Some Converse Theorems in Newtonian Field Theory and Their Applications

Abstract

Two converse theorems related to a family of homogeneous homothetic bodies and connected to the theory of the Newtonian field are proved. In both of them the function characterising the attraction is unknown and it is demonstrated that this function in the first theorem is given by the one characterising the Newtonian field and in the second theorem it is given by this latter function with the addition of a linear function of distance. A second unknown function appears in the second theorem and it is proved that it is a linear function of the volume of the bodies. Moreover, in both the theorems it is proved that the unknown shape of the bodies must be spherical. The conjecture is made that the two theorems are still true without the hypothesis that the unknown function characterising the attraction has a pole. The full significance of the two theorems is briefly illustrated by the application of the second theorem to the case of a finite homogeneous fluid, the behaviour of which is isotropic with respect to an element of it. Among other results, it is found that the shape of the fluid is necessarily spherical and the forces at a distance which are exerted among the elements of the fluid are expressed by Newton's Law of gravitation. AMS Classification 31B20, 31B99, 76A02, 76A99