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On the analytic calculation of motions of artificial earth satellites by conform transformations. (Zur analytischen Bahnberechnung künstlicher Erdsatelliten mittels konformer Transformationen.) (German. English summary) Zbl 0834.70003

Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften. Reihe C: Dissertationen. 440. München: Verlag der Bayer. Akad. d. Wiss. in Komm. bei der Beck’schen Verlagsbuchh. x (1995).
Summary: The motions of satellites affected by conservative and nonconservative forces in the Newtonian mechanics are interpreted as geodesic motions. The dynamical geometrical properties of the motions are analyzed. In particular, the motions of satellites in the Newtonian Earth’s gravitational field are explained as the minimal geodesic flows on Maupertuis’ manifolds according to the Maupertuis’ variational principle of least action. The Euclidean space, in which satellites move, is a conformal map of a Maupertuis’ manifold. The Maupertuis’ manifolds are in detail discussed. Their embedding is solved according to the Brinkmann’s embedding theorem and illustrated for the elliptic Kepler motions and orbits perturbed by the Earth’s oblateness.
To solve analytically the problems of satellite motions, the conformal mapping of KS transformation type is introduced. In particular, a set of canonical KS elements for elliptic Kepler motions is derived. By virtue of this conformal transformation, the equations of the Kepler motion are regularized and transformed into linear differential equations. At the same time, the equations of the elliptic Kepler motion are also stabilized. These canonical KS elements are free from all singularities which occur in the classical Kepler elements. Especially, the solutions of the perturbation computation with the KS elements keep away from the strictly resonance-like amplification. An analytical solution of first order due to perturbations of an anisotropic terrestial gravitational field, the direct lunisolar influence, the tides of the earth, the relativistic gravitational effects, the atmospheric drag and the direct solar radiation presure is given and compared with a numerical integration on the orbits of GPS, ERS1 and LAGEOS1.

MSC:

70-08 Computational methods for problems pertaining to mechanics of particles and systems
70M20 Orbital mechanics
49S05 Variational principles of physics
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