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Citation: Ortiz Ortiz, R.D.; Marín
Ramírez, A.M.; Oviedo de Julián, I.
Asymptotic Antipodal Solutions as
the Limit of Elliptic Relative
Equilibria for the Two- and n-Body
Problems in the Two-Dimensional
Conformal Sphere. Mathematics 2024,
12, 1025. https://doi.org/10.3390/
math12071025
Academic Editor: Ravi P. Agarwal
Received: 29 January 2024
Revised: 17 March 2024
Accepted: 27 March 2024
Published: 29 March 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
mathematics
Article
Asymptotic Antipodal Solutions as the Limit of Elliptic Relative
Equilibria for the Two- and n-Body Problems in the
Two-Dimensional Conformal Sphere
Rubén Darío Ortiz Ortiz 1,* , Ana Magnolia Marín Ramírez 1and Ismael Oviedo de Julián 2
1Grupo Ondas, Departamento de Matemáticas, Universidad de Cartagena, Sede San Pablo,
Cartagena de Indias 130001, Colombia; amarinr@unicartagena.edu.co
2Unidad Azcapotzalco, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana,
Cd. de México 02128, Mexico; iodj@azc.uam.mx
*Correspondence: rortizo@unicartagena.edu.co; Tel.: +57-312-669-1568
Abstract: We consider the two- and
n
-body problems on the two-dimensional conformal sphere
M2
R
,
with a radius
R>
0. We employ an alternative potential free of singularities at antipodal points. We
study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions
under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of
relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate
radius–mass relationship, these particles experience an unrestricted and free movement in alignment
with the geodesic of the canonical Killing vector field in
M2
R
. An even number of bodies with pairwise
conjugated positions, arranged in a regular
n
-gon, all with the same mass
m
, move freely on a
geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium.
Also, a center of mass formula is included. A relation is found for the relative equilibrium in the
two-body problem in the sphere similar to the Snell law.
Keywords: conformal sphere M2
R; the two-body problem; relative equilibria; antipodal points
MSC: 34D05; 70F15; 53Z05
1. Introduction
The study of the n-body problem on curved spaces, especially on spheres, introduces
unique challenges and phenomena not present in Euclidean spaces. The work of Borisov
et al. [
1
] provides essential insights into the dynamics of bodies on spaces of constant
curvature, foundational for our investigation.
In the realm of celestial mechanics, understanding the intricacies of motion in non-
Euclidean geometries is crucial. The research conducted by Ortega-Palencia and Reyes-
Victoria [
2
] and further expanded by Ortega Palencia et al. [
3
] delves into the n-body
problem in spaces of constant positive and negative curvature, offering valuable perspec-
tives that inform our approach.
Our study is also informed by the analysis presented by Diacu et al. [
4
], which
explores the n-body problem in spaces of constant curvature. Their findings provide
a broader context for our work, emphasizing the diversity of dynamics that different
geometric settings can induce.
The foundational principles laid out by Abraham and Marsden [
5
] in their seminal
work on mechanics provide the theoretical underpinnings for analyzing dynamical systems
in a geometric context, essential for our study.
Additionally, the exploration of antipodal equilibria in the two-dimensional sphere by
Ortega-Palencia et al. [
6
] and their further discussion in the arXiv preprint [
7
] offer impor-
tant precedents for our study. These works highlight the peculiarities of motion in curved
spaces and the significance of the chosen potential in determining the system’s behavior.
Mathematics 2024,12, 1025. https://doi.org/10.3390/math12071025 https://www.mdpi.com/journal/mathematics
Mathematics 2024,12, 1025 2 of 17
Through this research, we aim to build on these foundational studies, extending the
understanding of the n-body problem in curved spaces and exploring new facets of relative
equilibria and their stability. Our work is particularly inspired by the recent developments
in the field and seeks to contribute to the ongoing dialogue within the scientific community
regarding the dynamics of celestial bodies in non-traditional settings.
We want to point out the possibility of having a relative equilibrium in antipodal
points, as stated in [8].
In celestial mechanics with curvature, specifically in the case of a sphere, there are
two types of singularities related to the classical cotangent potential [
8
]. One refers to
collisions, while the other refers to antipodal points.
Various researchers, including those cited as [
9
,
10
], have explored the behavior of
collisions in spaces with non-zero constant curvature by applying classical regularization
methods from Newtonian mechanics. Their findings align with those previously obtained
in this domain, marking the beginning of investigations into dynamic types.
Additionally, in classical celestial mechanics, an alternative system that helps un-
derstand certain movements is studied, such as the planar three-body problem with an
attractive potential of 1
/r2
[
11
,
12
]. According to the Lagrange–Jacobi identity, where
¨
I=
4
H
, for any solution that is bounded, it is required to possess zero energy and maintain
a constant moment of inertia
I
. This condition holds when the energy at the initial state
is zero and its derivative is zero, the solution is bounded, as observed in the Newtonian
potential 1/r.
The work on how to carry out the study of behavior analysis around a geometric
singularity, that is, by antipodal points, is just beginning. In this work, a geometric method
is proposed to be able to study this type of singularity in the antipodal points for the
problem of two bodies in a conformal sphere of dimension two, with one alternative
potential that satisfies the Laplace–Beltrami equality [
13
] and preserves the periodic orbits
as the classic cotangent potential in the curved problem.
Our study focuses on the motion of two interacting point particles with masses
m1
and
m2
on the two-dimensional conformal sphere
M2
R=b
C=C∪{∞}
. Consider the complex
variable wand its conjugate ¯
w, upon which we define a conformal metric of the form
ds2=4R4
(R2+|w|2)2dw d ¯
w, (1)
where Ris a constant parameter determining the conformal properties of the metric.
