Choreographic three bodies on the lemniscate
by
Hiroshi Fukuda
School of Administration and Informatics, University of Shizuoka, 52-1 Yada, Shizuoka 422-8526, Japan
Coauthors: Toshiaki Fujiwara (Faculty of General Studies, Kitasato University, Kitasato 1-15-1, Sagamihara, Kanagawa 228-8555, Japan), Hiroshi Ozaki (Department of Physics, Tokai University, 1117 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan)

We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period T on the lemniscate, x(t) = (e_x+e_y cn(t))sn(t)/(1+cn²(t)) parameterized by the Jacobian elliptic functions sn and cn with modulus k² = (2+3^1/2)/4, conserve the center of mass and the angular momentum, where e_x and e_y are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of squares of the curvature, the product of distances and the sum of squares of distances between bodies. We find that they satisfy the equation of motion under the potential energy

å i < j  (1/2 lnrij -31/2/24 rij2)

or

å i < j  1/2 lnrij - å i  31/2/8 ri2,

where r_ij the distance between the body i and j, and r_i the distance from the origin. The first term of the potential energies is the universal gravitation in two dimensions but the second term is a mutual repulsive force or a repulsive force from the origin, respectively. Then, geometric construction methods for the positions of the choreographic three bodies are given.

Date received: June 11, 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-95.