West Virginia University
Morgantown, WV, USA
email: gingold@math.wvu.edu
University of Pennsylvania
Philadelphia, PA, USA
email: jocelynq@seas.upenn.edu
Harry Gingold, Jocelyn Quaintance
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 3, 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.
DOI: 10.58997/ejde.2025.33
Show me the PDF file (3483 KB), TEX file for this article.
|
Harry Gingold West Virginia University Morgantown, WV, USA email: gingold@math.wvu.edu |
|---|---|
|
Jocelyn Quaintance University of Pennsylvania Philadelphia, PA, USA email: jocelynq@seas.upenn.edu |
Return to the EJDE web page