Choreographic Three Bodies on the Lemniscate
Toshiaki Fujiwara, Hiroshi Fukuda, Hiroshi Ozaki

TL;DR
This paper presents a new choreographic three-body solution on the lemniscate, demonstrating conserved quantities and deriving equations of motion under specific potential energies involving logarithmic and quadratic terms.
Contribution
It introduces a novel elliptic function parametrization for choreographic three bodies on the lemniscate and analyzes their conserved quantities and equations of motion.
Findings
Bodies conserve center of mass and angular momentum.
They satisfy equations of motion under combined logarithmic and quadratic potentials.
Geometric methods for constructing their positions are provided.
Abstract
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+cn^2(t)) parameterized by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/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 sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij} -sum_{i}sqrt{3}/8 r_{i}^2, 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 Newton's gravity in two dimensions…
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