On the planar central configurations of rhomboidal and triangular four- and five-body problems
M. Shoaib A. R. Kashif I. Szucs-Csillik



Not to appear in Nonlearned J., 45.

Rhomboidal and triangular four- and five-body problems
Shoaib et al.



Abu Dhabi Men's College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, United Arab Emirates

Department of Mathematics, Capital University of Sciences and Technology, Islamabad, Pakistan

Astronomical Institute of the Romanian Academy, Cireşilor 19, 400487 Cluj-Napoca, Romania



We consider a symmetric five-body problem with three unequal collinear masses on the axis of symmetry.
The remaining two masses are symmetrically placed on both sides of the axis of symmetry.
Regions of possible central configurations are derived for the four- and five-body problems.
These regions are determined analytically and explored numerically.
The equations of motion are regularized using Levi-Civita type transformations and then the phase space is investigated for chaotic and periodic orbits by means of Poincaré surface of sections.






§ INTRODUCTION


Studying the four- and five-body problems are very important as (for example) it is known that approximately two-third of the stars in our Galaxy are part of
multi-stellar systems.
This paper focuses on Central Configurations (CC), a very important topic in  the gravitational n-body problems (cf.
<cit.>, <cit.>, <cit.> and <cit.>). 
A central configuration, subject of this paper, in the n-body problem is a particular position of the n particles, where the position and acceleration vectors are proportional with the same constant of proportionality (see <cit.> and <cit.>).
CC are useful in understanding the nature of solutions near singularities. They are also useful in providing explicit homographic solutions of the equations of motion and families of periodic solutions.
 






















In this paper, we explore the central configurations of rhomboidal and triangular four- and five-body problems.
In 2012, Bakker and Simmons gave a linear stability analysis of a rhomboidal four-body
problem and have shown that isolated binary collisions are regularizable at
the origin. <cit.> had earlier studied the same problem to
regularize binary collisions. In a 1993 paper, Lacomba and Perez-Chavela
studied it's escape and capture orbits.
Furthermore, <cit.> studied the existence and linear stability of periodic orbits
for equal masses and <cit.> used the Poincaré sections to find regions of stability
for the rhomboidal four-body problem.  <cit.> show that if the diagonals of a four-body convex central configuration are perpendicular, then the configuration must be a rhombus. We study a similar problem in theorem 3 in which we explicitly show the regions where such a configuration exists. Other studies on the rhomboidal four-body problem include
<cit.>,  <cit.>, <cit.> and <cit.>.

 <cit.>  studies a restricted five-body problem having
three bodies of equal mass, m, placed on the vertices of the equilateral
triangle; they revolve in the plane of the triangle around their
gravitational centre in circular orbits under the influence of their mutual
gravitational attraction; at the centre a mass of β m is present where
β≥ 0. A fifth body of negligible mass compared to m moves in the
plane under the gravitational attraction of the other bodies. They discuss the
existence and location of the Lagrangian equilibrium points and show that
there are 9 Lagrangian equilibrium points.

<cit.>  discusses  the relative equilibria for a special case of the 5-body
problem. He considers a configuration which consists of four bodies at the
vertices of a rhombus, with opposite vertices having the same mass, and a
central body. He shows that there exist a one parameter family of degenerate
relative equilibria where the four equal masses are positioned at the
vertices of a rhombus with the remaining body located at the centre.
 <cit.> discuss the same problem
but in the post Newtonian field of Manev. They prove the existence of
mono-parametric families of relative equilibria for the masses (m_0,1,m,1,m), where m_0 is the central mass, and prove that the
Manev five-body problem with masses (m_0,1,m,1,m) admits relative
equilibria regardless of the value of the mass of the central body. A
continuum of such equilibria (as in the Newtonian field) does not exist in
the Manev rhomboidal five-body problem.
<cit.> found infinitely many periodic orbits for a rhomboidal
five-body problem when two pairs of masses are placed at the vertices of a
rhombus and a stationary fifth mass is placed at the origin.
Moreover, <cit.> derive regions of central configurations for the same
model. <cit.> and <cit.> study various aspects of
restricted rhomboidal five-body problem with four positive masses on
the vertices of a rhombus and the fifth infinitesimal mass in the plane of the
four masses. <cit.> study the central configurations of a rhombus like five-body problem with four equal masses and a fifth infinitesimal mass. <cit.> and <cit.> derive regions of central configuration for a special case of rhomboidal five-body problem with two axes of symmetry and two pairs of equal masses and fifth mass at the centre of mass. They show the existence of regions with a continuous family of central configurations. In this paper, in theorem 3, we show that if the fifth mass is moved from the centre of mass to anywhere on the axes of symmetry then there are no central configurations. Other studies on the rhomboidal and triangular five-body problem include <cit.>, <cit.> and <cit.>

In the present paper we set up a five-body problem which has three collinear
unequal masses on the axis of symmetry. The remaining two masses are
symmetrically placed on both sides of the axis of symmetry. First four of the
masses effectively form a rhombus, with the fifth mass placed anywhere on the axis of symmetry.
For a second case, the five masses will form a triangle, with
one of the masses moved up on the axis of symmetry. As a particular case, we also discuss a four-body configuration with a zero central mass. The
paper is organized as follows: in section 2 we set up the problem and list
the main results, in sections 3, 4 and 5 we give analytical proofs and numerically
explore these results. In section 6, we give the Hamiltonian
equations of motion, regularize the singularities and then use
Poincaré surface of sections to investigate the phase space, analyzing the
effect of changing the mass of the central body on the stability of rhomboidal five-body problem. Conclusions are given in section 7.






§ MAIN RESULTS




Let us consider n-1 mass points (m_0, m_1, ..., m_n-1), m_i>0, i=0,n-1, the position vectors of n-1 mass points r_i, i=0,n-1, and the inter-body distances r_ij, i,j,=0,n-1.

Using <cit.> notation, it is known that this n-body system forms a planar non-collinear central configuration if the following holds:

    f_ij=∑_k=0,k≠ i,j^n-1m_k(R_ik-R_jk)Δ _ijk=0,

where R_ij=1/r_ij^3 and Δ _ijk=(r_i-r_j)∧
(r_i-r_k). The Δ _ijk represent the areas of the
triangle determined by (r_i-r_j) and (r_i-r_k). For n=5 we obtain the general non-collinear five-body problem with the following ten CC equations:

    f_01   =   m_2(R_02-R_12)Δ _012+m_3(R_03-R_13)Δ
    _013+  
       +   m_4(R_04-R_14)Δ _014,  
    
    f_02   =   m_1(R_01-R_21)Δ _021+m_3(R_03-R_23)Δ
    _023+  
       +   m_4(R_04-R_24)Δ _024,  
    
    f_03   =   m_1(R_01-R_31)Δ _031+m_2(R_02-R_32)Δ
    _032+  
       +   m_4(R_04-R_34)Δ _034,  
    
    f_04   =   m_1(R_01-R_41)Δ _041+m_2(R_02-R_42)Δ
    _042+  
       +   m_3(R_03-R_43)Δ _043,  
    
    f_12   =   m_0(R_10-R_20)Δ _120+m_3(R_13-R_23)Δ
    _123+  
       +   m_4(R_14-R_24)Δ _124,  
    
    f_13   =   m_0(R_10-R_30)Δ _130+m_2(R_12-R_32)Δ
    _132+  
       +   m_4(R_14-R_34)Δ _134,  
    
    f_14   =   m_0(R_10-R_40)Δ _140+m_2(R_12-R_42)Δ
    _142+  
       +   m_3(R_13-R_43)Δ _143,  
    
    f_23   =   m_0(R_20-R_30)Δ _230+m_1(R_21-R_31)Δ
    _231+  
       +   m_4(R_24-R_34)Δ _234,  
    
    f_24   =   m_0(R_20-R_40)Δ _240+m_1(R_21-R_41)Δ
    _241+  
       +   m_3(R_23-R_43)Δ _243,


    f_34   =   m_0(R_30-R_40)Δ _340+m_1(R_31-R_41)Δ
    _341+  
       +   m_2(R_32-R_42)Δ _342.



