# Existential Properties of Algebraic Integrals of a Rigid Body

Authors: A. I. Ismail; W. S. Amer
Journal: Advances in Astronomy
 (2022)

---

## Abstract

In this article, we consider kinematical considerations of a rigid body rotating around a given fixed point in a Newtonian force field exerted by an attractive center with a rotating couple about their principal axes of inertia. The kinematic equations and their well-known three elementary integrals of the problem are introduced. The existence properties of the algebraic integrals are considered. Besides, we search as a special case of the fourth algebraic integral for the problem of the rigid body’s motion around a fixed point under the action of a Newtonian force field with an orbiting couple. Lagrange’s case and Kovalevskaya’s one are obtained. The large parameter is used for satisfying the existing conditions of the algebraic integrals. The comparison between the obtained results and the previous ones is arising. The numerical solutions of the regulating system of motion are obtained utilizing the fourth-order Runge-Kutta method and are plotted in some figures to illustrate the positive impact of the imposed forces and torques on the behavior of the body at any time.


---

## Body

## 1. Introduction

In recent decades, many studies that are concerned with the motion of a rigid body (RB) have emerged, which was prevalent in the middle of the last century. The importance of this problem is due to its wide uses in practical life. In the current years, the employment of modern technology in studying this problem has pushed it forward once again.The principle of angular momentum [1] is considered the basic principle for obtaining the equations of motion (EOM) of this problem under the action of a Newtonian force field (NFF) in addition to the effect of a gyrostatic couple [2]. The principles of conservation of total energy, the projections of the body’s angular momentum in the direction of the fixed vertical axis, and the geometric constraint are used to obtain its three famous elementary integrals [3]. In [4], the author utilized the small parameter for satisfying the existence of some integrable cases of Euler–Poisson’s equations for an RB that is influenced by an NFF and a uniform one. The existence of the periodic solutions of an RB applying the small parameter with the application of gyro dynamic was considered in [5–10]. In [5], the chaotic behavior of the equation of Rayleigh–Duffing in the existence of gyro motion is examined. The small parameter method is applied in [6–10] to obtain the periodic solutions of different rotatory movements of the RB, while the Krylov-Bogoliubov-Mitropolsky technique (KBMT) is employed in [11–14] to gain the asymptotic solutions of the same body in the uniform filed, in addition to the gyro moment vector’s action. Moreover, the method of averaging is used in many studies, e.g., [15–20] to obtain the averaging system of the governing one for different cases of the rotational motion of asymmetric RB such as in a uniform field of force, in the existence of gyro moment, in an NFF, and in the presence of a point charge on the axis of dynamic symmetry. Recently, in [21–25], the large parameter method is utilized to obtain the approximate solutions of this problem under certain initial conditions. The numerical results of the flywheel motion are found in [26] when the body is influenced by NFF in addition to the gyro moment vector, while the vibrating motion of the RB is investigated numerically in [27] for the position of relative equilibrium.In [28], a generalization of two-dimensional mechanical integrable systems with twenty arbitrary parameters is considered. Some applications of these systems on the dynamics of RB problems have been presented when the body rotates around a certain fixed point, while in [29], the authors have examined the conditional dynamics of an RB attached with an asymmetric rotor through the axis of dynamic symmetry. The case of the axially symmetric magnetized gyrostat with electric charges is considered.In [30], some generalized cases for integrable problems in the Euclidean plane and the pseudosphere are studied, where some specified values for the free parameters of the integrable system are considered. In [31], the authors presented the EOM for systems of rigid bodies in the presence of potential fields. The Hamiltonian formalism is given for these systems in integrable cases. Also, the nonintegrability cases for different problems for the rigid bodies’ dynamics are given. In [32], the problem of a rotating RB around a fixed point is considered. The authors presented a new solving procedure type for the equations of Euler-Poisson. Arnold, Kozlov, and Neishtadt in [33] proved that the EOM of a heavy RB is integrable only in cases of Euler, Lagrange, Kovalevskaya, Chaplygin, and Goryachev Chaplygin. Thus, the natural way to find a new integrable problem in this field is to generalize the well-known ones. This motivates us to study the effect of NFF on the integrability of Lagrange’s case and Kovalevskaya’s one which represents the novelty of the present work. To achieve this purpose, the large parameter is used to investigate the existence of a fourth first integral for a gyrostatic couple. A comparison between the obtained results and the previous ones is presented. Numerical outcomes of the controlling system are achieved using the fourth-order Runge-Kutta method to show the positive impact of the affected forces and torques on the body’s motion. The significance of the present work can be found in many applications of the gyroscopic theory in various fields, such as engineering, physics, and industrial applications.