We define the corresponding differentiable structure for this space in these coordinates
(for more details, refer to [14,15]).
By aligning the vector field within the Lie algebra that corresponds to the associ-
ated subgroup with the gravitational field in the cotangent space, we determine the
time-dependent algebraic criteria (
t
) required for solutions to achieve a state of rela-
tive equilibrium.
The techniques used in [8,16] are employed in this analysis.
In celestial mechanics on
M2
R
, it is common to use the cotangent potential as an exten-
sion of the Newtonian potential. This paper presents arguments justifying the introduction
of a suitable variant of this potential.
This document is organized as follows:
For an overview of the equations of motion, see Section 2, we have developed a
revised potential to tackle the issue, successfully overcoming the challenge of singularities
at antipodal points.
We express the equations of motion for the problem in complex coordinates in
M2
R
,
following the approach used in [8,16].
For insights into elliptic relative equilibria and their properties, consult Section 3:
utilizing the newly introduced potential, we formulate the algebraic equations that define
the elliptic relative equilibria in the general problem context.
Mathematics 2024,12, 1025 3 of 17
For a detailed exploration of the two-body problem, refer to Section 4: we classify
the relative equilibria for the two-body problem, following the approach of Borisov et al.
in [17], but considering the new potential.
Regarding the analysis of antipodal points, see Section 5: for any antipodal pair of
points in
M2
R
, we successfully derive limiting solutions by applying a regularized version
of the original equations of motion.
2. Dynamics Formulation and Conditions for Equilibrium
As discussed in [
8
], utilizing the stereographic projection method, the authors formu-
late the motion equations pertinent to this problem. This involves projecting the sphere
(with a radius of
R
) from its embedding in
R3
onto the complex plane
C
, which utilizes the
specified metric (1).
In Ref. [
5
], the classical motion equations for particles with positive masses
mk
,
k=1, . . . , n
are discussed, situated within a Riemannian or semi-Riemannian manifold
characterized by coordinates
xk
,
k=
1,
. . .
,
N
, endowed with a metric
(gij )
and an as-
sociated connection
Γi
jk
. The connection here, specifically the Levi-Civita connection, is
not arbitrary. It is determined by the specified metric (1) through the Christoffel sym-
bols, which are uniquely defined by the metric to ensure the connection is torsion-free
and metric-compatible.
Replacing the Levi-Civita connection with an arbitrary one could lead to a different
set of Christoffel symbols, altering the motion equations and, consequently, the particles’
trajectories and equilibrium states. Such changes would reflect a fundamental shift in the
geometric structure of the space in which the particles are moving.
These particles move under the influence of a pairwise-acting potential U.
Theorem 1. Consider a system of
n
particles with positive masses
mk
,
k=
1,
. . .
,
n
, situated
within a Riemannian or semi-Riemannian manifold characterized by coordinates
xk
,
k=
1,
. . .
,
N
,
and endowed with a metric
(gij )
as specified in (1) and an associated Levi-Civita connection
Γi
jk
.
The equations of motion for these particles under the influence of a pairwise-acting potential
U
are
given by
D˙
xi
dt =¨
xi+∑
l,j
Γi
lj ˙
xl˙
xj=∑
k
mkgik ∂U
∂xk, (2)
where
i=
1, 2,
. . .
,
N
. If the potential
U
is constant across a connected domain, the particle
trajectories align with the geodesics of the manifold defined by the metric (1).
Proof.
Given the manifold’s metric
(gij )
specified in (1) and the associated Levi-Civita
connection
Γi
jk
, the covariant derivative
D˙
xi
dt
accounts for the curvature dictated by (1)
and ensures that the acceleration
¨
xi
is defined in a coordinate-independent manner. The
equation of motion incorporates both the intrinsic geometry of the manifold, represented
by the Christoffel symbols
Γi
lj
, and the external force derived from the potential
U
. When
U
is constant, the term
∑kmkgik ∂U
∂xk
vanishes, indicating that the particles’ acceleration is
solely dictated by the manifold’s geometry as defined by (1), thus aligning their trajectories
with geodesics.
Remark 1. It is noted that in Equation (2) the covariant derivative of
˙
xi
is represented on the
left-hand side, whereas the gradient of the potential within the specified metric is depicted on the
right-hand side. Should the potential remain constant, particle trajectories align with geodesics. If
a set of particles moves along a geodesic solution curve, then the right-hand side of Equation (2)
vanishes, such that the solution of the potential is constant on one connected domain.
In this section, we introduce the motion equations for the
n
-body problem within the
conformal sphere denoted as M2
R.
Mathematics 2024,12, 1025 4 of 17
2.1. Introducing the Novel Potential
Exploring the
n
-body problem on the sphere, we reference a potential frequently
encountered in contemporary research, expressed as follows:
U(θ) = cot(θ), (3)
in which
θ
is the angle at the center of the sphere, delineated by the position vectors of the
particles. This potential is attractive because dU
dθ(θ) = −csc2(θ)<0, 0 <θ<π.
Here, we introduce a slight modification to the potential (3) and define it as the
new potential:
U(θ) = cotθ
2, (4)
which remains attractive over the entire interval (0, 2π).
Theorem 2. Consider a pair of particles with masses
mk
and
mj
positioned at locations
Ak
and
Aj
on the sphere
S2
R
. If
wk
and
wj
are their respective stereographic projections onto
M2
R
, then the
potential Uk j
Rexperienced by the particles is given by
Ukj
R=mkmj
1
Rcotdkj
2R=mkmj|R2+wk¯
wj|
R2|wk−wj|. (5)
Proof.
Given the particles at
Ak
and
Aj
on
S2
R
, the geodesic distance
dkj
between them is
related to the angle
θkj
at the origin by
dkj =Rθk j
. From the trigonometric identity on the
sphere, we have
cos(θkj ) = Ak·Aj
R2.