Dziobek equations <cit.> for a five-body problem when
the five masses have position vectors 𝐫_0=(0,w), 𝐫
_1=(-1,0), 𝐫_2=(0,s), 𝐫_3=(1,0), 𝐫
_4=(0,-t), where s,t,w∈ℝ are
@̌mathfonts

    f_01   :=    m_1(R_03-R_13)Δ _013+m_2(R_02-R_12)Δ
    _012+  
       +   m_4(R_04-R_14)Δ _014=0,  
    
    f_12   :=    m_0(R_10-R_20)Δ_120+m_1(R_13-R_23)Δ_123+
    
       +   m_4(R_14-R_24)Δ_124=0,  
    
    f_14   :=    m_0(R_10-R_40)Δ _140+m_1(R_13-R_43)Δ
    _143+  
       +    m_2(R_12-R_42)Δ _142=0.




Using the definition of R_ij, Δ _ijk and 𝐫_i (i=0,1,2,3,4) we obtain

    R_01   =   R_03=1/(1+w^2)^3/2,   R_02=1/
    |s-w|^3,  
    
    R_13   =   1/8,R_12=R_23=1/(1+s^2)^3/2,R_24=
    1/(s+t)^3,  
    
    R_14   =   R_34=1/(1+t^2)^3/2,   R_04=1/
    |w+t|^3,

and

    Δ _ijk   =   -Δ _jik=-Δ _ikj=-Δ _kji,  
    Δ _ijk   =   Δ _jki=Δ _kij,  
    Δ _ijk   =   0,   if   i=j   or  i=k  or  j=k, 
    Δ _012   =   Δ _023=w-s,  Δ _013=2w,  
    Δ _014   =   Δ _043=w+t,  Δ _132=2s,  Δ
    _143=2t,  
    Δ _142   =   Δ _324=s+t,  Δ _024=0.

Using the symmetry of the problem through (<ref>-<ref>), substituting m_1=m_3 in Eqs. <ref>-<ref>, and applying the corresponding relations
from equations (<ref>) and (<ref>) we see that f_01
and f_03 are identical (Eqs. <ref> and <ref>). Similarly, f_12 is identical to f_23 (Eqs. <ref> and <ref>), and f_14
is identical to f_34 (Eqs. <ref> and <ref>). The
substitution of R_10=R_30, R_12=R_32, and R_14=R_34
implies that f_13=0. The substitution of Δ _024=Δ
_042=Δ _240=0 in equations <ref>, <ref> and <ref> gives 
@̌mathfonts

    f_02 :=((R_01-R_21)Δ _021+(R_03-R_23)Δ _023)m_1,       
    
    f_04 :=((R_01-R_41)Δ _041+(R_03-R_43)Δ _043)m_1,       
    
    f_24 :=((R_21-R_41)Δ _241+(R_23-R_43)Δ _243)m_1.

Since R_01=R_03, R_23=R_21, R_41=R_43, and Δ
_023=-Δ _021, Δ _041=-Δ _043, Δ _241=-Δ
_243 we get f_02= f_04= f_24=0. We have shown that  f_01=f_03, f_12=f_23, f_14=f_34, f_02= f_04=f_24=f_13=0. Consequently, f_01, f_12 and f_14
are the only necessary equations for general five-body problem (corresponding to <ref>, <ref>, <ref>). This completes the proof of Lemma 1.





Consider two pairs of equal masses on the vertices of a
rhombus with a fifth mass on one of the axis of symmetry. The five positive masses
have their position vectors 𝐫_0=(0,w), 𝐫_1=(-1,0), 𝐫_2=(0,t), 𝐫_3=(1,0), 𝐫_4=(0,-t),
where t ∈ℝ^+,w∈ℝ\{0}. Then their is no  central configuration of this type.




 Let 𝐫_1=(-1,0), 𝐫_2=(0,s), 𝐫_3=(1,0), 𝐫_4=(0,-t), where s,t∈ℝ^+. Then, the configuration 𝐫=(𝐫_1,𝐫_2,𝐫
_3,𝐫_4) given in figure 1 will form a central configuration in the region
@̌mathfonts

    R_μ_2μ_4(s,t)    =   {(s,t)|(0.268<t≤ 0.577 ∧
       ∧   1-t^2/2t < s < √(3)) ∨ (0.577 < t < √(3)∧
       ∧   √(t^2+1)-t < s < √(3))}.







 Let 𝐫_0=(0,w), 𝐫_1=(-1,0), 𝐫_2=(0,s), 𝐫_3=(1,0), 𝐫_4=(0,-t),
where s,t,w∈ℝ.



  (a) The configuration 𝐫=(𝐫_0,𝐫_1,
𝐫_2,𝐫_3,𝐫_4) will form a central
configuration with @̌mathfonts

    μ _0   =   m_0/m_1=A_1C_2C_3-(B_1
    B_3C_2+B_2C_1C_3)/A_2C_1C_3+
    A_3B_1C_2,


    μ _2   =   m_2/m_1=A_3B_2C_1-(A_1
    A_3C_2+A_2B_3C_1)/A_2C_1C_3+
    A_3B_1C_2,


    μ _4   =   m_4/m_1=A_2B_1B_3-(A_1
    A_2C_3+A_3B_1B_2)/A_2C_1C_3+
    A_3B_1C_2,

where

    A_1   =   (R_03-R_13)Δ _013,    A_2=(R_10-R_20)Δ
    _120,  
    
    A_3   =   (R_10-R_40)Δ _140, 
    
    B_1   =   (R_02-R_12)Δ _012,    B_2=(R_13-R_23)Δ
    _123,  
    
    B_3   =   (R_13-R_43)Δ _143, 
    
    C_1   =   (R_04-R_14)Δ _014,    C_2=(R_14-R_24)Δ
    _124,  
    
    C_3   =   (R_12-R_42)Δ _142.



  (b) The mass ratios μ _0>0, μ _2>0 and μ _4>0 form
a central configuration in

    R(t,w)=R_μ _0∩ R_μ _2∩ R_μ _4,

where R_μ _0, R_μ _2 and R_μ _4 are respectively
given by equations (<ref>), (<ref>), and (<ref>).







§ PROOF OF THEOREM <REF>


In order to prove Theorem 2, we write Dziobek's equations of Lemma 1 in a matrix form, after we substitute m_2=m_4:

    [
    [    0  B_0  C_0;  A_1  B_1  C_1;  A_2 -B_1 -C_1 ]
    ] ·[
    [ m_0; m_1; m_2 ]
    ] =[
    [ 0; 0; 0 ]
    ] ,

where

    B_0   =   (R_03-R_13)Δ _013,B_1=(R_13-R_23)Δ _123, 
    
    A_1   =   (R_10-R_20)Δ _120,A_2=(R_10-R_40)Δ _140, 
    
    C_1   =   (R_14-R_24)Δ _124, 
    
    C_0   =   (R_02-R_12)Δ _012+(R_04-R_14)Δ _014.

