Non-Conservation 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 rα (α 6= 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 α 6= −2, 2, 4 is impossible. And motions which satisfy the same conditions for α = 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 α 6= −2 is proved. We also prove that the motion which satisfies conditions (I)–(III) with general masses under the Newtonian potential α = −1 is impossible. PACS numbers: 45.20.Dd, 45.50.Jf, 45.50.Pk, 95.10.Ce