More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits

摘要

The famous three-body problem can be traced back to Isaac Newton in 1680s. Inthe 300 years since this "three-body problem" was first recognized, only threefamilies of periodic solutions had been found, until 2013 when \v{S}uvakov andDmitra\v{s}inovi\'c [Phys. Rev. Lett. 110, 114301 (2013)] made a breakthroughto numerically find 13 new distinct periodic orbits, which belong to 11 newfamilies of Newtonian planar three-body problem with equal mass and zeroangular momentum. In this paper, we numerically obtain 695 families ofNewtonian periodic planar collisionless orbits of three-body system with equalmass and zero angular momentum in case of initial conditions with isoscelescollinear configuration, including the well-known Figure-eight family found byMoore in 1993, the 11 families found by \v{S}uvakov and Dmitra\v{s}inovi\'c in2013, and more than 600 new families that have been never reported, to the bestof our knowledge. With the definition of the average period $\bar{T} = T/L_f$,where $L_f$ is the length of the so-called "free group element", these 695families suggest that there should exist the quasi Kepler's third law $\bar{T}^* \approx 2.433 \pm 0.075$ for the considered case, where $\bar{T}^*=\bar{T} |E|^{3/2}$ is the scale-invariant average period and $E$ is its totalkinetic and potential energy, respectively. The movies of these 695 periodicorbits in the real space and the corresponding close curves on the "shapesphere" can be found via the website:http://numericaltank.sjtu.edu.cn/three-body/three-body.htm