For the above system of equation, in order to have a non-trivial solution, the determinant of the
augmented matrix must be zero. After adding row 2 with row 3 of the
augmented matrix, the third row reduces to {A_1+A_2,0,0}.





Hence, f_A_1A_2=A_1+A_2=0 will imply the existence of non-trivial
solution (the case m_0=0 will be discussed in detail in section 4). Therefore, from the remaining two equations

    μ _1=m_1/m_2=-C_0/B_0, and  μ _0=
    m_0/m_2=C_1B_0-C_0B_1/B_0A_2

we obtain that they will form a central configuration with respect to the geometric constraint

    f_A_1A_2   =   (R_01-R_02)Δ _012+(R_01-R_04)Δ _014
       =   t-w/|t-w|^3-t+w/|t+w|^3+2w/(w^2+1)^3/
    2=0.

For positive solutions we must have f_A_1A_2=0 and μ _1>0, μ _2>0.



  a. If t>|w|, w∈ℝ\{0} then f_A_1A_2=0
can be written as a polynomial in two variables t and w such that t≠± w.

    G_1(w)     :=     2t^4w-4t^2w^3+4tw(w^2+1)^3/2+  
       +   2w^5=0.

The polynomial G_1(w) has two real negative roots and two complex roots.
As we require t>0, therefore this case is not interesting.


  b. If t<w,w>0  then f_A_1A_2=0 can be written as a
polynomial in two variables t and w such that t≠± w≠ 0.

    G_2(w)   :=   t^4w-t^2(2w^3+(1+w^2)^3/2)+  
       +   w^2(w^3-(1+w^2)^3/2)=0.

The polynomial G_2(w) is quadratic in t^2 and will have four roots
as functions of w.
@size8
@̌mathfonts

    t(w)=±√((w^2+1)^3/2/2w+w^2±√(
    (w^2+1)^3+8w^3(w^2+1)^3/2)/2w).



The function @size8
@̌mathfonts

    f_1(w)=(w^2+1)^3/2/2w+w^2-√(
    (w^2+1)^3+8w^3(w^2+1)^3/2)/2w

is negative for all values of w, see figure <ref>a. The function 
@size8
@̌mathfonts

    f_2(w)=(w^2+1)^3/2/2w+w^2+√(
    (w^2+1)^3+8w^3(w^2+1)^3/2)/2w

is positive for w>0 (see figure <ref>b). Then G_2(w) has two real
roots with one positive and one negative root. The negative root is not interesting.

 The necessary condition for the existence of central
configurations is satisfied at

    t_1(w)=√(f_2(w)).



It is straightforward to see from figure <ref> that μ
_0(w)>0 when w<√(3), and μ _1(w)>0 when w>√(3). So,
there is no common region where both μ _0(w) and μ _1(w) are
positive (Fig. 5).



  c. If t<|w|, w<0 then f_A_1A_2=0 can be written as a
polynomial in two variables t and w such that t≠± w≠ 0. 



    G_3(w)   :=   t^4w+t^2( ( w^2+1) ^3/2-2w^3) +
       +    w^2(w^3+( w^2+1)^3/2)=0.



The polynomial G_3(w) is quadratic in t^2 and have the following
four real roots as functions of w.

    t_2(w)=-√(f_3(w)),   t_3(w)=√(f_3(w)),  
    
    t_4(w)=-√(f_4(w)),   t_5(w)=√(f_4(w)),

where
@size8
@̌mathfonts

    f_3(w)=-(w^2+1)^3/2/2w+w^2-√(
    (w^2+1)^3/2·(1-8w^3))/2w,


    f_4(w)=-(w^2+1)^3/2/2w+w^2+√(
    (w^2+1)^3/2·(1-8w^3))/2w.


Two of the above roots i.e. t_3(w) and t_5(w) are positive, and t_2(w) and t_4(w) are negative. These roots are given in figure <ref>.

By the close inspection of figure <ref> it can be seen that t_3(w)>|w| for all w<0. As a result, this root is invalid as we have
the additional constraint of t<|w| for this special case. The
positive root t_5(w) remains less than |w| for w<0 when it is
defined in (-4,-1.8) and (-0.4,0).

The mass ratio μ_1 is positive when w∈ (-0.2,0) as B_0 and C_0
have opposite signs in this interval, see figure (<ref>a). In the
same interval A_2B_0 and B_0C_1-B_1C_0 have opposite signs and hence μ_0 is negative, no central configurations at t=t_5(w) when w∈
(-0.4, 0). For w<-1.8, C_0 is an increasing function of w with its
absolute minimum, 0, occurring when w → -∞. Hence, C_0>0 for all w<-1.8. Similarly, B_0 is also positive. Therefore, μ_1<0 for w<-1.8
(see figure <ref>b). This completes the proof of Theorem <ref>.










§ PROOF OF THEOREM <REF>


After substituting m_0=0 in the equations of Lemma 1 and using the
resulting symmetries, the following reduced system of Dziobek's equations is
obtained:

    (R_13-R_23)Δ _123m_1+(R_14-R_24)Δ _124m_4   =   0,
    
    
    (R_13-R_14)Δ _134m_1+(R_12-R_24)Δ _124m_2   =   0,

where

    R_23   =   (1+s^2)^-3/2,   R_14=(1+t^2)^-3/2, 
    
    R_24   =   (t+s)^-3, 
    Δ _123   =   -2s,  Δ _124=-(s+t),  Δ _134=-2t.

As the above system is under determined, therefore we define μ
_2=m_2/m_1 and μ _4=m_4/m_1. Simultaneous solution of
equations (<ref>) gives

    μ _2   =   (s^2+1)^3/2t(s+t)^2f_5(t)/
    4(t^2+1)^3/2f_6(s,t), 
    μ _4   =   sf_7(s)(t^2+1)^3/2(s+t)^2/
    4(s^2+1)^3/2f_8(s,t),

where



    f_5(t)   =   ((t^2+1)^3/2-8),
    
    f_6(s,t)   =   ((s^2+1)^3/2-(s+t)^3),
    
    f_7(s)   =   ((s^2+1)^3/2-8),
    
    f_8(s,t)   =   ((t^2+1)^3/2-(s+t)^3).


To find the central configuration (CC) regions, where all the masses are
positive, we need to find regions in the st-plane, where both μ _2
and μ _4 are positive. As f_5(t) is a monotonically increasing
function of t for all t>0, with a single zero at t=√(3), therefore
it is straightforward to see that f_5(t)<0, when t<√(3). Solving
f_6(s,t)=0 for variable t, we get t(s)=-s+√(s^2+1). It is easy to check
that f_6(s,t)<0 when t(s)>-s+√(s^2+1). The two functions
f_5(t) and f_6(s,t) are never simultaneously positive, therefore
μ_2>0 in

    R_μ _2={(s,t)|t>0∧ s>0∧√(s^2+1)-s<t<√(3)}.

Similarly, f_7(s)=0, when s=√(3) and f_8(s,t)=0, when s(t)=-t+
√(t^2+1). Similar μ _4>0 in

    R_μ _4={(s,t)|s>0∧ t>0∧√(t^2+1)-t<s<√(3)}.

The intersection of R_μ _2 and R_μ _4 is R_μ _2μ
_4 given by the equation (<ref>) (see Fig. <ref>).


