Proceedings of the Symposium on Celestial Mechanics and N-body Dynamics in Chiba, 2014

Eds. M. Saito, M. Shibayama and M. Sekiguchi

Three-tangents theorem in three-body motion in

three-dimensional space

Hiroshi Ozaki1,

Tetsuya Taniguchi2, Hiroshi Fukuda2, and Toshiaki Fujiwara2

1Tokai University, Education Program Center, Shimizu Campus

2Kitasato University, College of Liberal Arts and Sciences

Abstract. In the general three-body problem on the plane, the

conservation of the center of mass and zero angular momentum has

a simple geometrical meaning: three tangent lines from the three

bodies meet at a point at each instant. It is called “three-tangents

theorem”. Kuwabara and Tanikawa extended this theorem to the

three-body motion in plane and in three-dimensional space with

non-zero angular momentum. In this short note, we will investi-

gate an alternative three-tangents theorem compared to Kuwabara

and Tanikawa’s in three-dimensional space.

1. Introduction

After the discovery of the ﬁgure-eight solution to the planar equal mass

three-body problem [1,2,3,4], equal mass N-body periodic solutions has

been paid match attention to. Recently new families of periodic solutions

for three equal masses moving under Newtonian gravity in a plane were

found by numerical simulation [5]. Both the ﬁgure-eight solution and new

solutions have zero total linear momentum and zero total angular momen-

tum. This fact leads to the following simple geometrical theorem [6]:

[Theorem 1] (three-tangents)If the total linear momentum and the

total angular momentum are zero, three tangent lines at bodies meet at a

point or three lines are parallel.

This theorem is proved without using the equation of motion of three

bodies. It means that the three-tangents theorem can be applied to wide

class of potential models. For example, it holds even for the three-body

1ozaki@tokai-u.jp

2tetsuya@kitasato-u.ac.jp, fukuda@kitasato-u.ac.jp, fujiwara@kitasato-u.ac.jp

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2

motion under the attractive logarithmic-potential accompanied by an ar-

tiﬁcial repulsive potential [6].

Kuwabara and Tanikawa [7] extended this theorem to the planar three-

body motion with non-zero total angular momentum. They showed that

[Theorem 2] (Extended three-tangents)If the total linear momentum

is zero and the total angular momentum is Lin the planar three-body mo-

tion, the area Stof the triangle formed with three tangent lines at the bodies

is given by

St·α=L2

2,

where αis the double area of the triangle formed with three momentum

vectors p1,p2,p3:

α=p1∧p2=p2∧p3=p3∧p1.

The symbol ∧is the exterior product of two vectors in two-dimensional

space.

Moreover, they applied the theorem to the three-dimensional case.

When the total angular momentum is projected to the yz-, zx-, and xy-

planes, and three tangent lines are also projected to each coordinate plane.

[Theorem 3] (three-tangents in three dimensions I )If the total linear

momentum is zero and the total angular momentum is Lin the three-

dimensional space, the area St=(St

yz , St

zx, St

xy)of the triangle formed with

projected three tangent lines to the yz-, zx-, and xy-planes is given by

St·α=L·L

2,

where L=(Lyz, Lz x, Lxy ), and α=(αyz , αzx, αxy ).

Being stimulated by Kuwabara-Tanikawa’s study, we investigated al-

ternative three-tangents theorem in three-dimensional space.

2. Alternative three-tangents theorem in three-dimensional space

We will set up a Cartesian coordinate system O-xyz. Suppose that three

bodies with masses m1, m2, m3are acting the inertial forces between any

two bodies obeying Newton’s third law in the space. According to the

equation of motions, three bodies are conﬁgured in the space keeping the

center of mass being ﬁxed in the Cartesian coordinate system. It is con-

venient to take the center of mass of three bodies as the origin Oin the

system O-xyz.

Also each body has its own momentum vector pi(i= 1,2,3) so that

they satisfy ipi=0. Three lines li(i= 1,2,3) are drawn along piso that

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each line passes through each position for each body. If one of momentum

vectors is zero vector, put a point in the space instead of drawing a line.

We will call l1, l2, l3tangent lines. Let us introduce a vector αwhich is the

double area of the triangle formed with three momentum vectors p1,p2,p3:

α=p1×p2=p2×p3=p3×p1,

where the symbol ×is the vector product of two three-dimensional vectors.