Applying the law of cosines for spherical trigonometry gives us
cotdkj
2R=sR2+Ak·Aj
R2−Ak·Aj
.
Using the stereographic projection, the dot product Ak·Ajin terms of wkand wjis
Ak·Aj=R2 2R2(wk¯
wj+¯
wkwj) + (R2− |wk|2)(R2−|wj|2)
(R2+|wk|2)(R2+|wj|2)!.
Substituting this into the cotangent expression and simplifying yields the potential
experienced by the particles as
Ukj
R=mkmj|R2+wk¯
wj|
R2|wk−wj|,
which completes the proof.
2.2. Equations of Motion
Designate
w∈(M2
R)n
as the collective position vector for
n
particles, each with mass
mi>0, situated at points wifor i=1, 2, . . . , n, within the space M2
R.
The set of singularities within
M2
R
for the
n
-body problem, as defined by the cotangent
relation, comprises the solutions to the equation
wj−wk=
0. Proceeding from this point,
the singular set is identified as
∆(C) = [
kj
∆(C)kj , (6)
Mathematics 2024,12, 1025 5 of 17
where
∆(C)kj =nw∈(M2
R)n|wk=wj,k=jo(7)
represents instances of mutual collisions between particles having masses mjand mk.
Theorem 3 (Dynamics of n-Body Problem in
M2
R
).Consider a set of
n
point particles with
masses
mi>
0situated at points
wi
within the Riemannian manifold
M2
R
, where
w∈(M2
R)n
represents the collective position vector of these particles. The dynamics of each particle is governed
by the following equation:
¨
wk−2¯
wk˙
w2
k
R2+|wk|2=(R2+|wk|2)2
4R6
n
∑
j=k
mj
(wj−wk)(R2+¯
wjwk)(R2+|wj|2)
|wj−wk|3|R2+¯
wjwk|, (8)
for k =1, 2, . . . , n.
Proof.
The proof follows from integrating the geodesic equations for
M2
R
and the gradients
computed from the potential
UR
, as detailed in the equations provided, into the dynamics
defined by the Vlasov–Poisson equations. The resulting second-order complex ordinary
differential equations dictate the motion of the particles, ensuring that the trajectories
remain within the specified domain, avoiding the singular set ∆(C).
Corollary 1 (Singular Set and Binary Collisions in
M2
R
).In the space
M2
R
, the singular set
∆(C)
for the n-body problem consists solely of points satisfying
wj−wk=
0for any pair of particles
j
,
k
where
j=k
, indicating binary collisions. Antipodal points satisfying
R2+wj¯
wk=
0are not
considered part of the singular set in terms of the equations of motion.
Proof.
The characterization of the singular set stems from the definition of mutual collisions
between particles, which are the only points where the potential
UR
becomes undefined
or singular. The exclusion of antipodal points from the singular set is due to the specific
structure of the potential and the forces it dictates, which remain well defined for antipodal
configurations.
3. Elliptic Relative Equilibria
The group
SU(
2
)
is given by the Lie algebra
su(
2
)
generated by three matrices; for our
purposes we only work with the complex matrix:
Theorem 4. In the two-dimensional conformal sphere
S2
R
, the one-parameter subgroup generated
by
exp(tX)
, where
X=i0
0−i
, induces a family of elliptic Möbius transformations. For any
point
w
in the disk
D2
R
of radius
R
in the complex plane, the trajectory under this transformation
is a circular path given by
f(t
,
w) = e2it w
, representing a rotation around the
z
-axis in
R3
. This
circular motion corresponds to the differential equation
˙
w=
2
iw
, signifying the dynamical system’s
relative equilibrium state.
Proof.
The exponential mapping applied to the line
{tX :t∈R}
yields the one-parameter
subgroup
exp(tX) = eit 0
0e−it
. The modulus of the off-diagonal elements,
|e±it |
, is equal
to 1, characterizing elliptic transformations since
e±it =±
1 for
t∈R
. The action of this
subgroup on a point
w
in
D2
R
is described by the Möbius transformation
f(t
,
w) = e2it w
,
which geometrically corresponds to a rotation around the
z
-axis. The trajectory of
w
under
this transformation is a circular path in
D2
R
, aligning with the differential equation
˙
w=
2
iw
.
This demonstrates that the system is in a state of relative equilibrium, as the trajectories are
circular paths dictated by the subgroup’s rotational action.
Mathematics 2024,12, 1025 6 of 17
The trajectories generated by the one-parameter subgroup
exp(tX)
in
C
, along with the
circular paths on the two-dimensional sphere situated in
R3
, are illustrated in
Figures 1and 2.
Figure 1. Isosceles solutions.
Figure 2. Right-angled solutions.
Now, we can start our analysis of the so-called solutions of elliptic relative equilibria,
derived from the influence of the canonical one-dimensional parametric subgroup within
SU(
2
)
, corresponding to the differential equation
˙
wk=
2
iwk
. There is a study of the action
of the subgroup in [2,8,17] for the classic cotangent potential.
Definition 1. An elliptic relative equilibrium for the
n
-body problem in
M2
R
is a solution
w(t) = (w1(t),w2(t),·· · ,wn(t))
of the equations of motion (8) that is invariant under the Killing
vector field ˙
wk=2i wk.
We now present the following lemma:
Theorem 5. In the case of
n
point particles, each with positive masses
mk
,
k=
1,
. . .