## 2. Methodology

Consider an RB of massM rotates about a certain fixed point O. Let OXYZ be the fixed frame in space and Oxyz be the rotating one fixed in the body and rotates with it. Assume that the body lies in a uniform field of gravity g and a Newtonian one which emerges from the attracting center at a distance R from O. Suppose a couple ℓ¯≡ℓ1,ℓ2,ℓ3 that effects on the body, in which it concerns with the moving coordinate system, see Figure 1. Consider an object dm at a given point px,y,z of the body, exerted by a Newtonian attraction force dF¯, with positions r¯ and ρ¯ concerning the points O1 and, O respectively.Figure 1
Force and couples' distribution.The EOM is derived in the form [2](1)Adpdt+C−Bqr+ℓ3q−ℓ2r=Y0γ3−Z0γ2+αC−Bγ2γ3dγ1dt=rγ2−qγ3,Y0=Mgy0,α=3gR,ABC,pqr,ℓ3ℓ1ℓ2x0y0z0,γ3γ1γ2,here and whatever the symbols ABC,… indicate canceled equations.This system represents the nonlinear differential EOM of the RB about a fixed point in an NFF with gyrostatic couple componentsℓ1,ℓ2, and ℓ3 effect on the axes Ox,Oy, and Oz, respectively. This system is of the first order of p,q,r,γ1,γ2, and γ3 which represents the angular velocity components and the projections of the fixed unit vector Z^¯ on the principal axes, respectively. The quantities A,B,C,x0,y0, and z0 represent the body’s moments of inertia and the mass center’s coordinates which are constants. The integration of this system yields the variables p,q,r,γ1,γ2, and γ3 which are functions of time t.System (1) permits us to write the following integrals, which are related to energy, angular momentum, and geometry:(2)Ap2+Bq2+Cr2−2X0γ1+Y0γ2+Z0γ3+αAγ12+Bγ22+Cγ32=const.,Ap+ℓ1γ1+Bq+ℓ2γ2+Cr+ℓ3γ3=const.,γ12+γ22+γ32=1.The EOM for an RB around a fixed point in a homogenous gravity field and their three first integrals follow from (1) and (2) by taking g as the acceleration of gravity, R⟶∞, and ℓi=0i=1,2,3 to obtain the corresponding ones in [34].


