Periodic orbits for Newtonian planar three-body problem

Xiaoming LI and Shijun LIAO

Shanghai Jiao Tong University, China

Background

The famous three-body problem can be traced back to Newton [1] in 1680s, and attracted many famous mathematicians and physicists such as Euler [2], Lagrange [3] and so on. Poincare [4] 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. His discovery of the so-called “sensitivity dependence on initial conditions” (SDIC) laid the foundation of modern chaos theory. It well explains why in the 300 years only three families of periodic orbits of three-body system were found by Euler [2] and Lagrange [3], until 1970s when the Broucke-Hadjidemetriou-Henon family of periodic orbits were found [5–9]. The famous figure-eight family was numerically discovered by Moore [10] in 1993 and rediscovered by Chenciner and Montgomery [11] in 2000. In 2013, Suvakov and Dmitrasinovic [12] made a breakthrough to find 13 new distinct periodic orbits by means of numerical methods, which belong to 11 new families. In 2017 Li and Liao [13] found more than six hundred new periodic orbits of three-body system with equal mass, and in 2018 Li et al. [14] further reported more than one thousand new periodic orbits of three-body system with unequal mass. In 2018, Li and Liao [15] reported more than two hundred new collisionless periodic orbits in free-fall three-body problem.

The following show the details of our newly found periodic orbits.

1. Periodic orbits with equal mass

FIG1. The trajectories of six new periodic orbits with equal mass. Blue line: body-1, red line: body-2, black line: body-3.

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 (Science China Physics, Mechanics & Astronomy, 60(12), 2017,dio:10.1007/s11433-017-9078-5) [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.

2. Periodic orbits with unequal mass

FIG2. The trajectories of six new periodic orbits with unequal mass. Blue line: body-1, red line: body-2, black line: body-3.

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. It was published by Publications of the Astronomical Society of Japan on May 21th, 2018(doi:10.1093/pasj/psy057) [PDF].

3. Collisionless periodic orbits in free-fall three-body problem

FIG3. The trajectories of six new collisionless free-fall three-body periodic orbits. Blue line: body-1, red line: body-2, black line: body-3.

On May 21th 2018, we reported 234 collisionless periodic free-fall three-body orbits with different mass ratios, including 231 entirely new collisionless periodic orbits. For more detail, please refer to arXiv:1805.07980 and website.

References

[1] I. Newton, Philosophiae naturalis principia mathematica (London: Royal Society Press, 1687).

[2] L. Euler, Novo Comm. Acad. Sci. Imp. Petrop. 11, 144 (1767).

[3] J.L. Lagrange, Prix de lacademie royale des Sciences de paris 9, 292 (1772).

[4] J. H. Poincare, Acta Math. 13, 1 (1890).

[5] R. Broucke, Celestial Mechanics 12, 439 (1975).

[6] J. D. Hadjidemetriou, Celestial Mechanics 12, 255 (1975).

[7] J. D. Hadjidemetriou and T. Christides, Celestial mechanics 12, 175 (1975).

[8] M. Henon, Celestial mechanics 13, 267 (1976).

[9] M. Henon, Celestial mechanics 15, 243 (1977).

[10] C. Moore, Phys. Rev. Lett. 70, 3675 (1993).

[11] A. Chenciner and R. Montgomery, Annals of Mathematics 152, 881 (2000).

[12] M. Suvakov and V. Dmitrasinovic, Phys. Rev. Lett. 110, 114301 (2013).

[13] X. Li and S. Liao, SCIENCE CHINA Physics, Mechanics & Astronomy 60, 129511 (2017). [PDF] arXiv:1705.00527v4

[14] X. Li, Y. Jing, and S. Liao, Publications of the Astronomical Society of Japan (published online, 2018 ,dio: 10.1093/pasj/psy057) [PDF] arXiv:1709.04775

[15] X. Li and S. Liao, Collisionless periodic orbits in the free-fall three-body problem, 2018, arXiv:1805.07980

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