SQU Journal for Science, 2020, 25(1), 61-77 DOI:10.24200/squjs.vol25iss1pp61-77 Sultan Qaboos University 61 Behavior of Small Variable Mass Particle in Electromagnetic Copenhagen Problem Abdullah A. Ansari International Center for Advanced Interdisciplinary Research (ICAIR), Sangam Vihar, New Delhi, India. Email: icairndin@gmail.com. ABSTRACT: The paper presents the behavior of the motion properties of the variable mass test particle (third body), moving under the influence of the two equal primaries having electromagnetic dipoles. These primaries move on the same circular path around their common center of mass in the same plane. We have determined the equations of motion of the test particle whose mass varies according to Jean's law. Using the system of equations of motion we have evaluated the locations of equilibrium points, their movements and basins of the attracting domain. Finally, we examine the stability of these equilibrium points, all of which are unstable. Keywords: Variable mass; Electromagnetic dipoles; Copenhagen problem; Attracting domain. سلوك جسيم ذو كتلة خاضعة لتغيرات صغيرة في مشكلة كوبنهاجن الكهرومغناطيسية عبدالجبار أنصاريعبدهللا تأثير جسمان اوليان متساويان لهما قطبان مغنتسيان. )الجسم الثالث( تحت حركة جسيم ذو كتلة متغيرة الورقة العلمية سلوك خصائص هذه تقدم ملخص:ال ة الجسيم حيث الجسمان األوليان يتحركان على نفس المصار الدائري مركزه مركزهما الكتلي المشترك المتواجد في نفس المصتح. قمنا بإتقييم معادالت حرك ا بحساب أماكن نقاط التوازن ، حركتهما و أحواض مجاالت الجذابة. في نفرض ان كتلته تتغير حسب قانون جان . باستعمال نضام معادالت الحركة، قمن آخر العمل قمنا بالتترق لمسألة إستقرار نقاط التوازن و وجدنا أن كلها غير مستقرة. كهرومغناطيسي، مسألة كوبنهاق، مجال ذو جاذبية. كتلة متغيرة، ثنائي الألقطابتاحية: مفالكلمات ال 1. Introduction elestial Mechanics is a part of applied mathematics; basically it is the study of the motion properties of satellites under the influence of celestial bodies. The motion of these satellites or spacecraft is affected by the celestial bodies in many ways, such as by the different shapes of the bodies (i.e. heterogeneous body, homogeneous body, irregular triaxial bodies, oblate body, finite straight segments, cylindrical shape, etc.), radiation pressure, resonances, magnetic dipoles, charged bodies, Yarkovskii effects, the albedo effect, variable mass effects, etc. For more details, see [1] and [2]. An artificial body (satellite or spacecraft) is also perturbed by the electromagnetic dipoles which are studied by [3-5]. [3] presented the motion properties of charged particles as equilibrium points and their stability in the field of two magnetic dipoles. [4] has investigated the three-dimensional equilibrium points in a magnetic-binary system and then, by using the method of characteristic exponent, examined their stability. In continuation of the work of [4], [5] studied the basins of attraction in the electromagnetic Copenhagen problem. The Copenhagen problem where two primary bodies of equal mass are moving on the same circular path in the same plane is also commonly studied in celestial mechanics. [6] used the methods of chaotic scattering to investigate this problem. [7] used studied the asymptotic orbits and terminations of families in the Copenhagen restricted three- body problem. [8] studied the Copenhagen problem with quasi-homogeneous potential or Manev-type potential and illustrated the equilibrium locations and the regions of permitted motion. [9] derived the translational-rotational