From the Kinematics of Elliptical Motion to the Law of Gravitation

The law of force governing the motion of a material point along an elliptical orbit is derived using a purely kinematic approach. Starting from the differential equations of motion in a Cartesian coordinate system, an angular equation of motion is obtained that directly leads to Kepler’s second law. It is shown that the acceleration is directed toward the focus of the ellipse and is inversely proportional to the square of the distance. The derived expressions are applied to the Earth–Moon system. Forces calculated using Newton’s second law are compared with those given by Newton’s law of universal gravitation. The relative discrepancy does not exceed 1.1%, confirming their equivalence.