Investigating total collisions of the Newtonian N-body problem on shape space

TL;DR

This paper explores the behavior of total collisions in the Newtonian N-body problem within shape space, revealing that singularities persist even without absolute scale, but a shape-dynamical perspective helps distinguish collision solutions.

Contribution

It demonstrates that total collisions remain singular in shape space and introduces a shape-dynamical approach to identify and analyze collision solutions without external scale reference.

Findings

Total collisions are singular in shape space.

Shape momenta behavior causes singularities.

Collision solutions form a measure-zero set.

Abstract

We analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn't exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.