equations of motion and investigated the stability of its equilibrium points. [10] have investigated equilibrium positions as well as the basins of attraction in the Copenhagen restricted three-body problem numerically. C mailto:icairndin@gmail.com ANSARI, A.A. 62 After careful study of the literature presented, we became interested in studying and extending this topic with variable mass. Variable mass is an interesting topic in celestial mechanics and dynamical astronomy, which is widely studied in the restricted problem (two-body, three-body, four-body and five-body) [11-21]. However, no article yet exists in the literature addressing the combination of electromagnetic dipole and variable mass. This paper is organized as follows. The literature review is presented in section 1. The equations of motion are presented in section 2 while section 3 contains the investigation of equilibrium points and basins of attracting domain. The stability of these equilibrium points are examined in section 4. Finally the manuscript completes with a conclusion in section 5. Figure 1. Geometric configuration of the electromagnetic Copenhagen problem 2. Equations of motion The well-known Copenhagen problem where two bodies of equal masses m1 and m2 both are taken as point masses which are moving in circular orbits on the same circular path around their common center of mass. In this paper two bodies are also taken as electromagnetic dipoles with magnetic moments Mi (i = 1, 2). Now we are interested to determine the equations of motion of the third variable mass (m) body of charge q (Figure 1) with dimensionless variables in the synodic coordinate system where the variation of mass of the test particle originates from one point and has zero momenta ([17] and [3]) which reveals as: �̇� 𝑚 (�̇� −𝜔 𝑦)+( �̈� −𝑓1�̇� + 𝑔1�̇�) = 𝜔 2𝑥 + 𝑞 𝜔 𝑚 𝑐 (𝑥 𝜕𝐴2 𝜕𝑥 −𝑦 𝜕𝐴1 𝜕𝑥 +𝐴2), �̇� 𝑚 (�̇� +𝜔 𝑥)+( �̈� −ℎ1�̇� + 𝑓1�̇�) = 𝜔 2𝑦 + 𝑞 𝜔 𝑚 𝑐 (𝑥 𝜕𝐴2 𝜕𝑦 −𝑦 𝜕𝐴1 𝜕𝑦 −𝐴1), �̇� 𝑚 �̇� +( �̈� −𝑔1�̇� + ℎ1�̇�) = 𝑞 𝜔 𝑚 𝑐 (𝑥𝜕𝐴2 𝜕𝑧 −𝑦𝜕𝐴1 𝜕𝑧 ). (1) where 𝑓1 = (2 𝜔 +�̂� 𝐶𝑢𝑟𝑙 𝐴) 𝑞 𝑚 𝑐 , 𝑔1 = (𝑗̂ 𝐶𝑢𝑟𝑙 𝐴) 𝑞 𝑚 𝑐 , ℎ1 = (𝑖̂ 𝐶𝑢𝑟𝑙 𝐴) 𝑞 𝑚 𝑐 , (2) VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 63 with 𝐴 = (𝐴1,𝐴2,𝐴3) = ( �̅�1 ×�̅�1 𝑟1 3 + �̅�2 ×�̅�2 𝑟2 3 ) , �̅�1 = (𝑥 −0.5,𝑦,𝑧), �̅�2 = (𝑥 +0.5,𝑦,𝑧), �̅�1 = (0,0,1), �̅�2 = (0,0,𝜆), 𝐴1 = (− 𝑦 𝑟1 3 − 𝜆 𝑦 𝑟2 3 ), 𝐴2 = ( 𝑥 −0.5 𝑟1 3 + 𝜆(𝑥 +0.5) 𝑟2 3 ), 𝐴3 = 0, and also ω is the mean motion of the system, λ and c are constants. We choose the unit of time and charge of dipoles such that ω = 1 and (q/mc) = 1, hence Equations (1, 2) become as �̇� 𝑚 (�̇� −𝑦)+( �̈� −𝑓2�̇� + 𝑔2�̇�) = Ω𝑥, �̇� 𝑚 (�̇� + 𝑥)+( �̈� −ℎ2�̇� + 𝑓2�̇�) = Ω𝑦, �̇� 𝑚 �̇� +( �̈� −𝑔2�̇� + ℎ2�̇�) = Ω𝑧, (3) with Ω = 1 2 (𝑥2 +𝑦2)+(𝑥𝐴2 −𝑦𝐴1) (4) 𝑓2 = (2 + �̂� 𝐶𝑢𝑟𝑙 𝐴) , 𝑔2 = (𝑗̂ 𝐶𝑢𝑟𝑙 𝐴), ℎ2 = (𝑖̂ 𝐶𝑢𝑟𝑙 𝐴), (5) Due to variation of the mass of the infinitesimal body, we cannot study the properties of this body directly. Therefore we will use Jean's law, 𝑑𝑚 𝑑𝑡 = −𝜆1𝑚 𝜆2, (6) where λ1 is constant coefficient and λ2 is within the limits 0.4 ≤ λ2 ≤ 4.4, for the stars of the main sequence. While for the rocket λ2 = 1, therefore the mass of the rocket m = m0 e -λ 1 t varies exponentially where m0 is the mass of the test particle at time t = 0. We will use the Meshcherskii space time transformations to preserve the dimension of the space and time 𝑥 = 𝜆3 −1/2 𝛼, 𝑦 = 𝜆3 −1/2 𝛽, 𝑧 = 𝜆3 −1/2 𝛾, 𝑑𝑡 = 𝑑𝜏 (7) where λ3 = m/m0, now we can express the components of velocity and acceleration as �̇� = 𝜆3 −1/2 (�̇� + 1 2 𝜆1𝛼), �̇� = 𝜆3 −1/2 (�̇� + 1 2 𝜆1𝛽), �̇� = 𝜆3 −1/2 (�̇� + 1 2 𝜆1𝛾), (8) �̈� = 𝜆3 −1/2 (�̈� + 𝜆1�̇� + 1 4 𝜆1 2𝛼), �̈� = 𝜆3 −1/2 (�̈� +𝜆1�̇� + 1 4 𝜆1 2𝛽), �̈� = 𝜆3 −1/2 (�̈� + 𝜆1�̇� + 1 4 𝜆1 2𝛾), (9) ANSARI, A.A. 64 After using Equations (6, 7, 8 and 9) in Equation (3), we get ( �̈� − 𝑓3�̇� + 𝑔3�̇�) = V𝛼 +𝑉1, ( �̈� − ℎ3�̇� + 𝑓3�̇�) = V𝛽 + 𝑉2, ( �̈� − 𝑔3�̇� + ℎ3�̇�) = V𝛾 + 𝑉3, (10) where, V = 1 2 (𝛼2 + 𝛽2)+ 𝜆1 2 8 (𝛼2 + 𝛽2 + 𝛾2)+ 𝜆3 −1/2 (𝛼𝐵2 − 𝛽𝐵1) (11) 𝑉1 = 𝜆3 3/2 𝜆1 2 (𝛽 𝜕𝐵2 𝜕𝛼 − 𝛽 𝜕𝐵1 𝜕𝛽 − 𝛾 𝜕𝐵1 𝜕𝛾 ), 𝑉2 = 𝜆3 3/2 𝜆1 2 (−𝛼 𝜕𝐵2 𝜕𝛼 +𝛼 𝜕𝐵1 𝜕𝛽 − 𝛾 𝜕𝐵2 𝜕𝛾 ), 𝑉3 = 𝜆3 3/2 𝜆1 2 (𝛼 𝜕𝐵1 𝜕𝛾 + 𝛽 𝜕𝐵2 𝜕𝛾 ), 𝑓3 = 𝜆3 3/2 (2 𝜆3 −3/2 + �̂�.𝐶𝑢𝑟𝑙 𝐵) , 𝑔3 = 𝜆3 3/2 (𝑗̂.𝐶𝑢𝑟𝑙 𝐵), ℎ3 = 𝜆3 3/2 (𝑖̂ .𝐶𝑢𝑟𝑙 𝐵), (12) 𝐵 = (𝐵1,𝐵2,𝐵3) , 𝐵1 = (− 𝛽 ℓ1 3 − 𝜆 𝛽 ℓ2 3 ), 𝐵2 = ( 𝛼 − 𝜆3 1/2 0.5 ℓ1 3 + 𝜆(𝛼 + 𝜆3 1/2 0.5) ℓ2 3 ), 𝐵3 = 0, ℓ1 = √(𝛼 − 𝜆3 1/2 0.5) 2 + 𝛽2 + 𝛾2, ℓ2 = √(𝛼 + 𝜆3 1/2 0.5) 2 + 𝛽2 + 𝛾2 (13) 3. Analysis of Equilibrium Points and Basins of Attracting Domain 3.1 Equilibrium points We will put all the derivatives with respect to time on the left hand side of the system (10) to zero, hence V𝛼 + 𝑉1 = 0, (14) V𝛽 + 𝑉2 = 0, (15) V𝛾 + 𝑉3 = 0. (16) After solving equations (14), (15) and (16), we can find the locations of equilibrium points. We have plotted the locations of equilibrium points numerically in α-β-plane as shown in Figures 2, 3 and 4. VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 65 (a) λ1 = 0 and λ3 = 1 (b) λ1 = 0.2 and λ3 = 0.4 (c) λ1 = 0.2 and λ3 = 0.8 (d) λ1 = 0.2 and λ3 = 1.4 Figure 2. Locations of equilibrium points at λ = 1 in α β-plane ANSARI, A.A. 66 (a) λ1 = 0 and λ3 = 1 (b) λ1 = 0.2 and λ3 = 0.4 (c) λ1 = 0.2 and λ3 = 0.8 (d) λ1 = 0.2 and λ3 = 1.4 Figure 3. Locations of equilibrium points at λ = 7 in α β-plane. VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 67 (a) λ1 = 0 and λ3 = 1 (b) λ1 = 0.2 and λ3 = 0.4 (c) λ1 = 0.2 and λ3 = 0.8 (d) λ1 = 0.2 and λ3 = 1.4 Figure 4. Locations of equilibrium points at λ = 15 in α β-plane. ANSARI, A.A. 68 (a) λ = 1 (b) λ = 7 (c) λ = 15 Figure 5. Movement of equilibrium points at λ1 = 0, λ3 = 1 (Blue) and λ1 = 0.2, λ3 = 0.4 (Green), 0.8 (Red), 1.4 (Cyan) in α β-plane. Figure 2 presents the locations of equilibrium points at λ = 1 where figure 2(a) presents the classical case (λ 1 = 0 and λ3 = 1) which coincides with [5], where three equilibrium points (L1, 2, 3) were also obtained i.e. only collinear equilibrium points. Out of these equilibrium points L2 is situated at the origin, which is also the center of mass of the primaries, while L1 and L3 lie left and right of the origin. Further, figures 2(b), 2(c) and 2(d) present the locations of equilibrium points when λ1 = 0.2, λ3 = 0.4, 0.8 and 1.4, respectively i.e. the variation of mass cases. In these three cases we observe that at λ1 = 0.2 and λ3 = 0.4, the two equilibrium points (L1 and L3) move toward the origin as well as moving anti-clockwise from the α-axis. As we increase the value of λ3, we find that these two equilibrium points move away from the origin keeping L2 at the origin. We can see the movement of the equilibrium points (L1 and L3) in figure 5(a). Again, figure 3 presents the locations of equilibrium points at λ = 7, where figure 3(a) presents the classical case which also coincides with [5], where seven equilibrium points were obtained. But as we increase the value of λ3 by VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 69 keeping the value of λ1 the same at 0.2, we find only five