,
n
, moving
within
M2
R
, the requisite condition for
w(t)
to qualify as an elliptic relative equilibrium solution
under (8) is encapsulated by the subsequent rational complex functional equations, which are
time-dependent:
16R6(R2− |wk|2)wk
(R2+|wk|2)3=
n
∑
j=1,j=k
mj(|wj|2+R2)(R2+¯
wjwk)(wj−wk)
|R2+¯
wjwk||wj−wk|3(9)
where the velocity at each point is given by
˙
wk=
2
iwk
, where
wk
represents the value of the
k
-th
component of the vector w.
Proof.
Through direct calculations, we find that from equation
˙
wk=
2
i wk
, we have
¨
wk=−4wk. Substituting this into Equation (8) yields Equality (9).
The subsequent finding outlines prerequisites for the particles’ initial placements to
yield an elliptic relative equilibrium solution for Equation (9). These solutions depend on
the fixed points and their velocities.
Mathematics 2024,12, 1025 7 of 17
Corollary 2. In line with Theorem 5, the initial positions
wk,0
,
k=
1,
. . .
,
n
fulfill a necessary and
sufficient criterion to produce an elliptic solution for the system (8), which remains invariant under
the Killing vector field ˙
wk =2iwk, through this set of algebraic equations:
16R6(R2− |wk,0|2)wk,0
(R2+|wk,0|2)3=
n
∑
j=k
mj(wj,0 −wk,0)(R2+¯
wj,0wk,0 )(R2+|wj,0|2)
|wj,0 −wk,0|3|R2+¯
wj,0wk,0 |. (10)
Furthermore, the required velocity for each particle is determined by the equation
˙
wk,0 =
2
iwk,0
,
where
k
ranges from 1to
n
, where
wk,0
represents the initial value of the
k
-th component of the
vector w.
Proof.
Take
wk=wk(t) = e2it wk,0
to represent the impact of the Killing vector field
˙
wk=2iwk
at the initial position
wk,0
, corresponding to a velocity of
˙
wk,0 =
2
iwk,0
. By apply-
ing a multiplication of Equation (10) with
e2it
and incorporating the identity
¯
wj(t)wk(t) =
¯
wj,0e−2itwk,0 e2it =¯
wj,0wk,0, the resultant system is derived:
16 R6(R2−|wk|2)wk
(R2+|wk|2)3=
n
∑
j=1,j=k
mj(wj−wk)(R2+¯
wjwk)(R2+|wj|2)
|wj−wk|3|R2+¯
wjwk|.
This demonstrates that
wk(t)
serves as a solution to (9). To establish the reverse
argument, one simply sets
t=
0 within the framework of (9), thereby concluding the
corollary’s proof.
References [
2
,
8
] provide illustrative examples of the two- and three-body problems
situated on the conformal sphere M2
Rwith the classical cotangent potential.
4. Equilibria States in the Two-Body Problem Context
In the following segment, it is shown that relative equilibria exist in the context of
the two-body problem, specifically when employing potential (5), where the bodies are in
motion on the same circle or on two different circles. These results are consistent with the
findings in [1,2,8] for the classical cotangent potential.
Theorem 6. For the values of initial condition positions
w1,0 =α
and
w2,0 =β
(with 0
<α
,
β<R
), for the two-body problem with equal masses on the conformal sphere
M2
R
, the system (10)
yields only two types of relative equilibrium solutions:
1.
Under the condition
3√3R3
2≥m
, the particles position themselves diametrically opposite on
the same circle, with −β=α, termed as isosceles solutions (refer to Figure 1).
2.
Given the condition 2
R3≥m
, the particles occupy positions on separate circles, with
β=R(α−R)
α+R, creating a right angle, known as right-angled solutions (refer to Figure 2).
3. Both types of relative equilibria coincide for the value of α= (√2−1)R.
The solutions for βare in the interval (−R, 0).
Proof.
Firstly, in positions
w1,0
and
w2,0
, we note that the system (10) for the two-body
problem can be expressed as the following algebraic system:
16R6(|w1,0|2−R2)w1,0
(R2+|w1,0|2)3=m2(w2,0 −w1,0 )(R2+¯
w2,0w1,0 )(R2+|w2,0|2)
|w2,0 −w1,0|3|R2+w1,0 ¯
w2,0|,
16R6(|w2,0|2−R2)w2,0
(R2+|w2,0|2)3=m1(w1,0 −w2,0 )(R2+¯
w1,0w2,0 )(R2+|w1,0|2)
|w1,0 −w2,0|3|R2+¯
w1,0w2,0 |. (11)
From Corollary 2, by performing a suitable rotation, a condition both necessary and
sufficient for the presence of invariant elliptic solutions influenced by the Killing vector field
˙
wk=
2
iwk
dictates that the initial positions
w1,0 =α
and
w2,0 =β
(where 0
<α
,
β<R
)
Mathematics 2024,12, 1025 8 of 17
must conform to system (11). Since
R2+βα >
0, when substituted into the system,
it becomes
16R6(α2−R2)α
(α2+R2)3=m2(β−α)(βα +R2)(β2+R2)
|β−α|3|R2+αβ|=m2(β−α)(R2+β2)
|β−α|3,
16R6(β2−R2)β
(R2+β2)3=m1(α−β)(R2+αβ)(R2+α2)
|α−β|3|R2+αβ|=m1(α−β)(α2+R2)
|α−β|3. (12)
By equalizing the right- and left-hand sides of (12) and replacing
w1,0 =α
,
w2,0 =β
,
the following relation is derived:
m1(α−R)(α+R)(R2+β2)2α+m2(β−R)(β+R)(R2+α2)2β=0. (13)
For
m=m1=m2
, the solutions to the system outlined in (12), derived from
Equation (13), are as follows:
−β=α,β=R(α−R)
α+R,β=R21
α,β=−R(α+R)
α−R,
which is readily apparent.
1.
