| dc.description |
We show that choreographic three bodies {x(t), x(t + T/3), x(t − T/3)} of period T on the lemniscate, x(t) ( + cn(t))sn(t)/(1 + cn<sup>2</sup>(t)) parametrized by the Jacobian elliptic functions sn and cn with modulus k<sup>2</sup> (2 + √3)/4, conserve the centre of mass and the angular momentum, where and 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 ∑<sub>i<j</sub>((1/2) ln r<sub>ij</sub> − (√3/24)r<sub>ij</sub><sup>2</sup>) or ∑<sub>i<j</sub>(1/2) ln r<sub>ij</sub> − ∑<sub>i</sub>(√3/8)r<sub>i</sub><sup>2</sup>, where r<sub>ij</sub> is the distance between bodies i and j, and r<sub>i</sub> 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. |