equilibrium points, which mean that due to variation of mass, the number of equilibrium points reduces to five. The movement of all equilibrium points due to change of λ3 can be seen in figure 5(b). In the same way, figure 4 represents the case when λ = 15, where figure 4(a) represents the classical case and shows five equilibrium points, out of which three (L1, L2 and L3) are collinear and two (L4 and L5) are non-collinear. But when we consider the variation of mass case, i.e. λ1 = 0.2, λ3 = 0.4, 0.8 and 1.4, we observe that L1 and L3 move downward from the α--axis. We can see the complete movement of these equilibrium points in figure 5(c). In this way, the variation parameters are working as catalytic agents. 3.2 Basins of the Attracting Domain To solve the multivariate functions, we will use a simple, fast and accurate N-R iterative method. Therefore we illustrate the basins of the attracting domain for our present model. This method is activated when an initial condition is given to the configuration plane, while it stops when the positions of the equilibrium points are reached with some predefined accuracy. The regions of convergence are composed of all the initial values that tend to a specific equilibrium point. This is one of the most important qualitative properties of the dynamical systems. It is illustrated in the following way: After classifying a dense uniform grid of 1024×1024 initial conditions, a multiple scan of the configuration plane was done. By setting the maximum number of iterations to be 500, we set the predefined accuracy as 10 -15 . Using the above iterative method, we plotted the basins of attracting domain for three cases. The mathematical formulae for our model in α-β-plane when γ = 0, is presented by: 𝛼𝑛+1 = α𝑛 −( (V𝛼+𝑉1)(V𝛽𝛽+𝑉2𝛽)−(V𝛽+𝑉2)(V𝛼𝛽+𝑉1𝛽) (V𝛼𝛼+𝑉1𝛼)(V𝛽𝛽+𝑉2𝛽)−(V𝛼𝛽+𝑉1𝛽)(V𝛽𝛼+𝑉2𝛼) ) (𝛼𝑛,𝛽𝑛) , (17) 𝛽𝑛+1 = β𝑛 −( (V𝛽+𝑉2)(V𝛼𝛼+𝑉1𝛼)−(V𝛼+𝑉1)(V𝛽𝛼+𝑉2𝛼) (V𝛼𝛼+𝑉1𝛼)(V𝛽𝛽+𝑉2𝛽)−(V𝛼𝛽+𝑉1𝛽)(V𝛽𝛼+𝑉2𝛼) ) (𝛼𝑛,𝛽𝑛) . (18) where αn, βn are the values of α and β coordinates of the n th step of the Newton-Raphson iterative process in Equations (17) and (18). The first and second derivatives are obtained from the right hand sides of the equations of motion (10). If the initial point converges rapidly to one of the equilibrium points then this point (α, β) will be a member of the basin of attracting domain. This process stops when the successive approximation converges to an attractor (in the dynamical system attractor means equilibrium point). For the classification of different equilibrium points on the plane, we used a color code. Here we have represented the basins of attracting domain at λ = 1, 7, and 15 (taken from [5]) and these are shown in figures 6(a), 7(a) and 8(a). Figures 6(b), 7(b) and 8(b) are the zoomed parts of figures 6(a), 7(a ) and 8(a) respectively. From figure 6(b), we observe that all the equilibrium points belong to cyan colored regions which extend to infinity. From figure 7(b), we observe that there are seven attracting points (equilibrium points) out of which L 1, L4, L5 and L7 belong to the cyan, blue, light green and green colored regions which are extended to infinity respectively while L2 belongs to the light blue colored region which covers a finite region. Again from figure 8(b), we observe that L1 belongs to the cyan colored region, L2, L4 and L5 belong to the light blue colored region, while L3 belongs to the light green region, all of these regions extending to infinity. ANSARI, A.A. 70 (a) λ = 1 (b) Zoomed part of figure (a) near primaries Figure 6. Basins of attracting domain at λ1 = 0.2, λ3 = 1.4 in α β-plane. (a) λ = 7 (b) Zoomed part of figure (a) near primaries Figure 7. Basins of attracting domain at λ1 = 0.2, λ3 = 1.4 in α β-plane. VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 71 (a) λ = 15 (b) Zoomed part of figure (a) near primaries Figure 8. Basins of attracting domain at λ1 = 0.2, λ3 = 1.4 in α β-plane. 