This document presents a new coordinate system for studying the three-body problem described by Lagrange's case where three particles move in conic sections forming an equilateral triangle. The coordinate system uses Euler angles and the principal moments of inertia as coordinates. This provides a symmetric treatment with no preferential status given to any particle. The equations of motion are derived in these new coordinates, making Lagrange's case exhibit the expected symmetry. Key equations relate the new coordinates to the distances between particles and the principal moments of inertia.

1. A NEW SOLUTION TO THE LAGRANGE’S CASE OF THE
THREE-BODY PROBLEM
E. PIÑA
Departamento de Física, Universidad Autónoma Metropolitana – Iztapalapa, P.O. Box 55 534,
Mexico, D.F., 09340 Mexico, e-mail: pge@xanum.uam.mx
(Received: 10 April 1998; accepted: 22 April 1999)
Abstract. The motion of three particles, interacting by gravitational forces, is studied in a new
coordinate system given by the principal axes of inertia, as determined by Euler angles, and using
the inertia principal moments and an auxiliar angle as coordinates.
The solution to the particular Lagrange case of the three-body problem is reviewed and solved in
these new coordinates.
Key words: three-body problem, Lagrange’s case, new coordinates
1. Introduction
In 1906, new planetoids were discovered travelling near the orbit of Jupiter with the
same period as Jupiter, keeping a distance of 60 degrees in longitude from Jupiter –
some preceding, some following; any of them forming an equilateral triangle with
Jupiter and the Sun (Pannekoek, 1989).
They were named Achilles, Patroclus, Hector, Nestor, and other heroes of the
Trojan war; the entire group is now called the Trojan asteroids.
The importance of these discoveries comes from the particular solution found
by Lagrange to the three-body problem when the particles move in conic sections
like the Newton’s two-body problem, the three particles placed at the vertices of
an equilateral triangle (Sommerfeld, 1964). A simple presentation of this problem
was published by Carathéodory (1993) and will be considered here with different
kind of coordinates.
New realizations of this Lagrange’s solution have been found in many different
places of the Solar System. The satellite Eureka forming a triangle with Mars and
the Sun was a recent discovery in 1990, but other discovered triads are formed by
planets with two moons. Besides these examples, perturbations of the equilateral
configuration around one or two points of the triangle, as in the horseshoe orbit,
allow explanation of the motion of many bodies near the planets. A recent account
of the experimental and theoretical work on the Trojan planets has been reviewed
by Érdi (1997).
The purpose of this work was to look for a more symmetric aproach to the
general three-body problem. The traditional treatments of this subject give a pref-
Celestial Mechanics and Dynamical Astronomy 74: 163–174, 1999.
© 1999 Kluwer Academic Publishers. Printed in the Netherlands.

2. 164 E. PIÑA
erential or different status to one particle with respect to the other two particles. The
studies by Boccaletti and Pucacco (1996), Szebehely (1974) and Whittaker (1965)
are typical and they contain many other references. The Jacobi’s coordinates that
are frequently used for this problem are a representative example of this.
Symmetric coordinates were introduced by Murnaghan (1936), Lemaître (1954),
and others. In particular, Murnaghan uses the distances between particles as co-
ordinates, and Lemaître closely related quantities.
In the next section, we present the equations of motion in a coordinate system
that provides no difference to any of the three particles. The resulting equations
may be useful by themselves. In these coordinates the Lagrange’s case exhibits the
expected symmetry of that case.
2. Symmetric Coordinates
In order to define new coordinates, symmetric with respect to the three particles,
consider the unit vector u in the perpendicular direction to the plane of the particles
in the inertial frame, parametrized by Euler angles acting as spherical coordinates:
u =


sin θ sin φ
− sin θ cos φ
cos θ

 . (1)
G is the rotation matrix transforming from the inertial frame into the plane of
the particles. The u direction is transformed to a constant direction by this rotation
u = G


0
0
1

 . (2)
G rotates to the frame of the principal inertia axes of the particles. One of the
principal inertia directions is u, but an extra Euler angle ψ should be incorporated
to rotate in the plane of the particles in order to obtain a diagonal matrix of inertia.
G is therefore parametrized by Euler angles.
Assuming the origin is at the center of mass, rj (j = 1, 2, 3) denote the coordin-
ates of the particles in the inertial frame and sj in the new frame
rj = G sj (3)
The third coordinate of the vectors sj in the new frame is zero
sj =


sj x
sj y
0

 , j = 1, 2, 3. (4)

