Particular superintegrability of 3-body (modified) Newtonian Gravity

TL;DR

This paper demonstrates that the 3-body Newtonian gravitational system exhibits maximal superintegrability along the Figure-8 trajectory, with explicit integrals of motion identified, extending to modified Newtonian gravity in 2D.

Contribution

It explicitly finds five additional Liouville integrals for the 3-body problem along the Figure-8 trajectory, establishing superintegrability in both Newtonian and modified gravity.

Findings

Five explicit Liouville integrals found for the 3-body system.

3-body choreographic motion on Figure-8 is maximally superintegrable.

Superintegrability conjectured for all 3-body potentials admitting Figure-8 motion.

Abstract

It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in ${\mathbb R}^3$ along the Remarkable Figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is shown they become constants of motion along this trajectory. Hence, 3-body choreographic motion on Figure-8-shape trajectory in ${\mathbb R}^3$ Newtonian gravity (Moore, 1993), as well as in ${\mathbb R}^2$ modified Newtonian gravity by Fujiwara et al, 2003, is maximally superintegrable. It is conjectured that any 3-body potential theory which admit Figure-8-shape choreographic motion is superintegrable along the trajectory.