4. Stability of Equilibrium Points To reveal the stability properties of the small body's motion in its vicinity (α0+ α1, β0+ β1, γ0 + γ1), where α1, β1, γ1 are small movements from the equilibrium points (α0, β0, γ0). The variational equations for the system (10) can be written as: 𝛼1̈ −𝑓3𝛽1̇ + 𝑔3𝛾1̇ = (𝑉𝛼1𝛼1 0 +𝑉1𝛼1 0 )α1 +(𝑉𝛼1β1 0 +𝑉1β1 0 )β1 +(𝑉𝛼1γ1 0 +𝑉1𝛾1 0 )γ1, 𝛽1̈ −ℎ3𝛾1̇ + 𝑓3𝛼1̇ = (𝑉β1𝛼1 0 +𝑉2𝛼1 0 )α1 +(𝑉β1β1 0 +𝑉2β1 0 )β1 +(𝑉β1γ1 0 +𝑉2𝛾1 0 )γ1, 𝛾1̈ −𝑔3𝛼1̇ + ℎ3𝛽1̇ = (𝑉γ1𝛼1 0 +𝑉3𝛼1 0 )α1 +(𝑉γ1β1 0 +𝑉3β1 0 )β1 +(𝑉γ1γ1 0 +𝑉3𝛾1 0 )γ1. (19) where the superscript 0 denotes the value at the corresponding equilibrium point. In the phase space, the above system (19) can be rewritten as: α̇1 = 𝛼2, β̇1 = 𝛽2, γ̇1 = 𝛾2, ANSARI, A.A. 72 α2̇ = (𝑉𝛼1𝛼1 0 +𝑉1𝛼1 0 )α1 +(𝑉𝛼1β1 0 +𝑉1β1 0 )β1 +(𝑉𝛼1γ1 0 +𝑉1𝛾1 0 )γ1 +𝑓3𝛽2 −𝑔3𝛾2, β2̇ = (𝑉β1𝛼1 0 +𝑉2𝛼1 0 )α1 +(𝑉β1β1 0 +𝑉2β1 0 )β1 +(𝑉β1γ1 0 +𝑉2𝛾1 0 )γ1 +ℎ3𝛾2 −𝑓3𝛼2, γ2̇ = (𝑉γ1𝛼1 0 +𝑉3𝛼1 0 )α1 +(𝑉γ1β1 0 +𝑉3β1 0 )β1 +(𝑉γ1γ1 0 +𝑉3𝛾1 0 )γ1 +𝑔3𝛼2 −ℎ3𝛽2. (20) We use Meshcherskii space-time inverse transformations to examine the stability of the equilibrium points because the mass and distance of the small particle change with time. 𝛼3 = 𝜆3 −1/2 𝛼1, 𝛽3 = 𝜆3 −1/2 𝛽1, 𝛾3 = 𝜆3 −1/2 𝛾1, 𝛼4 = 𝜆3 −1/2 𝛼2, 𝛽4 = 𝜆3 −1/2 𝛽2, 𝛾4 = 𝜆3 −1/2 𝛾2. (21) With the help of equation (21), the system (20) can be written as follows: �̇� = 𝑀 𝑋, where �̇� = ( α3̇ β3̇ γ3̇ α4̇ β4̇ γ4̇ ) , 𝑋 = ( 𝛼3 𝛽3 𝛾3 𝛼4 𝛽4 𝛾4 ) and 𝑀 = ( 1 2 𝜆1 0 0 𝑉𝛼1𝛼1 0 +𝑉1𝛼1 0 𝑉β1𝛼1 0 +𝑉2𝛼1 0 𝑉γ1𝛼1 0 +𝑉3𝛼1 0 0 1 2 𝜆1 0 𝑉𝛼1β1 0 +𝑉1β1 0 𝑉β1β1 0 +𝑉2β1 0 𝑉γ1β1 0 +𝑉3β1 0 0 0 1 2 𝜆1 𝑉𝛼1γ1 0 +𝑉1𝛾1 0 𝑉β1γ1 0 +𝑉2𝛾1 0 𝑉γ1γ1 0 +𝑉3𝛾1 0 1 0 0 1 2 𝜆1 −𝑓3 𝑔3 0 1 0 𝑓3 1 2 𝜆1 −ℎ3 0 0 1 −𝑔3 ℎ3 1 2 𝜆1 ) . The characteristic equation for the matrix M is 𝜆4 6 +𝛼5𝜆4 5 +𝛼4𝜆4 4 +𝛼3𝜆4 3 +𝛼2𝜆4 2 +𝛼1𝜆4 +𝛼0 = 0, (22) where 𝛼0 = 1 64 (−8(2(−4(𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 )+4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 )) VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 73 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )−2(−4( (𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 ) +4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 ) ) (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 ) +2(−4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 ))(𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )) +8ℎ3 (−4(𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 ))𝜆1) +8𝑔3 (−4(𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 )+4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 ))𝜆1) +32𝑓3𝑉2𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 −32𝑓3𝑉𝛽1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 +8(4ℎ3𝑉1𝛽1 0 +4𝑔3𝑉1𝛽1 0 +4ℎ3𝑉𝛼1𝛽1 0 +4𝑔3𝑉𝛽1𝛽1 0 )(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 +32𝑓3𝑉𝛽1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 −32𝑓3𝑉1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 −32𝑓3𝑉𝛼1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 +32𝑓3𝑉1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 −8(4ℎ3𝑉1𝛼1 0 +4𝑔3𝑉2𝛼1 0 +4ℎ3𝑉𝛼1𝛼1 0 +4𝑔3𝑉𝛽1𝛼1 0 )(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 −32𝑓3𝑉2𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 +32𝑓3𝑉𝛼1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 −16𝑓3ℎ3𝑉1𝛾1 0 𝜆1 2 −16𝑓3𝑔3𝑉2𝛾1 0 𝜆1 2 −16𝑓3 2𝑉3𝛾1 0 𝜆1 2 −16𝑓3ℎ3𝑉𝛼1𝛾1 0 𝜆1 2 −4ℎ3(4ℎ3𝑉1𝛼1 0 +4𝑔3𝑉2𝛼1 0 +4ℎ3𝑉𝛼1𝛼1 0 +4𝑔3𝑉𝛽1𝛼1 0 )𝜆1 2 −4𝑔3(4ℎ3𝑉1𝛽1 0 +4𝑔3𝑉2𝛽1 0 +4ℎ3𝑉𝛼1𝛽1 0 +4𝑔3𝑉𝛽1𝛽1 0 )𝜆1 2 +4(−4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 ))𝜆1 2 −16𝑓3𝑔3𝑉𝛽1𝛾1 0 𝜆1 2 −16𝑓3ℎ3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 2 −16𝑉1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 2 −16𝑉𝛼1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 2 −16𝑓3𝑔3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 2 −16𝑉2𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 2 −16𝑉𝛽1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 2 −16𝑓3 2𝑉𝛾1𝛾1 0 𝜆1 2 +16𝑉1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 2 +16𝑉2𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 2 +16𝑉𝛼1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 2 +16𝑉𝛽1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 2 −8𝑓3𝑉1𝛽1 0 𝜆1 3 +8𝑔3𝑉1𝛾1 0 𝜆1 3 +8𝑓3𝑉2𝛼1 0 𝜆1 3 −8ℎ3𝑉2𝛾1 0 𝜆1 3 −8ℎ3𝑉𝛽1𝛾1 0 𝜆1 3 −8𝑓3𝑉𝛼1𝛽1 0 𝜆1 3 +8𝑔3𝑉𝛼1𝛾1 0 𝜆1 3 +8𝑓3𝑉𝛽1𝛼1 0 𝜆1 3 −8𝑔3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 3 +8ℎ3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 3 +4𝑓3 2𝜆1 4 +4𝑔3 2𝜆1 4 +4ℎ3 2𝜆1 4 −4𝑉1𝛼1 0 𝜆1 4 −4𝑉2𝛽1 0 𝜆1 4 −4𝑉3𝛾1 0 𝜆1 4 −4𝑉𝛼1𝛼1 0 𝜆1 4 −4𝑉𝛽1𝛽1 0 𝜆1 4 −4𝑉𝛾1𝛾1 0 𝜆1 4 +𝜆1 6) ANSARI, A.A. 74 𝛼1 = 1 64 (−16ℎ3(−4(𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 )) −16𝑔3(−4(𝑉1𝛾1 0 +𝑉𝛼1𝛾1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 )−64𝑓3𝑉2𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 ) +4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛾1 0 +𝑉𝛽1𝛾1 0 ))+64𝑓3𝑔3𝑉2𝛾1 0 𝜆1 −16(4ℎ3𝑉1𝛽1 0 +4𝑔3𝑉2𝛽1 0 +4ℎ3𝑉𝛼1𝛽1 0 +4𝑔3𝑉𝛽1𝛽1 0 )(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 ) −64𝑓3𝑉𝛽1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )+64𝑓3𝑉1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 ) +64𝑓3𝑉𝛼1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )+64𝑓3𝑓3𝑉3𝛾1 0 𝜆1 +64𝑓3ℎ3𝑉𝛼1𝛾1 0 𝜆1 +16(4ℎ3𝑉1𝛼1 0 +4𝑔3𝑉2𝛼1 0 +4ℎ3𝑉𝛼1𝛼1 0 +4𝑔3𝑉𝛽1𝛼1 0 ))(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 ) −64𝑓3𝑉1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )+64𝑓3𝑉2𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 ) −64𝑓3𝑉𝛼1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )+64𝑓3𝑉𝛽1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )+64𝑓3ℎ3𝑉1𝛾1 0 𝜆1 +16ℎ3(4ℎ3𝑉1𝛼1 0 +4𝑔3𝑉2𝛼1 0 +4ℎ3𝑉𝛼1𝛼1 0 +4𝑔3𝑉𝛽1𝛼1 0 )𝜆1 +16𝑔3(4ℎ3𝑉1𝛽1 0 +4𝑔3𝑉2𝛽1 0 +4ℎ3𝑉𝛼1𝛽1 0 +4𝑔3𝑉𝛽1𝛽1 0 )𝜆1 −16(−4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 ))𝜆1 +64𝑓3ℎ3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 +64𝑓3𝑔3𝑉𝛽1𝛾1 0 𝜆1 +64𝑉1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 +64𝑉2𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 +64𝑉𝛼1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 +64𝑓3𝑔3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 +64𝑉𝛽1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 +64𝑓3𝑓3𝑉𝛾1𝛾1 0 𝜆1 −64𝑉1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 −64𝑉2𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 −64𝑉𝛼1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1 +64𝑉𝛽1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )𝜆1)+48𝑓3𝑉1𝛽1 0 𝜆1 2 −48𝑔3𝑉1𝛾1 0 𝜆1 2 −48𝑓3𝑉2𝛼1 0 𝜆1 2 +48ℎ3𝑉2𝛾1 0 𝜆1 2 +48𝑓3𝑉𝛼1𝛽1 0 𝜆1 2 −48𝑔3𝑉𝛼1𝛾1 0 𝜆1 2 −48𝑓3𝑉𝛽1𝛼1 0 𝜆1 2 +48ℎ3𝑉𝛽1𝛾1 