Consider t<0 in the setup of Theorem <ref>, guaranteeing a
triangular four-body arrangement, then the configuration
(m_1, m_2, m_1, m_4) will form a triangular central configuration in 

@̌mathfonts

    TR_t_-(s,t)    =   {(s,t)|(-3.73<t<-√(3)∧
       ∧    h_1(t)<s<√(3)) ∨ (-√(3) <t≤ -1 ∧
       ∧    (0<s<h_1(t) ∨√(3)<s<h_2(t))) ∨
       ∨    (-1 <t<-1/√(3)∧√(3)<s<h_2(t)) 
       ∨    (-1/√(3) <t<0∧ h_2(t)<s<√(3))},

where

    h_1(t)=1-t^2/2t,   h_2(t)=√(t^2+1)-t.




The proof of the Corollary 5 follows the same procedure as the Theorem 3.
Therefore, we only give a sketch of this proof and leave the details
to the interested readers.

Using the same procedure as above we can show that the mass m_2 is
positive in the region
@̌mathfonts

    TR_m_2(s,t)    =   {(s,t)|(0<s<1/√(3)∧(-√(3)<t<H(s)
    
       ∨    -s<t<0)) ∨ (1/√(3)<s<√(3)∧
       ∧    (H(s)<t<-√(3)∨ -s<t<0)) ∨
       ∨    (s>√(3)∧ (H(s)<t<-s∨
       ∨    -√(3)<t<0))},

where

    H(s)=-√(s^2+1)-s.


Similarly, m_4 is positive in the region
@size8
@̌mathfonts

    TR_m_4(s,t)    =   (t<-√(3)∧ (0<s<√(3)∨ -t<s<h_2(t))) 
       ∨    (-√(3)<t<-1/√(3)∧ (0<s<-t∨
       ∨   √(3)<s<h_2(t))) ∨ (-1/√(3)<t<0∧
       ∧    (0<s<-t∨ h_2(t)<s<√(3))).

The intersection of TR_m_2(s,t) and TR_m_4(s,t) give TR_t_-(s,t).








§ PROOF OF THEOREM <REF>






 §.§ Proof of Theorem <ref>a


Equations (<ref>), (<ref>) and (<ref>)
of Lemma 1 define the CC of rhomboidal or triangular five-body problems.
Let μ _0=m_0/m_1, μ _2=m_2/m_1, μ _4=m_4/m_1 and rewrite the equations from Lemma 1 as

    A_1m_1+B_1m_2+C_1m_4   =   0, 
    
    A_2m_0+B_2m_1+C_2m_4   =   0, 
    
    A_3m_0+B_3m_1+C_3m_2   =   0,

where

    A_1   =   (R_03-R_13)Δ _013,A_2=(R_10-R_20)Δ _120, 
    
    A_3   =   (R_10-R_40)Δ _140, 
    
    B_1   =   (R_02-R_12)Δ _012,B_2=(R_13-R_23)Δ _123, 
    
    B_3   =   (R_13-R_43)Δ _143, 
    
    C_1   =   (R_04-R_14)Δ _014,C_2=(R_14-R_24)Δ _124, 
    
    C_3   =   (R_12-R_42)Δ _142.

Divide the above equations by m_1 and write them in the following
matrix form:

    [
    [   0 B_1 C_1; A_2   0 C_2; A_3 C_3   0 ]
    ] ·[
    [ μ _0; μ _2; μ _4 ]
    ] =-[
    [ A_1; B_2; B_3 ]
    ].

After a number of row operation, a simultaneous solution of the above system
is obtained as below.

    μ _0   =   A_1C_2C_3-(B_1B_3C_2+B
    _2C_1C_3)/A_2C_1C_3+A_3B_1C
    _2,  
    μ _2   =   A_3B_2C_1-(A_1A_3C_2+A
    _2B_3C_1)/A_2C_1C_3+A_3B_1C
    _2, 
    μ _4   =   A_2B_1B_3-(A_1A_2C_3+A
    _3B_1B_2)/A_2C_1C_3+A_3B_1C
    _2.

This completes the proof of Theorem 4a.



 §.§ Proof of Theorem <ref>b


The required values of R_ij and Δ _ijk can be generated from
Lemma 1 by the substitution of s=1.

 The function

    μ _0=A_1C_2C_3-(B_1B_3C_2+B_
    2C_1C_3)/A_2C_1C_3+A_3B_1C_
    2

attains positive value in the region

    R_μ _0=(R_D∩ R_N_μ _0)∪ (R_D^c∩ R_N_μ
    _0^c),

where R_D^c and R_N_μ _0^c are the complements of the
regions R_D and R_N_μ _0.



Let N_μ _0=A_1C_2C_3-(B_1B_3C_2+B
_2C_1C_3) and D=A_2C_1C_3+A_3B_
1C_2. For μ _0 to be positive, N_μ _0 and D must
have the same sign.

The denominator D will be positive when both factors A_2C_1C
_3 and A_3B_1C_2 are positive or at least one of them is
positive, such that the positive part is greater than the absolute value of
the negative part. Both these factors are positive in

    R_da(t,w)    =   {(t,w)|(0<w≤ 0.41∧ d_1(w)<t<1)  
       ∨   (0.41<w<0.58∧ d_1<t<0.5 · d_2(w))  
       ∨   (w>2.41∧ 0.41<t<1)},

where

    d_1(w)=√(w^2+1)-w,   d_2(w)=(1-w^2)/w.

Similarly, when



  * A_2C_1C_3>0 and A_3B_1C_2<0 then A_2C_1C_3 > |A_3B_1C_2| in the following region. 

    R_db(t,w)    =   {(t,w)|(0.4<t≤ 0.58 ∧
       ∧    0.6· d_2(t) <w<1)∨
       ∨    (0.58<t<1 ∧ d_1(t)<w<1)}.




  * A_2C_1C_3<0 and A_3B_1C_2>0 then
A_3B_1C_2 > |A_2C_1C_3| in the following region. 

    R_dc(t,w)    =   {(t,w)|(w≤ -1.49∧ 0<t<0.23) 
       ∨   (-1.49<w<-1∧
       ∧    0<t<0.2 · d_2(w)) ∨
       ∨   (1<w<2.41∧ 0.41<t≤ 1)}.


Therefore, the denominator D is positive in  (see Fig. <ref>a):

    R_D=R_da(t,w)∪ R_db(t,w)∪ R_dc(t,w).


The numerator of μ _0, N_μ _0 is positive in  (see Fig. <ref>b and Appendix):

    R_N_μ _0=R_cN_μ _0(t,w)∪ R_dN_μ _0(t,w).

Therefore, μ _0>0 in the intersection of R_D
and R_N_μ _0, and the
intersection of their complements. Region R_μ _0 is shown in figure
<ref>c. This completes the proof of Lemma <ref>.






 The function

    μ _2=A_3B_2C_1-(A_1A_3C_2+A_2B_3C_1)/A_2C
    _1C_3+A_3B_1C_2

attains positive values in the region

    R_μ _2=(R_D∩ R_N_μ _2)∪ (R_D^c∩ R_N_μ
    _2^c),

where  R_D^c and R_N_μ _2^c are complements of the
regions R_D and R_N_μ _2 respectively.



Let N_μ _2=A_3B_2C_1-(A_1A_3C_2+A_2B_3C_1). For μ _2 to be positive, N_μ _2 and D must have the same sign. We
have given the complete analysis of D in Lemma <ref>. Now, we need
to find regions where N_μ _2 is positive. It is proved in the appendix that   N_μ _2(t,w)>0 in

    R_N_μ _2=R_aN_μ _2(t,w)∪ R_bN_μ _2(t,w)∪
    R_cN_μ _2(t,w).