3. LAGRANGE’S CASE OF THE THREE-BODY PROBLEM 165
Figure 1. The Euler angles in the new frame of reference. The origin is at the general center of mass.
Vectors x1, x2, x3, represent the orthonormal basis of the inertial system. Vectors a1, a2, a3, are the
principal inertia directions. a3 is the perpendicular direction to the plane of the particles spanned by
the two directions a1 and a2.
The angular velocity in Euler angles in the rotated system is equal to
ω
ω
ω = φ̇


sin θ sin ψ
sin θ cos ψ
cos θ

 + θ̇


cos ψ
− sin ψ
0

 + ψ̇


0
0
1

 . (5)
That is used to write the velocity of particle j
ṙj = G(ω
ω
ω × sj + ṡj ) (6)
and the expression of the kinetic energy
T =
1
2
X
j
mj ṡ2
j + ω
ω
ω ·
X
j
mj sj × ṡj
+
1
2
ω
ω
ωT

1
X
j
mj s2
j −
X
j
mj sj sT
j

ω
ω
ω, (7)
where 1 is the unit matrix. The inertia matrix appears inside the square bracket of
Equation (7)
I = 1
X
j
mj s2
j −
X
j
mj sj sT
j =


I1 0 0
0 I2 0
0 0 I3

 . (8)

4. 166 E. PIÑA
Since three particles form a plane body the inertia moment around the axis 3 is
the sum of the other two moments:
I3 = I1 + I2. (9)
The origin of coordinates is at the center of mass, and the coordinates of the
particles will obey the equations
X
j
mj sj = 0, (10)
that is, in terms of the nonzero components of (4)
m1s1 x + m2s2 x + m3s3 x = 0, m1s1 y + m2s2 y + m3s3 y = 0. (11)
These equations suggest to consider the vectors in the mass space
Sx = (s1 x, s2 x, s3 x) and Sy = (s1 y, s2 y, s3 y) , (12)
that are orthogonal to the mass vector
m = (m1, m2, m3). (13)
Introduce now two constant vectors in this space a and b, orthogonal to the mass
vector, and orthogonal to each other
a · m = b · m = a · b = 0 . (14)
Then the vectors (12) are linear combinations of these two constant vectors
Sx = αa + βb and Sy = γ a + δb. (15)
The rotated frame has been chosen in the principal directions of inertia, this fact
being expressed in terms of components as
m1 s1 x s1 y + m2 s2 x s2 y + m3 s3 x s3 y = 0 , (16)
and expressed in vector form as
(s1 x, s2 x, s3 x)


m1 0 0
0 m2 0
0 0 m3




s1 y
s2 y
s3 y

 = 0. (17)
We introduce the following notation for the matrix
M =


m1 0 0
0 m2 0
0 0 m3

 . (18)
In order to complete the definition of vectors a and b assume
b M aT
= 0 (19)

5. LAGRANGE’S CASE OF THE THREE-BODY PROBLEM 167
which fixes the two directions of a and b in the plane orthogonal to m, and also
assume
a M aT
= b M bT
= 1 (20)
that normalizes the magnitudes of the two vectors. In the Appendix, we present
the computation of these vectors in terms of the masses and some of the properties
necessary to obtain the equations presented here.
Computation of the two inertia moments give us
I2 =
X
j
mj s2
j x = α2
+ β2
(21)
and
I1 =
X
j
mj s2
j y = γ 2
+ δ2
. (22)
Replacing (15) in (16), and taking into account the Equations (19) and (20),
produces
α γ + β δ = 0 . (23)
These α, β, γ, δ parameters become related to the principal inertia moments by
α β
γ δ
=
R2 0
0 R1
cos σ sin σ
− sin σ cos σ
, (24)
where σ is an angle, and
R2
1 = I1 , (25)
and
R2
2 = I2 . (26)
At time t = 0, R2 is positive semidefinite, and R1 is real. As time evolves R1 or
R2 can become zero, when the three particles are collinear, and a change of sign
could occur at that moment. The Equations (21)–(23) are identically satisfied by
this representation.
In the following, the quantities R1, R2, and σ are used as coordinates. One
computes, in the new coordinates
1
2
m1 ṡ2
1 + 1
2
m2 ṡ2
2 + 1
2
m3 ṡ2
3
= 1
2
(α̇2
+ β̇2
+ γ̇ 2
+ δ̇2
) = 1
2
Ṙ1
2
+ 1
2
Ṙ2
2
+ 1
2
(R2
1 + R2
2)σ̇2
(27)
and also
X
j
mj sj × ṡj = −