0 𝜆1 2 +48𝑔3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 2 −48ℎ3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 2 −32𝑓3 2𝜆1 3 −32𝑔3 2𝜆1 3 −32ℎ3 2𝜆1 3 +32𝑉1𝛼1 0 𝜆1 3 +32𝑉2𝛽1 0 𝜆1 3 +32𝑉3𝛾1 0 𝜆1 3 +32𝑉𝛼1𝛼1 0 𝜆1 3 +32𝑉𝛽1𝛽1 0 𝜆1 3 +32𝑉𝛾1𝛾1 0 𝜆1 3 −12𝜆1 5) 𝛼2 = 1 64 (−64𝑓3ℎ3𝑉1𝛾1 0 −64𝑓3𝑔3𝑉2𝛾1 0 −64𝑓3ℎ3𝑉1𝛾1 0 −64𝑓3𝑓3𝑉3𝛾1 0 −64𝑓3ℎ3𝑉𝛼1𝛾1 0 −16ℎ3(4ℎ3𝑉1𝛼1 0 +4𝑔3𝑉2𝛼1 0 +4ℎ3𝑉𝛼1𝛼1 0 +4𝑔3𝑉𝛽1𝛼1 0 ) −16𝑔3(4ℎ3𝑉1𝛽1 0 +4𝑔3𝑉2𝛽1 0 +4ℎ3𝑉𝛼1𝛽1 0 +4𝑔3𝑉𝛽1𝛽1 0 )−64𝑓3𝑔3𝑉𝛽1𝛾1 0 VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 75 +16(−4(𝑉1𝛽1 0 +𝑉𝛼1𝛽1 0 )(𝑉2𝛼1 0 +𝑉𝛽1𝛼1 0 )+4(𝑉1𝛼1 0 +𝑉𝛼1𝛼1 0 )(𝑉2𝛽1 0 +𝑉𝛽1𝛽1 0 )) −64𝑓3𝑓3𝑉𝛾1𝛾1 0 −64𝑓3ℎ3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )−64𝑉1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 ) −64𝑉𝛼1𝛾1 0 (𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )−64𝑓3𝑔3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )−64𝑉2𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 ) −64𝑉𝛽1𝛾1 0 (𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )+64𝑉1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )+64𝑉2𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 ) +64𝑉𝛼1𝛼1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )+64𝑉𝛽1𝛽1 0 (𝑉3𝛾1 0 +𝑉𝛾1𝛾1 0 )−96𝑓3𝑉1𝛽1 0 𝜆1 +96𝑔3𝑉1𝛾1 0 𝜆1 +96𝑓3𝑉2𝛼1 0 𝜆1 −96𝑓3𝑉𝛼1𝛽1 0 𝜆1 +96𝑔3𝑉𝛼1𝛾1 0 𝜆1 +96𝑓3𝑉𝛽1𝛼1 0 𝜆1 −96ℎ3𝑉𝛽1𝛾1 0 𝜆1 −96ℎ3𝑉2𝛾1 0 𝜆1 −96𝑔3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )𝜆1 +96ℎ3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )𝜆1 +96𝑓3 2𝜆1 2 +96𝑔3 2𝜆1 2 +96ℎ3 2𝜆1 2 −96𝑉1𝛼1 0 𝜆1 2 −96𝑉2𝛽1 0 𝜆1 2 −96𝑉3𝛾1 0 𝜆1 2 −96𝑉𝛼1𝛼1 0 𝜆1 2 −96𝑉𝛽1𝛽1 0 𝜆1 2 −96𝑉𝛾1𝛾1 0 𝜆1 2 +60𝜆1 4), 𝛼3 = 𝑓3𝑉1𝛽1 0 −𝑔3𝑉1𝛾1 0 −𝑓3𝑉2𝛼1 0 +ℎ3𝑉2𝛾1 0 +𝑓3𝑉𝛼1𝛽1 0 −𝑔3𝑉𝛼1𝛾1 0 −𝑓3𝑉𝛽1𝛼1 0 +ℎ3𝑉𝛽1𝛾1 0 +2𝑉2𝛽1 0 𝜆1 +𝑔3(𝑉3𝛼1 0 +𝑉𝛾1𝛼1 0 )−ℎ3(𝑉3𝛽1 0 +𝑉𝛾1𝛽1 0 )−2𝑓3 2𝜆1 −2𝑔3 2 −2ℎ3 2𝜆1 +2𝑉1𝛼1 0 𝜆1 +2𝑉𝛽1𝛽1 0 𝜆1 +2𝑉𝛼1𝛼1 0 𝜆1 +2𝑉𝛾1𝛾1 0 𝜆1 − 5 2 𝜆1 3, 𝛼4 = 𝑓3 2 +𝑔3 2 +ℎ3 2 −𝑉1𝛼1 0 −𝑉2𝛽1 0 −𝑉3𝛾1 0 −𝑉𝛼1𝛼1 0 −𝑉𝛽1𝛽1 0 −𝑉𝛾1𝛾1 0 + 15 4 𝜆1 2, 𝛼5 = −3𝜆1. We have solved equation (22) numerically for three different values of parameter λ (1, 7, 15) and evaluated characteristic roots which are given in tables (1-3). We observed from these roots given in these three tables 1, 2, 3 that all the equilibrium points are unstable because at-least one characteristic root is either a positive real number or positive real part of the complex characteristic root, while [5] found in his investigation that some equilibrium points are stable in some interval values for the parameter λ. In this way we can say that these variation parameters convert to all the equilibrium points as unstable. Therefore it is easy to say that these variation parameters have great impact on this dynamical system. Table 1. Corresponding characteristic roots of equilibrium points in α-β-plane at λ = 1, λ1 = 0.2, λ3 = 1.4. S.N. Equilibrium Point (α, β) Roots Nature 1 1.8922290573, - 0.2249047710 + 0.1238455712 ± 0.9648772652 i + 1.8829013676, - 1.7305925101 + 0.1000000000, + 0.1000000000 Unstable 2 0.0000000000, 0.0000000000 + 0.1919166735 ± 15.3247492922 i + 2.9831536654, + 0.0999999999 - 2.9669870125, + 0.1000000000 Unstable 3 1.8922290573, 0.2249047710 + 0.1238455712 ± 0.9648772652 i + 1.8829013676, + 0.1000000000 + 0.0999999999, - 1.7305925101 Unstable ANSARI, A.A. 76 Table 2. Corresponding characteristic roots of equilibrium points in α-β-plane at λ = 7, λ1 = 0.2, λ3 = 1.4. S.N. Equilibrium Point (α, β) Roots Nature 1 - 2.9288319681, - 0.5956468434 + 0.1244710980 ± 0.7705292447 i + 1.7304695528, - 1.5794117488 + 