## 3. New Treatments for the Algebraic Integrals

For the motion of an RB about a specified fixed point under the impact of an NFF, the existence of the fourth algebraic integral exists only for the cases analogous to Eu1er′s case [35] and Lagrange’s one [36] for the corresponding motion of an RB. In this situation, we search about the fourth first algebraic integral for the RB problem under the action of an NFF with a gyrostatic couple ℓ3 about the z-axis; see Figure 1.The system of EOM for the RB rotatory movement around a fixed point in presence of a central NFF with a gyrostatic couple componentℓ3 about the principal z-axis can be acquired directly from the system of equations (1) at zero values of the other two components as in [37]. Therefore, one can get the corresponding three famous integrals from integrals (2) at ℓ1=ℓ2=0. It must be noted that these integrals do not explicitly include the time t, even though the last Jacobi's multiplier is equal to unity. Consequently, the presence of the fourth integral, which has been discovered to be algebraic, for this system demands that(3)r=r0=Const,forA=B,X0=Y0=0,which allows the problem to be reduced to quadrature [38].If the origin points of the coordinate systems and the center of mass coincide, i.e.,x0,y0,z0=0¯ and if ℓ1=0,ℓ2=0 are satisfied, then the system of EOM (1) will turn to the forms of [37, 39] as follows:(4)Adpdt+C−Bqr+qℓ3=αC−Bγ2γ3,Bdqdt+A−Cpr−pℓ3=αA−Cγ3γ1,Cdrdt+B−Apq=αB−Aγ1γ2,dγ1dt=rγ2−qγ3,γ2γ3γ1.Based on the above system, one gets(5)ddtA2p2+B2q2+Cr+ℓ32−2αℓ3B−Aγ1γ2=αBCdγ12dt+ACdγ22dt+ABdγ32dt.This case cannot be integrated due to the existence of the term−αℓ3B−Aγ1γ2 unless ℓ3=0. Therefore, for the case of Euler A<B<C or A>B>C besides x0,y0,z0=0¯, the fourth algebraic integral cannot be found. In other circumstances, the question arises as to whether the existence of a fourth algebraic integral is conceivable. Although it is well established that the ellipsoid of inertia must be a necessary ellipsoid of rotation, this is not a sufficient requirement for the existence of the new fourth algebraic integral [40].In what follows, we show for Lagrange's caseA=B,ℓ3≠0, and α≠0, the fourth algebraic integral is possible only if x0=y0=0, which is analog to Lagrange's conditions in the classical issue of the heavy RB motion [36].


## 4. Change of Variables

The purpose of the current section is to look at the EOM (1) from another point of view. Therefore, let us introduce new variablesyjj=1,2 and zj in place of the variables p,q, and γj as follows(6)y1=p+iq,y2=p−iq,z1=γ1+iγ2,z2=γ1−iγ2,i=−1,and replace t by −it.Based on the abovementioned statement and the substitution ofℓ1=ℓ2=0,A=B, and y0=0, we can rewrite the system of equations (1) in the form(7)Ady1dt=−A−Cy1r+Z0z1−X0γ3+αA−Cγ3z1+y1ℓ3,Bdy2dt=A−Cy2r−Z0z2+X0γ3−αA−Cz2γ3−y2ℓ3,Cdrdt=0.5X0z2−z1,dz1dt=y1γ3−rz1,dz2dt=rz2−y2γ3,dγ3dt=0.5y2z1−y1z2.This system admits the following first integrals:(8)Ay1y2+Cr2−X0z1+z2−2Z0γ3−αA−Cγ32=const.,Ay1z2+y2z1+2Cr+ℓ3γ3=const.,z1z2+γ32=const.Now, we introduce an arbitrary large parameterλ∝1lro where ro is assumed to be sufficiently small utilizing the substitution of.λ−2y1,λ−1y2,λ−1r,λ−3z1,λ−2z2,λ−3γ3,tλ,λ−1ℓ3 in place of y1,y2,r,z1,z2,γ3,t,ℓ3, thus system (7) and its first integrals (8) are rewritten in the forms(9)Ady1dt=−A−Cy1r+Z0z1−X0γ3+λ−3αA−Cγ3z1+y1ℓ3,Bdy2dt=A−Cy2r−Z0z2+λ−1X0γ3−λ−3αA−Cz2γ3−y2ℓ3,Cdrdt=0.5X0z2−λ−1z1,dz1dt=λ−1y1γ3−rz1,dz2dt=rz2−y2γ3,dγ3dt=0.5y2z1−y1z2,(10)Cr2−X0z2+λ−1Ay1y2−X0z1−2Z0γ3−λ−4αA−Cγ32=h1,Ay1z2+y2z1+2Cr+ℓ3γ3=h2,z1z2+γ32=h3,where h1,h2, and h3 are certain arbitrary constants.It must be noted that the zero values ofα and ℓ3 for system (9) yield the presented governing system of equations in [41]. The author demonstrated that the fourth algebraic integral may exist exclusively in the Lagrange circumstances A=B,x0=y0=0 and Kovalevskaya A=B=2C,y0=z0=0 for the rotary motion of a heavy RB with an inertia’s ellipsoid concerning the fixed point is a rotation’s ellipsoid [42, 43]. The proof was performed utilizing the first three terms of the expansion of the general integral of the acquired system of equations into a power series for the parameter λ∝1/r0 which is small when r0 is sufficiently large.The right sides of the derived equations of the system of motion, as well as their first integrals, are polynomials about the variablesy1,y2,r,z1,z2,γ3, and λ−1 (where λ is assumed to be large), which we analyzed in this work. According to the suitable manner of [3], we are going to demonstrate that the fourth algebraic integral exists for the mentioned cases in presence of the large parameter.Because the first three parts of the general integral’s expansion of the equations of system (9) into such a power series for the variable, 1/λ is independent of α or even ℓ3, in addition to the right-hand sides of system (9), and also the relations (10), are polynomials for y1,y2,r,z1,z2,γ3, and 1/λ; the outcomes [41] must be regarded as the precondition for the formation of a novel fourth integral of the RB problem under investigation.