0
0
1

 2 R1 R2 σ̇. (28)

6. 168 E. PIÑA
The kinetic energy as a function of the coordinates then becomes
T = 1
2
Ṙ1
2
+ 1
2
Ṙ2
2
+ 1
2
(R2
1 + R2
2)σ̇2
− 2 R1 R2σ̇ω3 +
+ 1
2
R2
1ω2
1 + 1
2
R2
2ω2
2 + 1
2
(R2
1 + R2
2)ω2
3. (29)
To express the potential energy we need the relation among the distances p, q,
r between particles and the new coordinates. The squares of the distances between
particles are linear combinations of quadratic terms in the α, β, γ, δ parameters


p2
q2
r2

 =
m1 + m2 + m3
m1 m2 m3
MAf



R2
1 sin2
σ + R2
2 cos2
σ
R2
1 cos2
σ + R2
2 sin2
σ
(R2
2 − R2
1) sin σ cos σ


 , (30)
where A is the matrix
A =




m1 a2
1 m1 b2
1 2 m1 a1 b1
m2 a2
2 m2 b2
2 2 m2 a2 b2
m3 a2
3 m3 b2
3 2 m3 a3 b3



 , (31)
and f is the involution
f =


0 1 0
1 0 0
0 0 −1

 . (32)
In the Appendix, I present the way I have derived these equations, some rela-
tions to obtain some of the coefficients of matrix A and its inverse matrix, and some
of the properties used to obtain the results in this paper.
The Euler’s equations expressing the conservation of the angular momentum
are
d
dt


R2
1 ω1
R2
2 ω2
(R2
1 + R2
2)ω3 − 2 R1 R2σ̇

 =


−R2
1ω2ω3 + 2ω2 R1 R2σ̇
R2
2ω1ω3 − 2ω1 R1 R2σ̇
−ω1ω2(R2
2 − R2
1)

 (33)
and the Lagrange’s equations for the other three coordinates are
d
dt
(R2
1 + R2
2)σ̇ − 2 R1 R2ω3
= −(R2
1 − R2
2)
G(m1 + m2 + m3)
2
1
p3
,
1
q3
,
1
r3
×
×Af


2 sin σ cos σ
−2 sin σ cos σ
− cos2
σ + sin2
σ

 , (34)

7. LAGRANGE’S CASE OF THE THREE-BODY PROBLEM 169
R̈1 = R1σ̇2
− 2R2σ̇ω3 + R1ω2
3 + R1ω2
1 −
−R1 G(m1 + m2 + m3)
1
p3
,
1
q3
,
1
r3
×
×Af


sin2
σ
cos2
σ
− sin σ cos σ

 , (35)
R̈2 = R2σ̇2
− 2 R1σ̇ω3 + R2ω2
3 + R2ω2
2 −
−R2 G(m1 + m2 + m3)
1
p3
,
1
q3
,
1
r3
Af


cos2
σ
sin2
σ
sin σ cos σ

 . (36)
3. The Lagrange’s Case of the Three Body-Problem
The equilateral triangle solution of the problem of three bodies was found by Lag-
range in 1772, in a memoir on this problem that was honoured with the prize of the
Paris Academy.
An elementary proof of this case was obtained by Carathéodory (1993) and is
reproduced in Sommerfeld’s Mechanics (1964).
The three-body problem can be solved in closed and elementary form if one
assumes that the triangle formed by the three particles always remains similar to
itself.
In the new coordinates, these restrictions are the following:
σ̇ = 0, (37)
R1 = λR̃1, (38)
R2 = λR̃2, (39)
p = λp̃, q = λq̃, r = λr̃, (40)
where λ is the scale factor associated to the similarity, and where R̃1, R̃2, p̃, q̃, r̃
are constants corresponding to λ = 1.
The equations of motion expressing the conservation of the angular momentum
vector become
d
dt
(λ2
ω1) = −λ2
ω2ω3 , (41)
d
dt
(λ2
ω2) = λ2
ω1ω3 , (42)