Region R_N_μ _2 is shown in figure <ref>a.
Consequently, μ _2>0 in the intersection of R_D and R_N_μ
_2, as well as the intersection of their complements. Region R_μ
_2 is shown in figure <ref>b. This completes the proof of
Lemma <ref>.








 The function

    μ_4=A_2B_1B_3-(A_3B_1B_2+A_1A_2C_3)/
    A_2C_3C_1+A_3B_1C_2

attains positive values in the region

    R_μ _4=(R_D∩ R_N_μ _4)∪ (R_D^c∩ R_N_μ
    _4^c).




Let N_μ _4=A_2B_1C_3-(A_3B_1B_2+A_1A_2C_3). For
positive μ _4, N_μ 4 and D must have the same sign. We
have given the complete analysis of D in Lemma <ref>. Now, we need
to find regions where N_μ _4 is positive. It is shown in the appendix that N_μ _4(t,w) is positive in  (see Fig. <ref>a):

    R_N_μ _4=R_aN_μ _4(t,w)∪ R_bN_μ _4(t,w)∪
    R_cN_μ _4(t,w).

Thus, μ _4>0 in the intersection of R_D and R_N_μ _4,
as well
as the intersection of their complements. Region R_μ _4 is shown in
figure <ref>b. This completes the proof of Lemma <ref>.



The central configuration region where all the masses have positive values
is determined by

    R(t,w)=R_μ _0∩ R_μ _2∩ R_μ _4.

The region R(t,w) is given in figure <ref>. This completes the
proof of Theorem <ref>b. To better understand the nature of the complicated CC regions in  figure <ref>, a number of relevant examples are given in figures  <ref> and  <ref>.






The considered variables t∈ (-1,0) and s=1 in Lemma 1, will give the five-body triangular configurations shown in figure 2.


 Consider t∈ (-1,0) in the setup of Theorem
<ref>, guaranteeing a triangular five-body arrangement for w>0. Then
the configuration (m_0, m_1, m_2, m_1, m_4) will
form a central configuration in region:

    R_t_-(t,w)=R_m_0(t,w)∩ R_m_2(t,w)∩ R_m_4(t,w).




Consider t<0 and solve the equation (<ref>) in the same way as in the
proof of Theorem <ref> to obtain the following regions of central
configurations for the isosceles triangular five-body problem, given in figure
2. We will give a sketch of the proof, and details to the
interested readers (as it follows the same procedure as in Theorem <ref>). The denominator is always negative when w>0. Therefore, the
mass ratios μ _0, μ _2 and μ _4 will be positive when their
respective numerators are negative. Thus, the central configuration regions
where μ _0, μ _2 and μ _4 are respectively positive, are given
below. 









    R_m_0   =   (-1<t<-w∧ 0<w<1)∨
       ∨   (1<w<1.73∧ (-1<t<d_2(w)∨
       ∨   0.5/w-0.5 w<t<0))∨ 1.5<w<p),


    R_m_2   =   (0<w<1∧ (-1<t<-w∨ -w<t<0)) 
       ∨   (-1<t<d_2(w)∧ 1.73205<w<2.41421) 
       ∨   t>d_2(w),


    R_m_4   =   (0<w<1∧ -w<t<0)∨
       ∨   (1<w<1.73205∧ -1<t<0)∨
       ∨    (1.73<w<q)∨ (1.6<w<r),

where

    p(t)    =   84.13t^5+301.01t^4+424.28t^3+295.98t^2+ 
       +   103.56t+16.68, 
    
    q(t)    =   -3.91t^3-7.57t^2-4.22t+1.22, 
    
    r(t)    =   305.66t^3+52t^2+0.25t+1.76.

The central configuration region for the triangular five-body problem is

    R_t_-(t,w)=R_m_0(t,w)∩ R_m_2(t,w)∩ R_m_4(t,w).

This region is given in figure <ref>d alongside R_m_0
(figure <ref>a),R_m_2(figure <ref>b) and
R_m_4(figure <ref>c). The CC region in the figure <ref> corresponds to the triangular solutions of the five-body problem.




Similarly, by taking s=√(3), the configuration in figure 2 will become
an equilateral triangle. The CC regions for this equilateral triangle five-body
configuration can be found in the same way as in Theorem <ref> or
Corollary <ref>. Figure  <ref> shows the physical position of masses in the case of triangular five-body central configurations.





 §.§ Note:


It is impossible to find analytically the CC regions in the full general cases,
i.e. without fixing s=1. However, in the current analysis, the majority of the
solutions are obtained. 




§ HAMILTONIAN FORMULATION AND NUMERICAL APPLICATIONS


In this section, we will introduce the Hamiltonian formalism for the
rhomboidal five-body problem. This five-body problem is introduced in Lemma 1. Then we are going to use
regularized equations and Poincaré surface of section to investigate the
effect of the increasing value of central mass on the stability of the
proposed five-body problem. Using Poincaré surface of section we will
study two special cases to identify chaotic regions and quasi-periodic
orbits, with a special focus on the varying central mass. The two cases are:



  a. Four equal masses with a varying central mass.


  b. Two pairs of equal masses at the vertices of the
rhombus with a varying central mass.


Two particular cases with only two degrees of freedom are known for the
four-body problem, namely the one-dimensional symmetric four-body problem and
the rhomboidal symmetric four-body problem (see <cit.>). These
two problems are simpler, but they still have chaotic solutions.
As <cit.> showed, symmetries played an essential role in searching for periodic orbits in most of the problems of celestial mechanics <cit.>.
The Hamiltonian
system in two degrees of freedom can be conveniently investigated using the
Poincaré surface of section. The Poincaré section plays
an important role in understanding the few-body problem, specifically the
existence of periodic orbits <cit.>.

Let us consider five point masses m_1(x_1=-x_3, 0), m_2(0, y_2), m_3 (x_3,0), m_4(0, -y_4), and  m_5
(0, y_5) in a fixed plane Oxy (see Fig. <ref>). Let us take m_1=m_3, and the generalized momenta as p_i^', where i=1,n, to get the Hamiltonian:
@size8

    H    =   ∑_i=1^5p^'_i^2/m_i-2m_1m_2/
    √(x_3^2+y_2^2)-2m_1m_4/√(x_3^2+y_4^2)
    -2m_1m_5/√(x_3^2+y_5^2)- 
       -   m_1^2/2x_3-m_2m_4/√((y_4+y_2)^2)-
    m_2m_5/√((y_5-y_2)^2)-m_4m_5/√(
    (y_4+y_5)^2).

Consider m_2=m_4 to be symmetric on the axis Oy, (y_2=y_4), and the mass m_5=m_0 stationary at the origin (y_5=0
). The Hamiltonian then becomes: 

    H    =   p^'_2^2/2m_2+p^'_3^2/2m_1
    -2m_1m_2/√(x_3^2+y_2^2)-  
       -   m_1(m_1+4m_0)/4x_3-m_2(m_2+4m_0)/4y_2.

