Proof of non-conservation of the moment of inertia of three-body figure-eight choreography
by
Toshiaki Fujiwara
Faculty of General Studies, Kitasato University, Kitasato 1-15-1, Sagamihara, Kanagawa 228-8555, Japan
Coauthors: Hiroshi Fukuda (School of Adminstration and Informations, University of Shizuoka, 52-1 Yoda, Shizuoka 422-8526, Japan), Hiroshi Ozaki (Department of Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan)

We investigate three-body motion under the interaction potential energy V_\alpha=\alpha^-1r^\alpha for \alpha =/= 0 and V₀=log r, where r represents the mutual distance between bodies.

We prove a theorem. Theorem: Equal mass three-body motion with the following three conditions for \alpha =/= -2, 2, 4 is impossible. The conditions are (I) the moment of inertia is conserved and non-zero, (II) the angular momentum is zero and (III) one body is on the center of mass at an instant.

Explicit solutions of the equal mass three-body motion with conditions (I)-(III) for \alpha = 2, 4 are given. They have collision and their shapes are not eight. The above theorem and this fact prove a theorem conjectured by Alain Chenciner. We would like to call this theorem the ``Saari-Chenciner's Theorem''. Saari-Chenciner's Theorem: The moment of inertia of three-body figure-eight choreography is conserved if and only if \alpha = -2.

Date received: May 27, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calo-04.