Periodic orbits for Newtonian planar three-body problem

Xiaoming LI and Shijun LIAO

Shanghai Jiaotong University, China

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. They are found by means of the search grid 1000 * 1000 for the initial velocities [0,1] * [0,1] within a time interval [0,100]. For more detail, please refer to arXiv:1705.00527v2.

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. They are found by means of the search grid 4000 * 4000 for the initial velocities [0,1] * [0,1] within a larger time interval [0,200]. More than 600 among them have been never reported, to the best of our knowledge. For more detail, please refer to arXiv:1705.00527v3.

On July 11th 2017, the final version was updated on ArXiv, mainly for the modification of the names of these 695 families of periodic orbits of the three-body problems. For details, please refer to arXiv:1705.00527v4. It was accepted by Science China-Physics, Mechanics & Astronomy for the publication on July 11th 2017. It was published online on September 11th 2017[PDF].

For the detailed characteristic parameters (such as the periods, the scale-invariant averaged periods, initial velocities and so on), the definitions and lengths of the so-called free group element (word) of each orbit, and the movies of these periodic orbits, please visit the websites:

(A) The free group element (word);

(B) The movies of periodic orbits in real space and on the shape sphere.

On September 13th 2017, we reported 1349 families of Newtonian periodic planar three-body orbits with unequal mass and zero angular momentum and the initial conditions in case of isosceles collinear configurations. These 1349 families of the periodic collisionless orbits can be divided into seven classes according to their geometric and algebraic symmetries. Among these 1349 families, 1223 families are entirely new, to the best of our knowledge. For more detail, please refer to arXiv:1709.04775 and website.


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