Represent the generalized coordinates as x_3:=q_1, and y_2:=q_2, and the generalized momenta as p_3^':=p_1, and p_4^':=p_2, then the corresponding Hamiltonian takes the form 

    H    =   p_1^2/2m_1+p_2^2/2m_2-2m_1m_2/
    √(q_1^2+q_2^2)-  
       -   m_1(m_1+4m_0)/4q_1-m_2(m_2+4m_0)/4q_2,

The corresponding canonical equations of motion are

    q_1   =   p_1/m_1,  
    
    q_2   =   p_2/m_2,  
    
    p_1   =   -2m_1m_2/(q_1^2+q_2^2)^3/2q_1-
    m_1(m_1+4m_0)/2q_1^2,  
    
    p_2   =   -2m_1m_2/(q_1^2+q_2^2)^3/2q_2-
    m_2(m_2+4m_0)/2q_2^2.

Since we have singularities in the equations of motion, we use a
regularization technique, called the double Levi-Civita transformation to
regularize the Hamiltonian (see <cit.>).

Introduce the fictitious time τ as

    dt/dτ=q_1q_2,

and the transformed coordinates as

    q_i=Q_i^2,i=1,2.


In order to achieve regularization we extend the coordinate transformation
to a canonical transformation by introducing the generating function W
(see <cit.>) as

    W=p_1Q_1^2+p_2Q_2^2.

From the generating function P_j=∑_i=1^2p_i∂
q_i/∂ Q_j, j=1,2, we obtain the transformed  momenta

    p_i=P_i/2Q_i,i=1,2,

where P_i are the new momenta.

The canonical symplectic form of equations (<ref>) will be
preserved, if the new Hamiltonian is

    H=H-h_0,

where h_0 is the constant of total energy.

Therefore, the transformed Hamiltonian  is given by 
    H   =   1/8( P_1^2Q_2^2/m_1+
    P_2^2Q_1^2/m_2) -1/4(m_1(4m_0+m_1)Q_2^2
       +   m_2(4m_0+m_2)Q_1^2)-2m_1m_2Q_1^2Q_2^2/√(
    Q_1^4+Q_2^4)-h_0Q_1^2Q_2^2,

and the transformed equations of motion are

    dQ_i/dτ   =   ∂H/∂ P_i,
    
    dP_i/dτ   =   -∂H/∂ Q_i,  
    i=1,2.

To construct the Poincaré surface of section and find the corresponding
periodic orbits (see <cit.>), we plot the motion from the 4D phase
space (Q_1, Q_2, P_1, P_2) in a "cut" plane (Q_1=0, Q_2, P_1>0, P_2). Since H is conserved,
any point in this surface of section will uniquely define the orbit.

To investigate the effect of central mass on the existence of quasi-periodic
orbits, consider m_1=m_2=m_3=m_4=1 and



  (i.) m_0=1,

  (ii.) m_0=3,

  (iii.) m_0=5.


Consider Q_1(0)=0 and P_1(0)>0, to investigate the motion in the
(Q_2,P_2) plane, where the corresponding constant total energy h_0
have the values -0.46, -0.9, and -1.3. Figures <ref>, <ref>, and <ref> show the results for the gradually increased
central mass. In addition, we also give a representative periodic orbit in
each case.





In the equal mass case, there are a couple of quasi-periodic orbits at and
around P_2=0 and Q_2=2.83 (Fig. <ref>). The rest of the
points in figure <ref> are indicative of chaotic behavior. In figure 
<ref>, the central mass is increased to m_0=3. It is clear from
figure <ref> that the region with periodic orbits has begun to
increase in size. Although the outer region is still chaotic but
in inner region appears some regular structures. This is indicative of some
underlying structure in the dynamics. At m_0=5, five times bigger than
the outer masses, we can see higher number of quasi-periodic orbits around Q_2=3. The chaotic behavior which was very clear in the equal mass case
has completely disappeared. This indicates that the increase in central mass
plays a stabilizing role. Similar behavior was reported by <cit.>
for the hierarchical stability of the Caledonian symmetric five-body problem
(CS5BP).

To complete the analysis given above, we discuss two more examples with
two pairs of equal masses at the vertices of the rhombus, and other example with one pairs of equal masses on the x-axis and three different masses on the y-axis (see Fig. <ref>, <ref> and <ref>).
Consider


  a. m_1=1, m_2=2, and the central mass m_0=0.05 (h_0=-0.9);


  b. m_1=1, m_2=1.1, and the central mass m_0=3 (h_0=-0.9).





  c. m_1=m_3=1, m_2=0.27, m_4=0.4, m_0=0.76
(h_0=-0.9).







As in case a., the surfaces in figures <ref> and <ref> show various types of orbits, including both circle-like
quasi-periodic orbits and island orbits. Note the deformation in some of the
circular orbits, indicating the presence of nearby island orbits. When the
central body is small, the surface of section presents very few
quasi-periodic orbits surrounded by chaotic region. As before, the increase
in central mass completely changes the dynamics of the problem.
A direct comparison of figures <ref> with figures <ref>
reveal the obvious effect of changing central mass on the stability and
existence of quasi-periodic orbits in the rhomboidal five-body problem
introduced in section 1.


In case c. (see Fig. <ref>a) three non-equal masses are distributed on the y-axis. We used the general form of the Hamiltonian (equation <ref>), and the notations: x_3:=q_1, y_2:=q_2, m_5=m_0.  y_4 and y_5 are functions of q_2. As in previous cases, we constructed the corresponding new Hamiltonian (this transformation is trivial) and took y_4=q_2-1.4 and y_5=q_2-2.45. In this case we obtain
the Poincaré surface of section shown in figure <ref>. The inner region of figure <ref> points to the existence of regular structure and some quasi periodic orbits. The outer region exhibits chaotic behaviour. 






§ CONCLUSIONS


In this paper we studied the central configurations of rhomboidal and triangular four- and five-body problems. (The configuration has only one symmetry with m_1=m_3≠ m_2≠ m_4≠ m_0). First, Dziobek's equations are derived for this particular set up. Then, we derive
regions of central configuration for various special cases of the general
problem. We prove that no central configurations are possible when two pairs of masses are placed at the vertices of a
rhombus, (which is symmetric both about the x-axis and the y-axis) and the fifth mass is placed anywhere on the axis of symmetry, except the origin. Using
analytical techniques, regions of central configurations are derived for all
other cases, including rhomboidal four- and five-body problems,
isosceles and equilateral triangular five-body problems. To complement the
analytical results, these regions are also explored numerically. Equations of
motion are regularized using the double Levi-Civita transformations, and from
the regularized equations we identify regions with quasi-periodic orbits. Moreover, we investigate the chaotic behavior in the phase space by means of the  Poincaré sections. We show that if we increase the central mass, this has a
stabilizing effect on the rhomboidal five-body configuration.




§ APPENDIX




 §.§ Derivation of R_N_μ _0: 


The numerator of μ _0, N_μ _0 is positive when A_1C_
2C_3 >0 and B_1B_3C_2+B_2C_1C_3
<0 or when both factors have opposite sign and the positive factor is
greater than the absolute value of the negative factor. All these
possibilities are listed below with the corresponding regions, where N_μ
_0>0




  * A_1C_2C_3>0 and B_1B_3C_2+B_
2C_1C_3<0: Using numerical approximation techniques it is
obtained that N_μ _0>0 in the following region:

    R_cN_μ _0(t,w)=R_aN_μ _0(t,w)∪ R_bN_μ _0(t,w),

where 

    R_aN_μ _0(t,w)    =   {(t,w)|(0<t<0.41∧ 0<w<1) 
       ∨   (0.41<t<1∧ w>2.41)∨ (-1.73 
       <   w≤ -1.5∧ 0.75· d_2(w)<t<1)}, 
    
    R_bN_μ _0(t,w)    =   (w<-1.8∧ 0<t<0.23)∨ (-1.43 
       <   w≤ -1∧ 0.56<t<1)∨ (-1<w 
       <   -0.41∧ 0.41<t<-w)∨ (2<w 
       <   2.41∧ 0.41<t<1)}.




