant role in solving the
differential equations of a geodesic. (See E. Lichtenegger (1987) for a review of the four fundamental
geodesic problems.)
Section E-31.
The differential equations of a geodesic can be written either as a system of two differential equations
of second order (Lagrange portrait) or as a system of four differential equations of first order (Hamilton
portrait) as long as we refer to two-dimensional surfaces. The Lagrange portrait and the Hamilton
portrait of a geodesic is presented in the first section.
Section E-32.
Recently, H. Goenner, E. Grafarend and R. J. You (1994) have shown that the Newton law balancing
inertial forces, in particular accelerations, and acting forces, in particular, those forces which are being
derived from a potential, can be interpreted as a set of three geodesics in a three-dimensional Riemann manifold. In this case, the three-dimensional Riemann manifold is parameterized by conformal
coordinates (isometric coordinates), in short, the three-dimensional Riemann manifold is said to be
conformally flat. In addition, the factor of conformality represents as a Maupertuis gauge the potential
of conservative forces. In turn, we try in the second section to express geodesics firstly by conformal
coordinates (isometric coordinates) and secondly by Maupertuis gauge, in particular, aiming at a
representation of geodesics in their Newton form (Newton portrait).
Section E-33, Section E-34.
The program to express the minimal geodesies as the Newton law is realized for geodesics in the
sphere S
2
R in the third section and in the ellipsoid-of-revolution E
2
A 1 ,A 2
in the fourth section. Extensive
numerical examples in the form of computer graphics are given.
Section E-35, Section E-36.
We refer to Section E-35 for a review of Maupertuis gauged geodesics parameterized in normal coordinates with respect to a local tangent plane. We refer to Section E-36 for a review of Lie series,
Maupertuis gauged geodesics, and the Hamilton portrait.
E-31 The Lagrange portrait and the Hamilton portrait of a geodesic
Let there be given a two-dimensional Riemann manifold {M
2 , g µν } with the standard metric G given
by G = {g µν } ∈ R
2×2 , symmetric and positive-definite, shortly a surface. For geodetic applications,
we shall assume the following important properties.