8. 170 E. PIÑA
and
d
dt
(λ2
ω3) = −
R̃2
2 − R̃2
1
R̃2
1 + R̃2
2
λ2
ω1ω2, (43)
and the Lagrange’s equations are
d
dt
(λ2
ω3) =
R̃2
1 − R̃2
2
R̃1R̃2
!
1
λ
G(m1 + m2 + m3)
4
1
p̃3
,
1
q̃3
,
1
r̃3
×
×Af


2 sin σ cos σ
−2 sin σ cos σ
− cos2
σ + sin2
σ

 , (44)
λ̈ = λ(ω2
3 + ω2
1) −
−
1
λ2
G(m1 + m2 + m3)
1
p̃3
,
1
q̃3
,
1
r̃3
Af


sin2
σ
cos2
σ
− sin σ cos σ

 , (45)
λ̈ = λ(ω2
3 + ω2
2) −
−
1
λ2
G(m1 + m2 + m3)
1
p̃3
,
1
q̃3
,
1
r̃3
Af


cos2
σ
sin2
σ
sin σ cos σ

 . (46)
From the first two equations of motion we integrate
λ4
(ω2
1 + ω2
2) = C2
(constant). (47)
The observation that the same quantity appears in the left-hand side of Equa-
tions (43) and (44) results in the compatibility condition
2λ4
ω1ω2 = λG(m1 + m2 + m3)
R̃2
1 + R̃2
2
2R̃1R̃2
1
p̃3
,
1
q̃3
,
1
r̃3
×
×Af


2 sin σ cos σ
−2 sin σ cos σ
− cos2
σ + sin2
σ

 , (48)
and also by Equations (45) and (46) one has another compatibility
λ4
(ω2
2 − ω2
1) = λG(m1 + m2 + m3)
1
p̃3
,
1
q̃3
,
1
r̃3
×
×Af


cos2
σ − sin2
σ
− cos2
σ + sin2
σ
2 sin σ cos σ

 . (49)

9. LAGRANGE’S CASE OF THE THREE-BODY PROBLEM 171
Introducing the auxiliar variable ξ such that
λ2
ω1 = C sin ξ, λ2
ω2 = C cos ξ , (50)
the Equation (47) is satisfied identically and the left-hand sides of (48) and (49)
become respectively equal to C2
sin 2ξ and C2
cos 2ξ, and both proportional to λ,
according to the right-hand sides of that same equations. Adding both equations
squared results in an unwanted λ = constant.
To avoid such a case and in order to have λ variable one requires
C = 0, ω1 = ω2 = 0 . (51)
Equations (45) and (46) become
λ̈ = λω2
3 −
1
2 λ2
G(m1 + m2 + m3)
1
p̃3
,
1
q̃3
,
1
r̃3
Af


1
1
0

 , (52)
and Equations (48) and (49) impose the conditions
0 =
1
p̃3
,
1
q̃3
,
1
r̃3
Af


2 sin σ cos σ
−2 sin σ cos σ
− cos2
σ + sin2
σ

 , (53)
and
0 =
1
p̃3
,
1
q̃3
,
1
r̃3
Af


cos2
σ − sin2
σ
− cos2
σ + sin2
σ
2 sin σ cos σ

 . (54)
Then one requires the vector (1/p̃3
, 1/q̃3
, 1/r̃3
)Af to be orthogonal to the
other two vectors in (53) and (54); this is possible, for general σ, only when
(1/p̃3
, 1/q̃3
, 1/r̃3
) is parallel to vector (1, 1, 1), namely when
p̃ = q̃ = r̃ , (55)
and particles move equidistantly.
The motion of each particle follows the Newton’s two-body behaviour. Coordin-
ate λ is proportional to the radial distance to the center of mass, and Equation (52)
dictates the corresponding evolution in time. Since the direction of the angular
velocity is constant one has
ω3 = ψ̇ , (56)
and Equation (43) became Kepler’s second law
λ2
ψ̇ = constant. (57)