For the initial scenario where
−β=α
, the result is the isosceles configurations, and
by substituting it into any of the equations in system (12), we obtain the relation
R6(R2−α2)α3
(R2+α2)4=m
64.
We consider the function
F(α) = R6(R2−α2)α3
(R2+α2)4,
defined in the interval
[
0,
R]
. It has a maximum value of
3√3R3
128
at the critical point
√3R
3.
A simple analysis shows that there are isosceles solutions for this problem if
3√3R3
128 ≥m
64
,
which proves the first item.
2.
In the subsequent case, consider
ℓ
as the geodesic distance between
α
and
β=R(α−R)
α+R
.
First, let us establish that
β
is always smaller than
α
within the interval
hR(α−R)
α+R,αi
.
To determine which is smaller in the interval
hR(α−R)
α+R,αi
, we compare
R(α−R)
α+R
with
α
.
• If α=R, then R(α−R)
α+R=0, which is clearly smaller than αsince α=R>0.
•
If
α<R
, then
α−R<
0. This implies that
R(α−R)
α+R
is negative, and therefore,
smaller than α, which is positive.
• We analyze the difference α−R(α−R)
α+R:
α−R(α−R)
α+R=α(α+R)−R(α−R)
α+R
=α2+αR−Rα+R2
α+R
=α2+R2
α+R.
Since
α2+R2>
0 and
α+R>
0 for
α
,
R>
0, the difference is positive, indicating
that αis greater than R(α−R)
α+R.
Mathematics 2024,12, 1025 9 of 17
Therefore, in all cases within the interval
hR(α−R)
α+R,αi
, the point
R(α−R)
α+R
is always
smaller than α.
By defining the arc
Γ
that connects points
β
and
α
through a parametrization given by
x(t) = t, (14)
y(t) = 0, (15)
over the interval R(α−R)
α+R,α, the geodesic distance ℓcan be calculated as follows:
ℓ=ZΓdℓ=2R2Zα
R(α−R)
α+R
dt
t2+R2(16)
=2Rarctanα
R−arctanR(α−R)
R(α+R) (17)
=2Rarctan α
R−α−R
α+R
1+α
Rα−R
α+R!(18)
=2Rarctan(1) = πR
2. (19)
Now, let us consider the asymptotic behavior of ℓas αapproaches Rand 0:
1. As
α
approaches
R
, the geodesic distance
ℓ
should theoretically approach 0 as the
two points converge.
2. As
α
approaches 0, the point
β
approaches
−R
, and the geodesic distance
ℓ
should
approach the maximum possible value on the circle, which is
πR
, corresponding to
half the circumference of the circle of radius R.
This analysis provides a deeper understanding of the geometric configuration of the
system and the behavior of the geodesic distance under different conditions.
This delineates the process and calculation of length
ℓ
for the specified path
Γ
within
the two-dimensional manifold M2
R.
Conversely, it is established that
ℓ=θR
, with
θ
denoting the angle between the
specified points in
M2
R
. Consequently,
θ=π
2
, classifying the solution as right-angled.
In this case, by substituting
β=R(α−R)
α+R
into any of the equations in the system (12),
we obtain the relation R4(R2−α2)α
(R2+α2)2=m
8.
We consider the function
G(α) = R4(R2−α2)α
(R2+α2)2,
defined in the interval
[
0,
R]
. It has a maximum value of
R3
4
at the critical point
(√2−
1
)R
. Once again, a simple analysis shows that there are right-angled solutions
for this problem if R3
4≥m
8, which proves the second item.
This concludes the proof.
In [
1
], in theorem 4.3 there is a discussion about the stability of the solutions obtained.
In this theorem, we establish a connection between the relative equilibria for the
two-body problem and Snell’s law of geometric optics.
Theorem 7. If two particles in the two-body problem are in relative equilibrium and the given
substitution conditions are met, then the relationship between their masses and positions is analogous
to Snell’s law, with the particularity that the “indices of refraction” are the masses and the “angles
of refraction” are related to the positions of the particles.
Mathematics 2024,12, 1025 10 of 17
Proof.
Consider the following equation derived from the polynomial equation for the
two-body problem (see Equation (13)):
m1α(α2−R2)
(α2+R2)2=−m2β(β2−R2)
(β2+R2)2(20)
Here,
tan(θ1) = α
R
and
tan(θ2) = β
R
. Let us substitute
α=Rtan(θ1)
and
β=Rtan(θ2)
into Equation (20):
m1Rtan(θ1)(R2tan2(θ1)−R2)
(R2tan2(θ1) + R2)2=−m2Rtan(θ2)(R2tan2(θ2)−R2)
(R2tan2(θ2) + R2)2
Simplifying the equation, we obtain
m1tan(θ1)(tan2(θ1)−1) = −m2tan(θ2)(tan2(θ2)−1)
Using the trigonometric identity
tan2(θ)−
1
=−cos(
2
θ)
, we can rewrite the
equation as
m1tan(θ1)cos(2θ1) = −m2tan(θ2)cos(2θ2)
Now, using the double-angle formula
sin(
2
θ) =
2
sin(θ)cos(θ)
, we can express
tan(θ)cos(2θ)in terms of sin(4θ):
m1sin(4θ1) = −m2sin(4θ2)
This relationship is analogous to Snell’s law, where the masses act as indices of refrac-
tion and the angles are related to the positions of the particles, thus proving the theorem.
5. Exploring Antipodal Solutions: Approaching through the Lens of Relative Equilibria
Within this segment, our focus shifts to the examination of antipodal solutions that
emerge in the context of the two-body problem when considering scenarios involving
bodies of identical mass.
Antipodal Solutions as a Limit of Relative Equilibria for Equal Masses
According to Theorem 6, there are two types of relative equilibria for equal masses,
and each type gives rise to a different solution as αapproaches R.
