3-Body Problems, Hidden Constants, Trojans and WIMPs

TL;DR

This paper introduces new constants of motion for the linearised restricted 3-body problem, generalizes classical solutions to include hidden constants, and explores quantum gravity effects on WIMP trajectories, offering insights into celestial mechanics and galaxy evolution.

Contribution

It presents novel constants of motion and a generalized Lagrange solution, along with a semi-classical quantum gravity framework for WIMPs, linking classical and quantum celestial dynamics.

Findings

New constants of motion for the linearised restricted 3-body problem.

Generalization of Lagrange's equilateral triangle solution to isosceles triangles.

A semi-classical quantum gravity model explaining WIMP trajectories.

Abstract

This work includes two new results - principally two new constants of motion for the linearised restricted 3-body problem (e.g. for the Trojan asteroids) and an important isosceles triangle generalisation of Lagrange's equilateral triangle solution of the restricted case leading to hidden constants for Hildans as well as Trojans. Both of these results are classical, but we also have included new results on Newtonian quantum gravity emanating from the asymptotics relevant for WIMPish particles, explaining the origin of systems like that of the Trojans. The latter result uses a generalisation of our semi-classical mechanics for Schr\"odinger equations involving vector as well as scalar potentials, presented here for the first time, thereby providing an acid test of our ideas in predicting the quantum curvature and torsion of WIMPish trajectories for our astronomical elliptic states. The combined effect is to give a new celestial mechanics for WIMPs in gravitational systems as well as new results for classical problems. As we shall explain, we believe these results could help to see how spiral galaxies evolve into elliptical ones. A simple classical consequence of our isosceles triangle result gives a Keplerian type $4^{\textrm{th}}$ Law for 3-body problems. This is confined to the Appendix.