Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 33, pp. 1-27.

Title: Spherical compactifications of central force equations

Authors: Harry Gingold (West Virginia Univ., Morgantown, WV, USA)
         Jocelyn Quaintance (Univ. of Pennsylvania,  Philadelphia, PA, USA)

Abstract:
A spherical compactification is a map between an unbounded set of  
$R^n$ and a bounded set on a sphere in $R^{n+1}$.
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 
$R^n$ 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 $R^n$ may be appropriately 
labeled by $\infty U$, where $U$ is a unit vector in $R^n$, and
are ``visualized'' as points 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 $z'(t)=H(z(t))$.


Submitted November 25, 2024. Published April 03, 2025.
Math Subject Classifications: 70F15, 85A04.
Key Words: Central force equations; 
Newton's celestial mechanics equations; $N$-body problem; finite critical point; 
 critical point at infinity; spherical compactifications; Kepler's problem;
 stereographic projection; ultra-extended R^n; metric analysis.