2002/10/24 by Toshiaki Fujiwara, Hiroshi Fukuda, Hiroshi Ozaki · 2 citations
Physics and Astronomy · Mathematics · #math-ph #math.MP #msc:70F07
paper · pdf · doi:10.1088/0305-4470/36/11/310
published as J. Phys. A36, 2791--2800 (2003)
arxiv created 2002/10/24 · arxiv updated 2012/01/17
We show that choreographic three bodies x(t), x(t+T/3), x(t-T/3) of period T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn2(t)) parameterized by the Jacobi's elliptic functions sn and cn with modulus k2 = (2+sqrt3)/4, conserve the center of mass and the angular momentum, where x-hat and y-hat are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of square of the curvature, the product of distance and the sum of square of distance between bodies. We find that they satisfy the equation of motion under the potential energy sumi<j(1/2 ln rij -sqrt3/24 rij2) or sumi<j1/2 ln rij -sumisqrt3/8 ri2, where rij the distance between the body i and j, and ri the distance from the origin. The first term of the potential energies is the Newton's gravity in two dimensions but the second term is the 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.