10. 172 E. PIÑA
Appendix
Vectors a and b were computed using the properties (14), (19) and (20). I started
from
M a = xa + ym, (A.1)
since it is orthogonal to b, and it results in
a = y
m1
m1 − x
,
m2
m2 − x
,
m3
m3 − x
. (A.2)
Using the orthogonality with m the quadratic equation for x appears:
0 = [(m1 + m2 + m3)2
− 2(m2m3 + m3m1 + m1m2)]x2
−
− [(m1 + m2 + m3)(m2m3 + m3m1 + m1m2) − 3m1m2m3]x +
+ (m1 + m2 + m3)m1m2m3. (A.3)
But only the larger root xa of this equation is used for vector a. As vector b
should obey a similar equation, the other root is used for b. One has
a = ya
m1
m1 − xa
,
m2
m2 − xa
,
m3
m3 − xa
(A.4)
and
b = yb
m1
m1 − xb
,
m2
m2 − xb
,
m3
m3 − xb
. (A.5)
Since these vectors are orthogonal to vector m some of its components should
be negative. Assuming m1 m2 m3, and taking into account also their mutual
orthogonality, we get
m1 xa m2 xb m3 (A.6)
and
a1 0, a2 0, a3 0, b1 0, b2 0, b3 0. (A.7)
Coordinates in the principal inertia frame are related by the equation
s1 x s2 x s3 x
s1 y s2 y s3 y
=
α β
γ δ
a1 a2 a3
b1 b2 b3
=
R2 0
0 R1
cos σ sin σ
− sin σ cos σ
a1 a2 a3
b1 b2 b3
. (A.8)

11. LAGRANGE’S CASE OF THE THREE-BODY PROBLEM 173
The relative positions of the particles are
s2 x − s3 x s3 x − s1 x s1 x − s2 x
s2 y − s3 y s3 y − s1 y s1 y − s2 y
=
α β
γ δ
a2 − a3 a3 − a1 a1 − a2
b2 − b3 b3 − b1 b1 − b2
, (A.9)
but using the explicit expressions for the components of the vectors a and b one
finds
s2 x − s3 x s3 x − s1 x s1 x − s2 x
s2 y − s3 y s3 y − s1 y s1 y − s2 y
=
r
m1 + m2 + m3
m1m2m3
α β
γ δ
−b1m1 −b2m2 −b3m3
a1m1 a2m2 a3m3
. (A.10)
From these equations one can compute the relative distances


p2
q2
r2

 =


(s2 x − s3 x)2
+ (s2 y − s3 y)2
(s3 x − s1 x)2
+ (s3 y − s1 y)2
(s1 x − s2 x)2
+ (s1 y − s2 y)2


=
m1 + m2 + m3
m1m2m3
×
×






m2
1 b2
1 m2
1 a2
1 −2 m2
1 a1 b1
m2
2 b2
2 m2
2 a2
2 −2 m2
2 a2 b2
m2
3 b2
3 m2
3 a2
3 −2 m2
3 a3 b3








α2
+ γ 2
β2
+ δ2
αβ + γ δ

 , (A.11)
that are used in the form (30).
The inverse of matrix (31) was necessary to obtain some results at Section 3.
One has
A−1
= (m1 + m2 + m3)


−b2 b3 −b3 b1 −b1 b2
−a2 a3 −a3 a1 −a1 a2
1
2
(a2b3 + a3b2) 1
2
(a3b1 + a1b3) 1
2
(a1b2 + a2b1)

 .
(A.12)
with the property
(1, 1, 0)A−1
= (1, 1, 1) (A.13)

12. 174 E. PIÑA
Acknowledgements
I acknowledge the partial financial support for this work from CONACYT, project
400200-5-1399 PE. I thank important remarks from Profs. D. Saari, F. Daicu, J.
Llibre, L. Jiménez and L. S. García Colín. I also thank the stimulating comments
by E. Lacomba, E. Pérez and T. de la Selva.
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