  * A_1C_2C_3>0 and B_1B_3C_2+B_
2C_1C_3>0: It is numerically confirmed that B_1B_3
C_2+B_2C_1C_3>A_1C_2C_3 when A
_1C_2C_3>0. Hence, N_μ _0<0 for all (t,w) such
that A_1C_2C_3>0.


  * A_1C_2C_3<0 and B_1B_3C_2+B_
2C_1C_3<0. In this case using numerical approximation
techniques we find that N_μ _0 is positive in the following region: 


    R_dN_μ _0(t,w)    =   {(t,w)|(-1.73<w<-1.5∧
       ∧   0<t<0.25)∨ (-1.5≤ w<-1∧
       ∧   0<t<0.2· d_2(w))∨ (0<t<0.41
       ∧    -t<w <0) ∨  (0.41<t<1 ∧
       ∧    0<w<1)}.



Hence, N_μ _0>0 in (see Fig. <ref>b):

    R_N_μ _0=R_cN_μ _0(t,w)∪ R_dN_μ _0(t,w).










 §.§ Derivation of R_N_μ _2: 


 The numerator of μ_2, N_μ _2, is always
positive when A_3B_2C_1>0 and A_1A_3C_2+A_2C_1C_3<0.
It is not necessarily positive when A_3B_2C_1>0 and A_1A_3C_2+A_2C_1C_3<0 or A_3B_2C_1<0 and A_1A_3C_2+A_2C_1C_3<0. It might be positive in a segment of
this region or might not be positive at all. We will list all these
possibilities below with the corresponding regions, where N_μ _2>0.



  * A_3B_2C_1>0 and A_1A_3C_2+A_2C_1B_3<0. Using
numerical approximation techniques we show that N_μ _2>0 in

    R_aN_μ _2(t,w)=aaN_μ _2(t,w)∪ baN_μ _2(t,w)∪
    caN_μ _2(t,w),

where 
    aaN_μ _2(t,w)    =   {(t,w)|(0<w≤ 0.41∧ d_1(w)< 
       <    t<1) ∨ (0.41<w<0.58 ∧
       ∧   d_1(w)<t<0.5· d_2(w)) 
       ∨   (w>√(3)∧ 0<t<d_1(w))},


    baN_μ _2(t,w)    =   {(t,w)|(w≤ -2.41∧ 0.463-
       -    0.079· w <t<1)∨ (-2.41<w<
       <    -1.73∧ 0.463-0.079w < 
       <    t < 0.5 d_2(w))},
    
    caN_μ _2(t,w)    =   {(t,w)|1<w<1.73∧
       ∧    0<t<d_1(w)}.




  * A_3B_2C_1>0 and A_1A_3C_2+A_2C_1B_3>0. Using
numerical approximation techniques we show that N_μ _2>0 in

    R_bN_μ _2(t,w)=abN_μ _2(t,w)∪ bbN_μ _2(t,w)∪
    cbN_μ _2(t,w),

where

    abN_μ _2(t,w)    =   {(t,w)|-1.73<w<-1 ∧
       ∧   0<t<0.5· d_2(w)},


    bbN_μ _2(t,w)    =   {(t,w)|1<w<1.73 ∧
       ∧   0<t<d_1(w)},


    cbN_μ _2(t,w)    =   {(t,w)|(w≤ -2.41∧
       ∧    0<t<1)∨ (-2.41<w<-1.73 
       ∧    0<t<0.5· d_2(w))}.



  * A_3B_2C_1<0 and A_1A_3C_2+A_2C_1B_3<0. Using
numerical approximation techniques we show that N_μ _2>0 in

    R_cN_μ _2(t,w)=acN_μ _2(t,w)∪ bcN_μ _2(t,w),

where
    acN_μ _2(t,w)    =   {(t,w)|(-2.4<w<-2∧
       ∧    0.5· d_2(w)<t<1)∨
       ∧    (1<w<1.73∧ d_1(w)<t<1)}, 
    
    bcN_μ _2(t,w)    =   {(t,w)|(1.31· t^2-2.012· t+0.9<
       <   w<0.58∧ 0<t<d_1(w))∨
       ∨    (1.31· t^2-2.012 · t+0.9≤ w<1 
       ∧    0<t<0.5 · d_2(w))∨
       ∨    (w>1.732∧ d_1(w)<t<1)}.

Therefore, N_μ _2(t,w)>0 in region

    R_N_μ _2=R_aN_μ _2(t,w)∪ R_bN_μ _2(t,w)∪
    R_cN_μ _2(t,w).





 §.§ Derivation of R_N_μ _4: 


N_μ _4 is always
positive when A_2B_1C_3>0 and A_3B_1B_2+A_1A_2C_3<0.
It is not necessarily positive when A_2B_1C_3>0 and A_3B_1B_2+A_1A_2C_3<0 or when both the factors are negative. It
might be positive in a segment of this region or might not be positive at
all. We will list all these possibilities below with the corresponding
regions, where N_μ _4>0.



  * When A_2B_1C_3>0, and A_3B_1B_2+A_1A_2C_3<0, N_μ _4>0 in region
    R_aN_μ _4   =   {(t,w)|(1.73<w<2.4∧ 0.<t<d_1)∨
       ∨    (-1<w<-0.41∧ -w<t<1)∨
       ∨    (0<w<1 ∧ d_1<t<1) ∨
       ∨    (w<-377.64t^4+677.58t^3-458.06t^2
       +   136.9t-17.79∧ 0.25<t<0.6) ∨
       ∨   (-1<w<-0.41∧ 0<t<0.05)∨
       ∨    (1.73<w<2.41∧ d_1<t<0.41)}.




  * When A_2B_1C_3>0, and A_3B_1B_2+A_1A_2C_3>0, N_μ _4>0 in region 
    R_bN_μ _4   =   {(t,w)|(0.1<w<1∧ 0.41<t<d_1) 
       ∨   (1<w<1.73∧ d_1<t<0.41) 
       ∨   (1.73<w<2.41∧ 0.41<t<1) 
       ∨   (-1<w<-0.41∧ -w<t<1) 
       ∨   (0<w<1∧ d_1<t<1) 
       ∨   (1<w<1.73∧ 0<t<d_1) 
       ∨   (1.31t^2-2t+0.9<w<1∧
       ∧    0<t<0.41) ∨
       ∨   (1<w<1.73∧ 0.41<t<1) 
       ∨   (1.73<w<2.41∧ d_1<t<0.41)}.




  * When A_2B_1C_3<0, and A_3B_1B_2+A_1A_2C_3<0, N_μ _4>0 in region 
    R_cN_μ _4   =   {(t,w)|(-0.41<w<0∧ 0<t<-w)∨
       ∨   (w<-1.3t^2-1.9t+3.44∧
       ∧    d_1<t<0.41) 
       ∨   (w>2.41∧ 0<t<d_1)}.

Therefore, N_μ _4(t,w)>0 in  (see Fig. <ref>a):

    R_N_μ _4=R_aN_μ _4(t,w)∪ R_bN_μ _4(t,w)∪
    R_cN_μ _4(t,w).







