A database of high precision trivial choreographies for the planar three-body problem I. Hristov1 R. Hristova12 I. Puzynin3

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A database of high precision trivial

choreographies for the planar three-body problem

I. Hristov1, R. Hristova1,2, I. Puzynin3,

T. Puzynina3, Z. Sharipov3, and Z. Tukhliev3

1Faculty of Mathematics and Informatics, Soﬁa University "St. Kliment Ohridski",

Soﬁa, Bulgaria

ivanh@fmi.uni-sofia.bg

2Institute of Information and Communication Technologies, Bulgarian Academy of

Sciences, Soﬁa, Bulgaria

3Meshcheryakov Laboratory of Information Technologies, Joint Institute for Nuclear

Research, Dubna, Russia

zarif@jinr.ru

Abstract. Trivial choreographies are special periodic solutions of the

planar three-body problem. In this work we use a modiﬁed Newton’s

method based on the continuous analog of Newton’s method and a high

precision arithmetic for a specialized numerical search for new trivial

choreographies. As a result of the search we computed a high precision

database of 462 such orbits, including 397 new ones. The initial condi-

tions and the periods of all found solutions are given with 180 correct

decimal digits. 108 of the choreographies are linearly stable, including 99

new ones. The linear stability is tested by a high precision computing of

the eigenvalues of the monodromy matrices.

Keywords: Three-body problem ·Trivial choreographies ·Modiﬁed

Newton’s method, ·High precision arithmetic

1 Introduction

A choreography is a periodic orbit in which the three bodies move along one

and the same trajectory with a time delay of T/3, where Tis the period of

the solution. A choreography is called trivial if it is a satellite (a topological

power) of the famous ﬁgure-eight choreography [1,2]. Trivial choreographies are

of special interest because many of them are expected to be stable like the ﬁgure-

eight orbit. About 20 new trivial choreographies with zero angular momentum

and bodies with equal masses are found in [3,4,5], including one new linearly

stable choreography, which was the ﬁrst found linearly stable choreography after

the famous ﬁgure-eight orbit. Many three-body choreographies (345) are also

found in [6], but they are with nonzero angular momentum and an undetermined

topological type.

In our recent work [7] we made a purposeful (on a small domain of initial

conditions) numerical search for ﬁgure-eight satellites (not necessarily choreogra-

phies). For numerical search we used a modiﬁcation of Newton’s method with a

arXiv:2210.00594v1 [math.NA] 2 Oct 2022

2 I. Hristov, R. Hristova, I. Puzynin et al.

larger domain of convergence. The three-body problem is well known with the

sensitive dependence on the initial conditions. To overcome the obstacle of deal-

ing with this sensitivity and to follow the trajectories correctly for a long time,

we used as an ODE solver the high order multiple precision Taylor series method

[8,9,10]. As a result we found over 700 new satellites with periods up to 300 time

units, including 45 new choreographies. 7 of the newly found choreographies are

shown to be linearly stable, bringing the number of the known linearly stable

choreographies up to 9.

This work can be regarded as a continuation of our recent work [7]. Now

we make a specialized numerical search for new trivial choreographies by using

the permuted return proximity condition proposed in [4]. We consider the same

searching domain and the same searching grid step as in [7]. Considering pretty

long periods (up to 900 time units), which are much longer than those in the

previous research, allows us to compute a high precision database of 462 trivial

choreographies, including 397 new ones. 99 of the newly found choreographies

are linearly stable, so the number of the known linearly stable choreographies

now rises to 108.

2 Diﬀerential equations describing the bodies motion

The bodies are with equal masses and they are treated as point masses. A planar

motion of the three bodies is considered. The normalized diﬀerential equations

describing the motion of the bodies are:

¨ri=

j=1,j6=i

(rj−ri)

kri−rjk3, i = 1,2,3.(1)

The vectors ri,˙rihave two components: ri= (xi, yi),˙ri= ( ˙xi,˙yi). The system

(1) can be written as a ﬁrst order one this way:

˙xi=vxi,˙yi=vyi,˙vxi=

j=1,j6=i

(xj−xi)

kri−rjk3,˙vyi=

j=1,j6=i

(yj−yi)

kri−rjk3, i = 1,2,3

(2)

We solve numerically the problem in this ﬁrst order form. Hence we have a vector

of 12 unknown functions X(t)=(x1, y1, x2, y2, x3, y3, vx1, vy1, vx2, vy2, vx3, vy3)>.

Let us mention that this ﬁrst order system actually coincides with the Hamilto-

nian formulation of the problem.

We search for periodic planar collisionless orbits as in [3,4]: with zero angular

momentum and symmetric initial conﬁguration with parallel velocities:

(x1(0), y1(0)) = (−1,0),(x2(0), y2(0)) = (1,0),(x3(0), y3(0)) = (0,0)

(vx1(0), vy1(0)) = (vx2(0), vy2(0)) = (vx, vy)

(vx3(0), vy3(0)) = −2(vx1(0), vy1(0)) = (−2vx,−2vy)

(3)

The velocities vx∈[0,1], vy∈[0,1] are parameters. We denote the periods of the

orbits with T. So, our goal is to ﬁnd triplets (vx, vy, T )for which the periodicity

condition X(T) = X(0) is fulﬁlled.

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Adatabaseofhighprecisiontrivialchoreographiesfortheplanarthree-bodyproblemI.Hristov1,R.Hristova1;2,I.Puzynin3,T.Puzynina3,Z.Sharipov3,andZ.Tukhliev31FacultyofMathematicsandInformatics,SoaUniversity"St.KlimentOhridski",Soa,Bulgariaivanh@fmi.uni-sofia.bg2InstituteofInformationandCommunicationTechnol...

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A database of high precision trivial choreographies for the planar three-body problem I. Hristov1 R. Hristova12 I. Puzynin3

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