Corollary 3. In the context of right-angled relative equilibria, as
α
approaches
R
a state of equilib-
rium for the system is achieved. In this state, one particle is positioned at the origin of coordinates,
while the other follows a geodesic circular path with a radius of
|w|=R
, moving at a velocity of
˙
w=2iw.
Proof.
The assertion is derived from the observation that as
α
tends towards
R
,
β
converges
to 0, as demonstrated by
lim
α→Rβ=lim
α→R
R(α−R)
α+R=0,
as αnears R.
Next, we examine the scenarios involving conjugate (antipodal) points within the
two-body problem, considering them as limiting cases of isosceles configurations. This
analysis becomes particularly relevant when
α
nears
R
and the criterion for the radius–mass
relationship, 3√3R3
2≥m, is met.
We present the following result, which demonstrates that the family of relative equi-
libria converges to the geodesic circle (equator), acting as a unified relative equilibrium
since it remains invariant under the canonical Killing vector field. However, while it does
Mathematics 2024,12, 1025 11 of 17
not serve as a solution to the overarching system (8), it constitutes a geodesic within
M2
R
,
where the potential exerted along this trajectory is nullified.
Nevertheless, a solution to the regularized system derived from (8) exists on
this geodesic.
Theorem 8. In the limit as
α
approaches
R
, the family of isosceles relative equilibria becomes an
equilibrium with equal masses, where the particles are located at antipodal points. These particles
move along the geodesic circle where
|w|=R
, possessing a velocity given by
˙
w(t) =
2
iw(t)
, and
behave as a single (limit) relative equilibrium.
Proof.
We note that for the two-body problem involving equal masses, the potential is
described as follows:
UR(w,¯w) = m2R2+w2¯
w1
R2|w2−w1|. (21)
Let
w1(t) = αe2it
and
w2(t) = −αe2it
constitute the elements of the relative
equilibrium
:
w(t) = (αe2it ,−αe2it ).
Substituting these values into Equation (21), we obtain
UR(w,¯w) = m2
R2|R2+ (−αe2it )(αe−2it)|
|2αe2it |
=m2
2α
(R2−α2)
R2.
This shows that the potential along the family of relative equilibria decreases and
converges to zero as αapproaches R.
If we denote z1(t) = Re2it and z2(t) = −Re2it as the components of the function:
z(t) = (z1(t),z2(t)) = (Re2it ,−Re2it), (22)
where each coordinate maps out the geodesic circle with radius
R
, moving at a velocity of
˙
z(t) = 2iz(t), then
lim
α→Rw(t) = lim
α→R(αe2it,−αe2it) = (Re2it,−Re2it ) = z(t).
Given the potential’s continuity and that conjugate points cease to be singularities for
the potential, it follows that
UR(z,¯z)=lim
α→RUR(w,¯w)
=lim
α→Rm2
2αR2R2−α2=0.
This illustrates that the potential becomes null along the geodesic circle (22).
Inserting
β=−Rand α=R
into Equation (13) confirms it as a solution to both
that equation and the regularized system derived from the relative equilibria condition
system (12), specified as follows:
Mathematics 2024,12, 1025 12 of 17
16R6α(α2−R2)
(R2+α2)3|R2+αβ|=m2(β−α)(R2+βα)(R2+β2)
|β−α|3,
16R6β(β2−R2)
(R2+β2)3|R2+αβ|=m1(α−β)(R2+αβ)(R2+α2)
|α−β|3.
Clearly, the function (22) does not resolve the general system (8). Nonetheless, through
straightforward substitution it fulfills the criteria of the regularized system derived from
(8) by circumventing singularities, including those arising from conjugate antipodal points:
¨
w1−2¯
w1˙
w2
1
R2+|w1|2!|R2+¯
w2w1|
(R2+|w1|2)2=m2(w2−w1)(R2+¯
w2w1)(R2+|w2|2)
4R6|w2−w1|3,
¨
w2−2¯
w2˙
w2
2
R2+|w2|2!|R2+¯
w1w2|
(R2+|w2|2)2=m1(w1−w2)(R2+¯
w1w2)(R2+|w1|2)
4R6|w1−w2|3.
This concludes the proof.
Corollary 4. In the limit as particles approach antipodal positions on the sphere, the interaction
potential between them tends to zero, and the particles behave as if they are in a relative equilibrium
state without mutual influence. This occurs along the geodesic circle where
|w|=R
, with the
particles possessing a velocity given by ˙
w(t) = 2iw(t).
Proof.
From the theorem’s proof, we note that as
α
approaches
R
, the potential
UR(w
,
¯w)
converges to zero. This indicates that the interaction force between the particles diminishes
and vanishes in the limit. Therefore, when the particles are at antipodal points, they move
along the geodesic circle without influencing each other, maintaining a constant velocity
˙
w(t) = 2iw(t), characteristic of a relative equilibrium in the system.
Corollary 5. Any pair of antipodal point particles with equal masses, satisfying the condition
on the radius–mass relation
3√3R3
2≥m
, can move freely along the geodesic associated with the
(finite) direction of the Killing vector field in the two-dimensional sphere M2
R.
Proof.
Given system (12) and polynomial Equation (13), we analyze the case where
m=m1=m2
. For the solutions outlined, we particularly focus on the scenario where
−β=α. Substituting this into the system yields the relation
R6(R2−α2)α3
(R2+α2)4=m
64. (23)
Consider the function
F(α) = R6(R2−α2)α3
(R2+α2)4
defined in the interval
[
0,
R]
. It reaches a
maximum at α=√3R
3with a value of 3√3R3
128 .
The condition for the existence of solutions is
3√3R3
128 ≥m
64
. Since this aligns with the
condition provided in the corollary,
3√3R3
2≥m
, the statement is proven for the specified
configuration.
