Three-Body Choreographies in Given Curves

TL;DR

This paper demonstrates that in certain symmetric and eight-shaped curves, choreographic motions of three bodies are uniquely determined by geometric constraints, extending classical two-body insights to specific three-body configurations.

Contribution

It introduces a geometric method to determine three-body choreographies on symmetric and eight-shaped curves, extending to generic curves without symmetry.

Findings

Unique choreographies on symmetric convex curves with non-zero angular momentum.

Existence of choreographies on eight-shaped curves similar to the figure-eight solution.

Method applicability to generic, non-symmetric curves with individual bodies and masses.

Abstract

As shown by Johannes Kepler in 1609, in the two-body problem, the shape of the orbit, a given ellipse, and a given non-vanishing constant angular momentum determines the motion of the planet completely. Even in the three-body problem, in some cases, the shape of the orbit, conservation of the centre of mass and a constant of motion (the angular momentum or the total energy) determines the motion of the three bodies. We show, by a geometrical method, that choreographic motions, in which equal mass three bodies chase each other around a same curve, will be uniquely determined for the following two cases. (i) Convex curves that have point symmetry and non-vanishing angular momentum are given. (ii) Eight-shaped curves which are similar to the curve for the figure-eight solution and the energy constant are given. The reality of the motion should be tested whether the motion satisfies an equation of motion or not. Extensions of the method for generic curves are shown. The extended methods are applicable to generic curves which does not have point symmetry. Each body may have its own curve and its own non-vanishing masses.