This relation is directly derived from ipi=0. Since three momentum

vectors p1,p2,p3are perpendicular to the double area α, they are always

parallel to a plane if all three momentum vectors are not zero vectors.

If α=0, either one of three momentum vectors is zero vector and others

are parallel with each other, or three momentum vectors are parallel with

one another, or all three momentum vectors are zero vectors.

On the other hand, If α6=0, three momentum vectors form a triangle

in the three-dimensional space. In this case, the following theorem is valid:

[Theorem 4] (three-tangents in three dimensions II )If the total linear

momentum is zero and three tangent lines are projected to a plane including

the total angular momentum in the three-dimensional space, the projected

three tangent lines meet at a point or three tangent lines are parallel.

3. Proof of Theorem 4

First, we will set up a Cartesian coordinate system O-x0y0z0. Three vectors

ˆ

x0= (1,0,0), ˆ

y0= (0,1,0), and ˆ

z0= (0,0,1) are taken as unit vectors along

x0-, y0-, and z0-axis, respectively. Let us choose a piece of plane τpassing

through the origin Oand being parallel to the total angular momentum L.

The normal vector of τis denoted by n. Now we set Cartesian coordinate

system O-x0y0z0so that two orthogonal x0- and y0-axis are on τ, the rest

z0-axis is perpendicular to τ.

Second, we will project position and momentum vectors (qiand pi)

of three bodies onto τ. The projected position and momentum vectors are

denoted by q0

iand p0

i. To project position and momentum vectors in the

three-dimensional space onto τ, it is convenient to introduce the projection

matrix Pin terms of nby P=I−ntn, where Iis the identity matrix.

Then we obtain the relations between qiand q0

i, similarly between piand

p0

i:

q0

i=Pqi

=qi−(qi·n)n,(1)

and

p0

i=Ppi

=pi−(pi·n)n.(2)

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Note that q0

iand p0

iare lying on the plane τat each instant. For the

system O-x0y0z0in which nis along z0-axis, the component representations

are written by

q0

i= (qix0, qiy0,0),p0

i= (pix0, piy0,0).

Since the center of mass Oof three bodies is projected onto itself, the

conservation laws of the center of mass and the total linear momentum are

valid for the Cartesian coordinate system O-x0y0z0. Thus we have iq0

i=0

and ip0

i=0at every instant. The total angular momentum is evaluated

in the system O-x0y0z0as follows:

L=

i

q0

i×p0

i+

i

(qiz0pix0−qix0piz0)ˆ

y0+

i

(qiy0piz0−qiz 0Piy0)ˆ

x0

=Lz0+Ly0+Lx0.

On the other hand the total angular momentum Lis always on the plane

τ, so only the z0component of Lvarnishes:

L0

z=

i

q0

i×p0

i

=

i

(qix0piy0−qiy0pix0)ˆ

z0=0.

Now let c0

tbe the intersection point of two projected tangent lines l0

1

and l0

2. The angular momentum L0

zabout the intersection point c0

tis also

zero: i(q0

i−c0

t)×p0

i=0.Also (q0

1−c0

t)×p0

1=0and (q0

2−c0

t)×p0

2=0.

Thus we have (q0

3−c0

t)×p0

3=0. This means that the projected tangent

three linesl0

1, l0

2and l0

3meet at the same point c0

ton τ.

Before closing our statement, we will represent the intersection point

of three tangent lines c0

ton τby the projected position and momentum

vectors. It is simply written by

c0

t=−[(q0

i×p0

i)·ˆ

z0]p0

j−[(q0

j×p0

j)·ˆ

z0]p0

i

(p0

i×p0

j)·ˆ

z0,(3)

where (i, j) = (1,2),(2,3),(3,1). Substituting (1) and (2) into (3), c0

tcan

also be written by qi,piand normal vector nin the three-dimensional

space.

Let us rotate τin the three-dimensional space for ﬁxed time. The

intersection point c0

tgoes to inﬁnity if the plane τis parallel to α(nis

perpendicular to α). This is easily veriﬁed because

(p0

i×p0

j)·ˆ

z0= (pi×pj)·n=α·n= 0.

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Acknowledgment

The research of one of the authors (HO) has been supported by Grand-in

-Aid for Science Research 25400408 JSPS.

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