6. Relative Equilibrium for the n-Body Problem
Corollary 6. An even number of point particles with pairwise conjugated positions
wj(t)
, arranged
in a regular
n
-gon and with equal masses, move freely on a geodesic with velocities
˙
wj=
2
iwj(t)
.
This geodesic movement behaves as a relative equilibrium.
Mathematics 2024,12, 1025 13 of 17
Proof.
We observe that from Remark 1, a necessary and sufficient condition for a set of
n
particles with masses
mk
to move along a geodesic solution curve, where the right-hand
side of Equation (2) vanishes along such a solution, is given by the system of equations, we
can approximate R2+¯
wjwkby R2+¯
wjwk+ϵfor some sufficiently small ϵ:
0=
n
∑
j<k
mj
(wj−wk)(R2+¯
wjwk+ϵ)(R2+|wj|2)
|wj−wk|3|R2+¯
wjwk+ϵ|=
n
∑
j<k
mj
(wj−wk)(R2+¯
wjwk+ϵ)
|wj−wk|3|R2+¯
wjwk+ϵ|, (24)
where the sum is taken over j,k=1, 2, · · · ,n.
Therefore, for the
n
-body problem with equal mass particles moving along a geodesic
circle, we have the equality
0=
n
∑
j<k
(wj−wk)(R2+¯
wjwk+ϵ)
|wj−wk|3|R2+¯
wjwk+ϵ|. (25)
Using the same method of applying a rotation for conjugate points into the line as
in Theorem 8for an
n
-even number of equal masses arranged in a regular
n
-gon along a
geodesic, and considering that the acting forces for any elements
wk
and
wj
cancel in pairs,
we obtain the equation
(wj−wk)(R2+¯
wjwk+ϵ)
|wj−wk|3|R2+¯
wkwj+ϵ|+(wk−wj)(R2+¯
wkwj+ϵ)
|wk−wj|3|R2+¯
wjwk+ϵ|=0. (26)
This leads us to the result, which complements the one obtained in Theorem 5 in [
4
]
for an odd number of particles with equal masses moving along a geodesic.
Remark 2. In remark 3 in [
4
], the authors parameterize the relative equilibria using a relation
between angular velocity, the equal masses of the bodies, and their positions on the sphere. This
leads to the observation that the velocity approaches infinity as the particles tend to the equator.
However, we believe it is not physically possible to speak of this scenario in our universe for the
following reasons:
1.
When the angular light velocity
˙
w(
0
) = ±
2
iα∞
is reached for the initial condition
w(0) = ±α∞
,
the masses corresponded to elementary particles and such quantities vanish.
2. The annular region
Ω={w∈M2
R|α∞<|w|<R+α∞}
containing the equator
|w|=R
does not admit any other real relative equilibria for
this problem
.
7. Center of Mass on a Two-Dimensional Sphere
In the study presented in [
6
], the concept of the center of mass for particles on a
spherical surface
S2
R
was explored. Here, we formalize this concept into a theorem and
provide a demonstration of the underlying principle.
Theorem 9. Given two particles on
S2
R
, it is always possible to map them to the equator determined
by their intersection with the
xy
-plane using an isometry, which is a composition of two rotations.
This mapping allows the problem of finding the center of mass on
S2
R
to be reduced to calculating
it on the one-dimensional sphere
S1
R
. When considering the stereographic projection of
S1
R
onto
the real axis, a point
P(x
,
y)
on
S1
R
corresponds to a point
u
on the real axis, where
u=Rx
R−y
and
P(0, R) = ∞. The inverse projection is given by
P−1(u) = 2R2u
u2+R2
R(u2−R2)
u2+R2!.
Mathematics 2024,12, 1025 14 of 17
Proof.
The proof involves demonstrating the isometry that maps the points on
S2
R
to
S1
R
.
First, we consider two rotations: one that aligns the axis passing through the two points
with the
z
-axis, and another that rotates the points to the
xy
-plane. This process does
not alter the relative distances between points, preserving the center of mass due to the
invariance under isometries.
Next, we consider the stereographic projection from
S1
R
to the real line. A point
P(x
,
y)
on
S1
R
is mapped to a point
u
on the real axis using the formula
u=Rx
R−y
. The
inverse mapping,
P−1(u)
, is necessary to retrieve the original points on the sphere from
their projections.
By applying these transformations, the calculation of the center of mass on
S2
R
is
effectively reduced to a simpler problem on
S1
R
and further to a calculation on the real line
via stereographic projection.
7.1. Arc Length from the South Pole
Theorem 10. Let
S1
R
be a sphere of radius
R
. The arc length
s
, extending from the south pole to a
specific point (x,y)on S1
R, whose stereographic projection is u, is calculated by
s=2Rarctanu
R.
More generally, the arc length
s
from point
Q1(x1
,
y1)
to point
Q2(x2
,
y2)
on
S1
R
, whose
stereographic projections are u1and u2, respectively, (with u1<u2), is given by
s=2Rarctanu2
R−arctanu1
R.
Proof.
For the first case, consider the integral representing the arc length from the south
pole to a point whose stereographic projection is u:
s=2R2Zu
0
1
t2+R2dt.
Using the definite integral of the arctangent function, we obtain
s=2R21
Rarctant
Ru
0
=2Rharctanu
R−arctan(0)i=2Rarctanu
R.
For the general case, the arc length between two points
Q1
and
Q2
is calculated by
considering the difference between the arc lengths from the south pole to each point:
s=2Rarctanu2
R−arctanu1
R.
This is directly deduced from the additive property of the arc length and the arctangent
function, providing a consistent way to calculate the distance along the sphere between
any two given points.
