Figure-eight choreographies of the equal mass three-body problem with Lennard-Jones-type potentials

We report on figure-eight choreographic solutions to a system of three identical particles interacting through a potential of Lennard-Jones-type $1/r^{12}-1/r^6$, where r is distance between the particles. By numerical search, we found there are a multitude of such solutions. A series of them are close to a figure-eight solutions to a homogeneous system with no 1/r ¹² term in the potential. The rest are very different, and have several points with large curvatures in their figure-eight orbits at which particles are repelled. Here figure-eight choreographies are periodic motions whose shape is symmetric in both horizontal and vertical axes, starting with an isosceles triangle configuration and going back to an isosceles triangle configuration with opposite direction through Euler configuration. Thus the lobe of such a figure-eight may be complex in shape, and need not be convex. (paper)