More than 600 among them have been never reported, to the best of our knowledge. They are found by means of the search grid 4000 * 4000 for the initial velocities * within a larger time interval. On May 30th 2017, we further reported 695 families (including the previous 164 ones) of planar periodic orbits of the three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration. For more detail, please refer to arXiv:1705.00527v2. They are found by means of the search grid 1000 * 1000 for the initial velocities * within a time interval. On April 28th 2017, we reported 164 families of planar periodic orbits of the three-body system with equal mass and zero angular momentum in case of initial conditions with isosceles collinear configuration, including the well-known Figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 100 new families that have been never reported. Blue line: body-1, red line: body-2, black line: body-3. The trajectories of six new periodic orbits with equal mass. The following show the details of our newly found periodic orbits. In 2018, Li and Liao reported more than three hundred new collisionless periodic orbits in free-fall three-body problem. further reported more than one thousand new periodic orbits of three-body system with unequal mass. In 2017 Li and Liao found more than six hundred new periodic orbits of three-body system with equal mass, and in 2018 Li et al. In 2013, Suvakov and Dmitrasinovic made a breakthrough to find 13 new distinct periodic orbits by means of numerical methods, which belong to 11 new families. The famous figure-eight family was numerically discovered by Moore in 1993 and rediscovered by Chenciner and Montgomery in 2000.
It well explains why in the 300 years only three families of periodic orbits of three-body system were found by Euler and Lagrange, until 1970s when the Broucke-Hadjidemetriou-Henon family of periodic orbits were found. His discovery of the so-called “sensitivity dependence on initial conditions” (SDIC) laid the foundation of modern chaos theory. Poincare found that the first integrals for the motion of three-body system do not exist, and besides orbits of three-body system are rather sensitive to initial conditions. The famous three-body problem can be traced back to Newton in 1680s, and attracted many famous mathematicians and physicists such as Euler, Lagrange and so on. Periodic orbits for Newtonian planar three-body problem Xiaoming LI and Shijun LIAO Shanghai Jiao Tong University, China Background
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