The significance of the Mathieu-Hill differential equation for Newtons apsidal precession theorem

Newtonʹs precession theorem in Proposition 45 of Book I of Principia relates a centripetal force of magnitude mor^n-3 as a power of the distance from the center to the apsidal angle theta, where theta is the angle between the point of greatest distance and the point of least distance. The formula theta = pi/radical n is essentially restricted to orbits of small eccentricity. A study of the apsidal angle for appreciable orbital eccentricity leads to an analysis of the differential equation of the orbit. We show that a detailed perturbative approach leads to a Mathieu-Hilltype of inhomogeneous differential equation. The homogeneous and inhomogeneous differential equations of this type occur in many interesting problems across several disciplines. We find that the approximate solution of this equation is the same as an earlier one obtained by a bootstrap perturbative approach. A more thorough analysis of this inhomogeneous differential equation leads to a modified Hill determinant. We show that the roots of this determinant equation can be solved to obtain an accurate solution for the orbit. This approach may be useful even for cases where n deviates noticeably from I. The derived analytic results were applied to the moon, Mercury, the asteroid Icarus, and a hypothetical object. We show that the differential equation that occurs in a perturbative relativistic treatment of the perihelion precession of Mercury also leads to a simplified form of the Mathieu-Hill differential equation.