0.1000000000, + 0.1000000000 Unstable 2 0.1420900714, - 0.0070146815 + 0.1993522521 ± 46.6873463284 i + 2.8958790218, + 0.1000000000 - 2.8945835262, + 0.0999999999 Unstable 3 0.4622777422, 0.7320519601 + 0.0897598108 ± 7.5033419383 i + 1.8018965165, + 0.0999999999 + 0.1000000000, - 1.5814161383 Unstable 4 0.3598259379, - 0.8004678161 + 0.0957408351 ± 6.8651530569 i + 0.1000000000, + 1.6062369651 - 1.3977186353, + 0.1000000000 Unstable 5 1.0704385208, - 1.5676578978 + 1.2156733907 ± 0.0888414341 i + 0.1000000000, - 0.8781507268 - 1.1531960546, + 0.1000000000 Unstable Table 3. Corresponding characteristic roots of equilibrium points in α-β-plane at λ = 15, λ1 = 0.2, λ3 = 1.4. S.N. Equilibrium Point (α, β) Roots Nature 1 - 3.5192043182, - 0.9124002233 + 0.1245222506 ± 0.6945789131 i + 1.6854000078, + 0.0999999999 - 1.5344445091, + 0.1000000000 Unstable 2 + 0.1939056387, - 0.0096053140 + 0.1999647710 ± 76.6270430841 i + 2.8730485051, - 2.8729780473 + 0.1000000000, + 0.1000000000 Unstable 3 1.9795166567, - 1.3700857646 + 0.1000000000, - 1.2570087540 - 0.5578040967, + 0.8068614037 + 1.4079514469, + 0.1000000000 Unstable 4 0.5347180082, 0.6049358108 + 0.0922083409 ± 17.2608437186 i + 1.8078564632, + 0.1000000000 - 1.5922731451, + 0.1000000000 Unstable 5 - 0.6759400828, + 0.4504694723 + 0.1005874123 ± 15.9957018321 i + 1.5269661797, - 1.3281410043 + 0.1000000000, + 0.1000000000 Unstable Conclusion In this paper we have investigated the effect of variation of mass of the infinitesimal body (test particle) under the influence of the electromagnetic Copenhagen problem. We have found that these variation parameters have great impact on the behavior of this dynamical system. In our investigation, firstly we have determined the equations of motion where both variation parameters λ1 and λ3 are clearly visible. Using this system of equations of motion we numerically illustrated the equilibrium points for three values of magnetic moment ( λ = 1, 7, 15) (taken from [5]) and respectively we found three, seven and five equilibrium points for the four different values of variation parameters λ 1 = 0, 0.2 and λ3 = 1, 0.4, 0.8, 1.4. The complete movements of these equilibrium points are presented in figure (5). We have also performed the Newton-Raphson basins of attracting domain in the above said three cases of λ and for fixed values of λ1 = 0.2 and λ3 = 1.4. The variations of attracting domain are presented in the figures 6, 7 and 8. For a clearer view, we have shown the figures in the zoomed parts of figures 6(a), 7(a) and 8(a) in figures 6(b), 7(b) and 8(b), respectively. In the following investigation we have explored numerically the stability of corresponding equilibrium points, for which we have evaluated the characteristic roots corresponding to the equilibrium points, and these are given in tables 1, 2 and 3. We observed from the tables that all the equilibrium points are unstable, which finding is clearly different from that of the investigation done by [5]. Conflict of Interest The author declares no conflict of interest. VARIABLE MASS IN ELECTROMAGNETIC COPENHAGEN PROBLEM 77 Acknowledgment The author is thankful to Prof. Rabah Kellil for his valuable suggestions to bring this manuscript up to the present form. References 1. Brouwer, Dirk and Clemence, Gerald M. Methods of celestial mechanics: Elsevier, 2013. 2. Szebehely, V. Theory of Orbits: Academic Press, New York, 1967. 3. 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