Spherical compactifications of central force equations

A spherical compactification is a map between an unbounded set of \(\mathbb{R}^n\) and a bounded set on a sphere in \(\mathbb{R}^{n+1}\). This article rigorously defines a parameterized family of spherical compactifcations and applies such compactifications to systems and solutions of ordinary differential equations (ODEs) associated with central force equations. Spherical compactification provides a means of embedding \(\mathbb{R}^n\) into a complete metric space. The compactified differential equation may have critical points that represent ``critical points at infinity'' of the original equation. These ``critical points at infinity'' in \(\mathbb{R}^n\) may be appropriately labeled by \(\infty U\), where \(U\) is a unit vector in \(\mathbb{R}^n\), and are ``visualized" as a point on the rim of a spherical compactifaction. To further legitimize objects of the form \(\infty U\) we develop a new calculus which interprets objects of the form \(\infty U_1 + \infty U_2\). We then utilize these spherical compactifications, which are of the form \(w(t) = \theta^{-1}(t)z(t)\), to transform a first order vector valued differential equation \(w'(t)=F(w(t))\) into the first order vector valued differential equation \(z'(t)=H(z(t))\) and provide two theorems which manifest the correspondence between finite critical points of \(w'(t)=F(w(t))\) and \(z'(t)=H(z(t))\). For more information see https://ejde.math.txstate.edu/Volumes/2025/33/abstr.html