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According to the Corollary 3 of Proposition vii from the First Book of Newton’s Principia, the forces responsible for planetary motions or binary star systems are central forces having the general form

(1)

The elementary work of such forces can be written as

(2)

and is an exact differential if the force vector satisfies the equation

(3)

Therefore these forces are conservative only if they are curl free: this is a known condition for the central forces, there is nothing new in it. However, this condition can be written only in terms of function Y(x, y):

(4)

and under this form it gives us a great deal of information. For instance, in the cases where

(5)

the central forces are conservative if

(6)

i.e. if the motion is in the direction of vector . A more explicit characterization of the conservativity can be obtained for the forces such that

(7)

In this case the condition (4) comes to

(8)

where λ is a number. Therefore the central force is conservative only in cases where the direction of motion is an eigenvector of the matrix of the quadratic form determining the trajectory. This can be the case for the initial conditions of the corresponding Kepler problem, but in general the motion does not take place along the axes of the trajectory. Therefore, in general, we are entitled to say that, judging by the trajectories of the cosmic bodies, the forces characterizing a gravitational field are not conservative forces.