Satellites and Riemannian Geometry

Details Satellites and Riemannian Geometry Satellites and Riemannian Geometry

Mar 1969

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1969cemec...1...36k&link_type=abstract

Celestial Mechanics, Volume 1, Issue 1, pp.36-45

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5

Scientific paper

In an axially symmetric three-dimensional Riemann-spaceg ik(u 1,u 2)-u 3 represents the cyclic parameter-, a gravitational potential ϕ(u 1,u 2) is given. For all masspoints with equal total energy and equal angular momentum there exists a function Ψ(u 1,u 2) by means of which the equations of motion can be reduced to a simple ordinary second-order differential equation. The function ϕ can be interpreted as the velocity with which the masspoint moves in the two-dimensional spaceu 1,u 2. Of particular interest is the case where the spaceu 1,u 2,u 3 is Euclidean. Ifu 1,u 2 are Cartesian coordinates in a planeu 3=const., and if the tangent vector of the trajectoryu 1(t)u 2(t) has the components cosω, sinω it is shown that the triple integral int int int ψ du^1 du^2 dω is an invariant integral in Cartan's sense, in other words, if the integral is extended over a domain in a meridian plane at timet=0, it keeps its value at any time.

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