A novel study on Kepler’s law and inverse square law of gravitation

Abstract

The Udwadia–Kalaba equation is a simple, aesthetic, and thought-provoking description of the world at a very fundamental level. It is about the way systems move. In this paper, we creatively apply the Udwadia–Kalaba approach to study heavenly bodies’ movements (especially on Kepler’s law and the inverse square law of gravitation). In an alternative way, we show that a heavenly body’s motion orbit can be an ellipse, a circle, a hyperbola, or a parabola and show the conservation of angular momentum. Furthermore, by applying the Udwadia–Kalaba approach, we use the constraint of motion orbit (ellipse, circle, hyperbola, or parabola) and the conservation of angular momentum constraint (or energy conservation constraint) and easily verify that any heavenly body’s motion complies with the inverse square law of gravitation. That is, we study Kepler’s law and Newton’s inverse square law in an analytical way, which makes the dynamicist more clear about the way heavenly bodies move and also makes the celestial mechanician more clear about the analytical mechanics (the Udwadia–Kalaba approach). Furthermore, for the students of dynamics and celestial physics, a different unique perspective is provided for them to study. At the end, we present the detailed process of applying the Udwadia–Kalaba approach to two imaginary cases to show its simplicity and efficiency.