2018/08/31 by Juan Carlos Lopez Vieyra · 1 citation
Physics and Astronomy · Mathematics · #math-ph #math.MP
paper · pdf · doi:10.1016/j.physleta.2019.03.004
Title was changed. Non essential changes were made to improve the presentation. 6 pages preprint Letter, 4 figures
arxiv created 2019/01/24 · arxiv updated 2019/05/22
In a remarkable paper of 2003 by Fujiwara et al. \citeFujiwara2003, a figure-eight three-body choreography on the algebraic lemniscate by Bernoulli was discovered. Such a choreography was found to be driven by the action of a pairwise potential V(rij) , depending only on the mutual relative distances rij, i,j=1,2,3. In the present Letter we show that two different choreographies of five bodies on the same algebraic lemniscate exist and correspond to solutions of ten coupled Newton equations of motion with a pairwise interaction potential. For each choreography the explicit form of the potential is found and ten constants of motion are presented, thus, it is superintegrable.