Evolution of the Moment of Inertia of Three-Body Figure-Eight Choreography

We investigate three-body motion in three dimensions under the interaction potential proportional to r^alpha (alpha \neq 0) or log r, where r represents the mutual distance between bodies, with the following conditions: (I) the moment of inertia is non-zero constant, (II) the angular momentum is zero, and (III) one body is on the centre of mass at an instant. We prove that the motion which satisfies conditions (I)-(III) with equal masses for alpha \neq -2, 2, 4 is impossible. And motions which satisfy the same conditions for alpha=2, 4 are solved explicitly. Shapes of these orbits are not figure-eight and these motions have collision. Therefore non-conservation of the moment of inertia for figure-eight choreography for alpha \neq -2 is proved. We also prove that the motion which satisfies conditions (I)-(III) with general masses under the Newtonian potential alpha=-1 is impossible.