7.2. Center of Mass
Theorem 11. Consider two masses
mk
,
k=
1, 2, located at points
Qk
,
k=
1, 2, respectively. Let
Qc(xc
,
yc)
be the point that satisfies the lever rule on the sphere
m1s1=m2s2
. Then, the following
equality holds:
arctanuc
R=1
m1+m2m1arctanu1
R+m2arctanu2
R. (27)
Mathematics 2024,12, 1025 15 of 17
Proof.
Given the lever rule on the sphere
m1s1=m2s2
, we can express this relationship as
2Rm1arctanuc
R−arctanu1
R=2Rm2arctanu2
R−arctanuc
R.
By simplifying this equation, we isolate arctanuc
R:
2Rm1arctanuc
R−2Rm1arctanu1
R=2Rm2arctanu2
R−2Rm2arctanuc
R.
2R(m1+m2)arctanuc
R=2Rm1arctanu1
R+2Rm2arctanu2
R.
arctanuc
R=1
m1+m2m1arctanu1
R+m2arctanu2
R.
This concludes the proof, demonstrating that the point
Qc(xc
,
yc)
indeed satisfies the
lever rule with respect to the given masses and positions on the sphere.
Theorem 12. Consider,
n
point masses
mk
,
k=
1,
. . .
,
n
situated at the coordinates
(xk
,
yk
,
zk)
,
k = 1, . . ., n
on
S2
R
, all lying on the same geodesic. Further, let their stereographic projections be
w1,w2, . . . , wnin C. Consider the equation
arctanwc
R=1
m
n
∑
k=1
mkarctanwk
R,
If we multiply both sides by R and take the limit as R →∞, we obtain
wc=1
m
n
∑
k=1
mkwk. (28)
This matches the expression for the center of mass in the Euclidean complex plane, specifically
within the complex plane (or R2), characterized by a Euclidean metric and null curvature.
Proof.
In a broader context, consider
n
point masses
mk
,
k=
1,
. . .
,
n
situated at the coordi-
nates
(xk
,
yk
,
zk)
,
k=
1,
. . .
,
n
on
S2
R
, all lying on the same geodesic. Let their stereographic
projections be
w1
,
w2
,
. . .
,
wn
in
C
. Then, if
wc
is their spherical center of mass, the following
relationship is satisfied:
arctanwc
R=1
m
n
∑
k=1
mkarctanwk
R,
where m=∑n
k=1mk.
Consider the equation given in the theorem. We multiply both sides by
R
to facilitate
the application of L’Hôpital’s rule:
R·arctanwc
R=R
m
n
∑
k=1
mkarctanwk
R.
As
R→∞
, we encounter an indeterminate form
∞·
0 on both sides of the equation.
To resolve this, we apply L’Hôpital’s rule by differentiating the numerator and the denomi-
nator with respect to
R
. This involves computing the derivative of
arctanw
R
with respect
to R, which gives −w
R2+w2.
Applying L’Hôpital’s rule, we have
lim
R→∞
arctanwc
R
1
R
=1
m
n
∑
k=1
mklim
R→∞
arctanwk
R
1
R
.
Mathematics 2024,12, 1025 16 of 17
After applying the rule and simplifying, we find
lim
R→∞
R2
R2+w2
c
wc=1
m
n
∑
k=1
mklim
R→∞
R2
R2+w2
k
wk,
which simplifies to the desired result as R→∞:
wc=1
m
n
∑
k=1
mkwk,
matching the expression for the center of mass in the Euclidean complex plane.
8. Conclusions
The document focuses on the relationship between spherical geometry and the n-body
problem on a two-dimensional conformal sphere,
M2
R
. The main conclusions drawn from
the results, theorems, and corollaries are summarized below:
1.
Radius–mass relationship: A specific condition related to the radius–mass relationship
of the particles is established. Under this condition, two antipodal point particles with
positive mass move unrestrictedly in a state of relative equilibrium along a geodesic
associated with the canonical Killing vector field in M2
R.
2.
Relative equilibrium: The exploration of how variations in the sphere’s radius and
the particles’ masses affect the behavior of relative equilibrium provides a deeper
understanding of the relationships between the system’s parameters and its dynamics.
3.
Geodesic movement: It is concluded that an even number of point particles, with pair-
wise conjugated positions and arranged in a regular n-gon with equal masses, move
freely on a geodesic with particular velocities, behaving as a relative equilibrium.
4.
Center of mass on the sphere: An approach to determining the center of mass on a
two-dimensional sphere is discussed, using stereographic projection and relating it to
the calculation of the center of mass on the one-dimensional sphere S1
R.
5.
Arc length: A method is provided to calculate the arc length from the south pole to a
specific point on
S1
R
, crucial for understanding the geometry involved in the n-body
problem on a sphere.
Author Contributions: Writing—review and editing, R.D.O.O., A.M.M.R. and I.O.d.J. All authors
have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Universidad de Cartagena, grant number 014-2022. The
APC was funded by the Universidad de Cartagena.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: This article does not rely on the analysis of numerical or experimental
datasets, given its focus on theoretical mathematical research. Therefore, no new data were created or
analyzed during the study.
Acknowledgments: This work was supported by the Universidad de Cartagena. We dedicate this
work to the memory of our dear friend and co-author, Pedro Pablo Ortega Palencia, who worked
hard on solving problems of celestial mechanics with curvature [
2
,
3
] and played a crucial role in the
publication of this work.
Conflicts of Interest: The authors declare no conflicts of interest. Our study is purely theoretical
and does not involve any empirical data that could potentially lead to conflicts related to data collec-
tion, analysis, or interpretation. We have maintained complete academic integrity and impartiality
throughout the research process. Our findings and conclusions are presented without any undue
influence from personal interests or funding sources.
Mathematics 2024,12, 1025 17 of 17
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