Moto di un satellite artificiale terrestre in un campo gravitazionale con simmetria assiale

The trajectory of motion of an Earth’s artificial satellite in an axialsymmetric gravitational field is interpreted as a geodesic flow in a conformally flat Riemannian manifold, here called Maupertuis’ manifold, according to the Maupertuis-Euler-Lagrange variational principle of least action. Two cases of space, namely the case of configuration space and impulse space, are studied. In both cases of space, the relation between the nature of curvature of the Maupertuis’ manifold and the orbital geometry is investigated. We also analyze the tidal matrices and discuss some properties of stability for the Kepler’s motions in the Maupertuis’ manifold by means of the equations of geodesic deviation. Finally, we give embedding spaces of Maupertuis’ manifolds according to the local and isometric embedding theorem.