"Geometrical" determination of the constants of motion in General Relativity

In recent time a theorem, due to E. Beltrami through which the integration of the geodesic equations of a curved manifold is obtained by means of a merely geometric method, has been revisited. This way of dealing with the problem is well in accordance with the geometric spirit of the Theory of General Relativity. In this paper we show another relevant consequence of this method. Actually, the constants of the motion, introduced in this geometrical way that is completely independent of Newton theory, are related to the conservation laws for "test particles" in the Einstein theory. These conservation laws may be compared with the conservation laws of Newton. In particular, by the conservation of energy (E) and the L, component of angular momentum, the equivalence of the conservation laws for the Schwarzschild field is verified and the difference between Newton and Einstein theories for the rotating bodies (Kerr metric) is obtained in a straightforward way.