## 5. Existing Conditions for the Fourth Integral

The goal of this section is to determine the indispensable conditions for the existence of an unprecedented fourth algebraic integral for system (7), which can be rewritten according to Kovalevskaya’s conditions in the form(11)2dy1dt=−y1r−c0γ3+αγ3z1+y1ℓ0,2dy2dt=y2r−c0γ3−αz2γ3−y2ℓ0,drdt=0.5c0z2−z1,dz1dt=y1γ3−rz1,dz2dt=rz2−y2γ3,dγ3dt=0.5y2z1−y1z2;c0=X0C,ℓ0=ℓ3C,with their the three first algebraic integrals(12)2y1y2+r2−c0z1+z2−αγ32=const,y1z2+y2z1+2r+ℓ0γ3=const,z1z2+γ32=const.Replacing in system (11), the quantities y1,y2,r,z2,t, and ℓ0 by λ−1/2y1,λ1/2y2,λ1/2r,λz2,λ−1/2t, and λ−3/2ℓ0, where λ is an arbitrary parameter which is considered to be large, to obtain the following system of equations(13)2dy1dt=−y1r−c0γ3+αγ3z1+λ−2y1ℓ0,2dy2dt=y2r−λ−1c0γ3−αz2γ3−λ−1y2ℓ0,drdt=0.5c0z2−λ−1z1,dz1dt=λ−1y1γ3−rz1,dz2dt=rz2−λ−1y2γ3,dγ3dt=0.5y2z1−y1z2,which is met with the following first algebraic integrals(14)r2−c0z2+λ−12y1y2−c0z1−αγ32=h1,y1z2+y2z1+r+λ−2ℓ0γ3=h2,z1z2+λ−1γ32=h3.Here,h1,h2, and h3 are certain arbitrary constants depending on λ. At λ=1, the above first integrals (14) will be identical to the integrals (2) when ℓ1=ℓ2=0.Substituting (14) about the quantities y1,z1,z2 into (13), we obtain(15)dγ3dr=y2z1−y1z2c0z2−λ−1z1,dy2dr=y2r+λ−1c0γ3−αz2γ3−λ−1y2ℓ0c0z2−λ−1z1.Based on [41], the fourth algebraic integral is found for system (11), then the previous system produces an algebraic integral in the form(16)Fh1,h2,h3,y2,γ3,r=const.,which can be extended as a power series of λ−1/p (p is an integer) in a very close region of the parameter λ⟶∞ to get(17)F0y2,γ3,r+λ−1/pF1y2,γ3,r+⋯+λ−1Fpy2,γ3,r+⋯=const,with coefficients F0,F1,…,Fp, which are algebraic functions of all their arguments.At least one of the quantitiesy2 or γ3 must be considered while expanding F0. Furthermore, if λ is large enough [34], the solutions y2r and γ3r of system (15) may be expanded according to a series of integral power of λ as follows:(18)y2=y20+λ−1y21+⋯,γ3=γ30+λ−1γ31+⋯,where the coefficients y20,y21,γ30, and γ31 will be determined from the next equations(19)dy20dr=y20r−αz20γ30c0z20,dγ30dr=y20z10−y10z20c0z20,(20)dy21dr=1c0z20y21r+c0γ30−αz21γ30+z20γ31−y20ℓ0−1z20z21−z10ry20−αz20γ30,dγ31dr=1c0z20y20z11−y21z10−y10z21−y11z20+−1z20z21−z10y20z11−y11z20.The expansions of the functionsy1,z1, and z2 take the forms(21)y1=y10+λ−1y11+⋯,z1=z10+λ−1z11+⋯,z2=z20+λ−1z20+⋯.These expansions besides the above ones (18) must satisfy the integrals (14).Substituting (18) into (17) to produce(22)F0y20,γ30,r+λ−1/pF1y20,γ30,r+⋯+λ−1y21∂F0y20,γ30,r∂y20+γ31∂F0y20,γ30,r∂γ30+⋯+λ−2⋯+⋯=const.According to the above equation, the first integrals of systems (19) and (20) must arise, which they can be formulated as follows:(23)F0y20,γ30,r=const,(24)y21∂F0y20,γ30,r∂y20+γ31∂F0y20,γ30,r∂γ30+F0y20,γ30,r=const.It must be mentioned that the left-hand side of (23) will depend necessarily upon at least one of the variables y20 or γ30.


