Particular superintegrability of 3-body (modified) Newtonian gravity

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 ${\bf R^3}$ 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 three-dimensional Newtonian gravity (Moore, 1993), as well as in two-dimensional 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.