Mathematical Analysis of Gravitational Motion and Orbital Stability in a Two-Body System

Abstract The Newtonian two-body problem provides an analytical basis for relating gravitational attraction to orbital geometry and stability. This paper presents a unified treatment of relative motion under an inverse-square central force, using the centre of mass transformation and a consistent set of orbital parameters. The Lagrangian formulation separates translational and relative motion, while conservation of angular momentum and mechanical energy reduces the dynamics to an effective radial potential. Circular equilibria are examined through the first and second derivatives of this potential, establishing the condition for local radial stability at fixed angular momentum. Transformation to the reciprocal radial coordinate produces Binet’s equation, whose Newtonian solution yields conic trajectories classified by eccentricity and specific orbital energy. Conservation of areal velocity also leads to the orbital period relation expressed by Kepler’s third law. The planetary quantities reproduced in the accompanying table illustrate this established proportionality and are not independent observations. The analysis shows how the main results of classical orbital mechanics follow from a common mathematical framework, while distinguishing the local stability of the idealized two-body orbit from long-term behaviour under additional gravitational and nonconservative perturbations.