Well Posed Origin Anywhere Consistent Systems in Celestial Mechanics

TL;DR

This paper addresses mathematical issues in celestial mechanics when the coordinate origin coincides with a point mass, proposing a translation-invariant relative system and deriving a new constant of motion.

Contribution

It introduces a translation-invariant relative system for celestial mechanics and derives a new constant of motion, resolving inadequacies in Newton's equations.

Findings

Identifies inadequacies in Newton's equations when origin coincides with a mass.

Proposes a system of relative differences free of these inadequacies.

Derives a new constant of motion indicating the 'restless' nature of the universe of relative differences.

Abstract

Certain measurements in celestial mechanics necessitate having the origin O of a Cartesian coordinate system (CCS) coincide with a point mass. For the two and three body problems we show mathematical inadequacies in Newton's celestial mechanics equations (NCME) when the origin of a coordinate system coincides with a point mass. A certain system of equations of relative differences implied by NCME is free of these inadequacies and is invariant with respect to any CCS translation. A new constant of motion is derived for the relative system. It shows that the universe of relative differences of the $N$-body problem is ``restless''.