The figure eight :
Initial conditions,thanks to C Simo. : (x1,y1) = -(x2,y2) and (x3,y3) = (0,0) ; (vx1,vy1) = (vx2, vy2) = -(vx3, vy3)/2; where (x1,y1) = (-0.97000436, 0.24308753) , (vx3,vy3) = (0.93240737, 0.86473146) and the three mass constants and Gravitational constant set to 1 . The eight viewed as a moving triangle:  >
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My favorite case of the three-body problem is the equal mass zero angular momentum negative energy three-body problem. ANIMATIONS of periodic solutions for my favorite case. Each solution depicted in these sites passes through the Euler configuration at some instant,. The Euler configuration is one in which the three lie on a line with one at the the midpoint of the other two. 100s by Suvakov & Dmitrasinovi. Brake orbits non-collision, by Li and Liao. Isosceles Solutions found by Nai-Chia Chen. Choreographies are solutions in which all N-bodies chase each other about the same curve . My work with Chenciner, Gerver and Simo following up on the rediscovery of figure eight initiated the discovery of a plethora of choreographies Over 300 planar three-body choreographies found by C. Simo. Graphics by P.Masson . (most have non-zero angular momentum) Minton's build-your-own choreographies Tour 2024 A tour of solutions I've used in public talks. a Popularization: PBS video on 3-body older N body sites not so well curated |