Springer Online Journal Archives 1860-2000
Abstract 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 $$\smallint \smallint \smallint \psi du^1 du^2 d\omega $$ 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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