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Newtonian Gravitational Force

The Newtonian force created by a body of mass M upon a body of mass m can be written in the vector form as

Here G is the so-called constant of gravitation and  is the position vector of m in an orthogonal reference frame with its origin in M. The characteristics of this force are obvious, but they will nevertheless be restated here only for the benefit of further development. First of all, as is oriented from M to m and as the masses of bodies are always positive one can see that the mathematical expression (1) reflects the fact that the gravitational force is always attractive being oriented from m to M. Secondly, this force acts always along the line joining the two bodies, i.e. it is a central force. Thirdly, its magnitude does not depend but on the distance between the two bodies, i.e. on the magnitude of the position vector

Finally we have the most important among the properties of this force: it is conservative, i.e. its elementary work, exerted while moving the mass m is an exact differential. This property of the force, to be explained in detail a little later, is related to the corresponding energy conservation property of the motion in a gravitation field, whence its name.

Now, in order to obtain the Kepler Laws we use the Law II of Newtonian System, to obtain the following equation of motion for the material point m:

Here K ≡ GM, and therefore the motion is ‘universal’ in the sense of its independence of the mass of moving body. A dot over a symbol means differentiation upon time; thus in equation (3) we have the second differential of the position vector. This is a direct generalization of the observed fact that all bodies fall with the same acceleration when situated close to the surface of the Earth. As well-known, the differential equation (3) gives the three Kepler Laws that describe the motion of planets around the Sun and of the Moon around Earth.

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