**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 (

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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