We thank the editor and the anonymous reviewer for their constructive comments, which helped us improve the presentation of the manuscript. We also thank Professor Mihail Barbosu for correcting and improving the language of the manuscript. I. Szücs-Csillik is partially supported by a grant of the Romanian Ministry of National Education and Scientific Research, RDI Programme for Space Technology and Advanced Research - STAR, project number 513, 118/14.11.2016.









99
[Alvarez-Ramírez, Corbera & Llibre (2016)]Martha2016 Alvarez-Ramírez, M., Corbera, M., & Llibre, J., 2016, On the central configurations in the spatial 5-body problem with four equal masses. Celestial Mechanics and Dynamical Astronomy, 124, 433

[Bakker & Simmons(2012)]bakker2012 Bakker, L., & Simmons, S.,
2012, 
Discrete and Continuous Dynamical Systems, 35, 2

[Burgos-Garcia & Delgado(2013)]burgos2013 Burgos-Garcia, J., &
Delgado, J., 2013, Astrophysics and Space Science, 345, 247

[Cheb-Terrab & Oliveira(1996)]cheb1996 Cheb-Terrab, E. S., &
Oliveira, H. P., 1996, 
Computer Physics Communications, 95 (2-3), 171

[Chen(2001)]Chen2001 Chen, K. C., 2001,

Archive for rational mechanics and analysis, 158
293

[Corbera & Llibre(2006)]Corbera2006 Corbera, M., & Llibre, J.,
2006,

Journal of mathematical physics, 47
122701

[Corbera, Cors & Roberts(2016)]Corbera2016Corbera, M., Cors, J. M., & Roberts, G. E., 2016, arXiv preprint arXiv:1610.08654

[Csillik(2003)]csillik2003 Csillik, I., 2003, Metode de
regularizare în mecanica cerească (Regularization methods in
Celestial Mechanics), Cluj-Napoca: Casa Cărţii de Ştiinţă


[Delgado & Perez-Chavela(1991)]Delgado1991 Delgado-Fernandez, J.,
& Perez-Chavela, E., 1991, The rhomboidal four-body problem. Global flow on
the total collision manifold, The Geometry of Hamiltonian Systems, New York: Springer

[Deng & Zhang(2014)]DengZhang2014 Deng C. & Zhang S., 2014,

Journal of Geometry and Physics, 83, 43

[Diacu(2003)]diacu2003 Diacu, F., 2003, lecture held at the Annual Session of the Astronomical Institute of Romanian Academy, 15-16 May 2003, Bucharest

[Dziobek(1900)]Dziobek1900 Dziobek, O., 1900, Astron. Nach., 152,
32

[Gidea & Llibre,(2010)]Marian2010Gidea, M., & Llibre, J., 2010, Celestial Mechanics and Dynamical Astronomy, 106(1), 89

[Hampton(2005)]Hampton2005 Hampton, M., 2005, Nonlinearity, 18(5), 2299.

[Hampton(2011)]Hampton2011 Hampton, M., & Jensen, A., 2011, Celestial Mechanics and Dynamical Astronomy, 109(4), 321

[Ji et al.(2000)]JI2000 Ji, J. H., Liao, X. H., & Liu, L., 2000,

Chinese Astronomy and Astrophysics, 24, 381

[Kulesza et al.(2011)]Kulesza2011 Kulesza, M., Marchesin, M., &
Vidal, C., 2011, 
Journal of Physics A: Mathematical and Theoretical, 44, 485204

[Lacomba & Perez-Chavela(1992)]Lacomba1992 Lacomba, E. A., &
Perez-Chavela, E., 1992

Celestial Mechanics and Dynamical Astronomy, 54, 343

[Lacomba & Perez-Chavela(1993)]Lacomba1993 Lacomba, E. A., &
Perez-Chavela, E., 1993, Celestial Mechanics and Dynamical Astronomy 57, 411

[Lee & Santoprete(2009)]Lee2009 Lee, T. L., & Santoprete, M., 2009, Celestial Mechanics and Dynamical Astronomy, 104(4), 369

[Llibre & Mello(2008)]Libre2008 Llibre, J., & Mello, L. F., 2008,

Celestial Mechanics and Dynamical Astronomy, 100, 141

[Llibre(2015)]Llibre2015 Llibre, J., 2015, Proceedings of the
American Mathematical Society, 143, 3587

[Llibre et al.(2015)]Libreetal2015 Llibre, J., Moeckel, R., & Simó, C., 2015, Central Configurations, Basel: Springer

[MacMillan & Bartky(1932)]MacMillan MacMillan, W. D. & Bartky, W.,
1932, 
Transactions of the American Mathematical Society, 34, 838, Dynamical
Astronomy, 57, 411

[Marchesin & Vidal(2013)]Marchesin2013 Marchesin, M., & Vidal,
C., 2013,

Celestial Mechanics and Dynamical Astronomy, 115, 261

[Marchesin(2017)]Marchesin2017 Marchesin, M., 2017, Astrophysics and Space Science, 362(1), 1

[Meyer & Hall(1992)]meyerhall1992 Meyer, K. R., & Hall, G. R., 1992, Introduction to Hamiltonian Dynamical Systems and the n-body problem,
New York: Springer

[Mioc & Blaga(1999)]MiocBlaga Mioc, V., & Blaga, C., 1999,  Hvar Observatory Bulletin, 23, 41

[Mioc & Barbosu(2003)]miocbarbosu2003 Mioc, V., & Barbosu, M., 2003, Serb. Astron. J., 167, 47

[Moeckel(2014)]Moeckel2014
Moeckel, R., 2014, 
Scholarpedia, 9(4), 10667

[Ollongren(1988)]Ollongren Ollongren, A, 1988, Journal of symbolic computation, 6(1), 117

[Perez-Chavela & Santoprete(2007)]Perez2007
Perez-Chavela, E., & Santoprete, M., 2007,

Arch. Ration. Mech. Anal. 185, 481

[Roberts(1999)]Roberts Roberts, G. E., 1999,  Physica D: Nonlinear Phenomena, 127, 141

[Shoaib et al.(2008)]Shoaib2008 Shoaib, M., Steves, B. A., & Sz
éll, A., 2008,

New Astronomy, 13, 639

[Shoaib et al.(2013)]Shoaib2013Shoaib, M., Sivasankaran, A., & Aziz, Y. A., 2013, Chaotic Modeling and Simulation, 3(3), 431

[Shoaib et al.(2012)]Shoaib2012 Shoaib, M., Faye, I., &
Sivasankaran, A., 2012, Some special solutions of the rhomboidal five-body
problem, In International Conference on Fundamental and Applied Sciences
2012: (ICFAS2012) (Vol. 1482, No. 1, pp. 496-501). AIP Publishing

[Shoaib et al.(2016)]Shoaib2016Shoaib, M., Kashif, A. R., & Sivasankaran, A., 2016, Advances in Astronomy, 2016.

[Simó(1978)]simo Simó, C., 1978,

Celestial Mechanics, 18, 165

[Szücs-Csillik(2016)]szucs1 Szücs-Csillik, I., 2016,
poster presented at the Symposium "Vistas in Astronomy, Astrophysics and Space Sciences"

[Waldvogel(2012)]Waldvogel2012 Waldvogel, J., 2012,

Celestial Mechanics and Dynamical Astronomy, 113, 113

[Yan(2012)]Yan2012 Yan, D., 2012,

Journal of Mathematical Analysis and Applications, 388, 942



