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 ${\mathbb{R}}^3$ 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.

4 pages, 1 figure, typos fixed, title modified, the correct value of elliptic modulus placed in p.2, left column

Cited by in corpus (2)

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