Celestial Mechanics Solutions where the Future is a Perfect Reflection of the Past

TL;DR

This paper demonstrates that Newton's celestial mechanics equations admit a continuum of symmetric solutions where future trajectories mirror past ones, including perturbed systems with no singularities and symmetric properties.

Contribution

It introduces a new class of symmetric solutions in celestial mechanics where trajectories are perfect reflections of past states, extending to perturbed systems with no singularities.

Findings

Existence of symmetric solutions with future as a reflection of past

Perturbed systems also exhibit symmetric solutions without singularities

All bodies evolve from a single spatial point

Abstract

Newton's equations of celestial mechanics are shown to possess a continuum of solutions in which the future trajectories of the N bodies are a perfect reflection of their past. These solutions evolve from zero initial velocities of the N bodies. Consequently, the future gravitational forces acting on the N bodies are also a perfect reflection of their past. The proof is carried out via Taylor series expansions. A perturbed system of equations of the N body problem is also considered. All real valued solutions of this perturbed system have no singularities on the real line. The perturbed system is shown to have a continuum of solutions that possess symmetry where the future velocities of the N bodies are a perfect reflection of their past. The positions and accelerations of the N bodies are then odd functions of the time. All N bodies then evolve from one location in space.