Superintegrability of $(2n+1)$-body choreographies, $n=1,2,3,\ldots, \infty$ on the algebraic Lemniscate by Bernoulli (inverse problem of classical mechanics)

TL;DR

This paper investigates the superintegrability of choreographies of odd-numbered bodies on the algebraic Lemniscate, explicitly finding maximal sets of integrals for specific cases and conjecturing maximal superintegrability for all odd numbers.

Contribution

It explicitly finds maximal Liouville integrals for 3- and 5-body choreographies on the algebraic Lemniscate and conjectures maximal superintegrability for all odd-numbered bodies.

Findings

Explicit maximal Liouville integrals for 3- and 5-body choreographies.

Conjecture of maximal superintegrability for all odd-numbered bodies.

Limit analysis suggests a one-dimensional liquid with infinite constants of motion.

Abstract

For one 3-body and two 5-body planar choreographies on the same algebraic Lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies $(2n+1)$ moving choreographically (without collisions) along given algebraic Lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit $n \rar \infty$ is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic Lemniscate and it is characterized by infinitely-many constants of motion.