## 6. Results and Discussion

To verify the formulated statement in the previous section, it is enough to demonstrate that system (13) includes a case in which finding the fourth algebraic integral is impossible. To realize this aim, consider the solution specified by the values of the arbitrary constants in the differential equations (15) as follows:(25)h1=a2,h2=h3=λ−2,where a is independent of λ. Making use of the expansions (18) and (21) into the first integrals (14), then we can write the solution under evaluation in the form(26)r2−c0z2+λ−12y1y2−c0z1−αγ32=a2,y1z2+y2z1+r+λ−2ℓ0γ3=λ−2,z1z2+λ−1γ32=λ−2.Therefore, the following expressions are obtained(27)z10=0,c0z20=r2−a2,y10=−c0rγ30r2−a2.In addition to the relations of the quantitiesz11,z21, and y11 as follows(28)y11z20=−y10z21+y20z11+rγ31,z11z20=−γ302−z10z21,c0z21=2y10y20−αγ302−c0z10.Then, equation (19), which employs (27), can be represented as follows:(29)dy20dr=rr2−a2y20−αc0γ30,dγ30dr=rr2−a2γ30,in which they are satisfied by the solutions(30)y20=−αc0rr2−a2,γ30=r2−a2.Now, the quantitiesy21 and γ31 must be determined. Therefore, substituting relations (27), (28), and (30) into equations (20) and transforming the variables to new ones Y and Γ according to the formulas(31)y21=r2−a2Y,γ31=r2−a2Γ,to obtain the following system:(32)dYdr=−αc0Γ+1r2−a22α2a4c0r2−a2+α2r2c0+2α2a2c0+c0+βrr2−a2,dΓdr=αrr2−a21−2a2r2−a2;β=ℓ0αc0.Integration of the last of equations yields(33)Γ=α2lnr2−a2+αa2r2−a2+const.The substitution from the previous expression into the first equation of (32) yields(34)Y=Slnr−ar+a−α2r2c0lnr2−a2+12βlnr2−a2+φ1r,where(35)S=12a3α2a2c0+c0.Here,φ1 and φjj=2,3,… in the following equations represent algebraic functions of r. Returning to the previous variables, we get(36)y21=12β−α2rc0r2−a2lnr2−a2+Sr2−a2lnr−ar+a+φ2r,γ31=α2r2−a2lnr2−a2+φ3r.As previously stated, if system (11) has the fourth algebraic integral, system (20) must also have the algebraic integral (24), which can be reduced utilizing the relations (30) and (36) to the form(37)S∂F0∂y20lnr−ar+a+12β−α2rc0∂F0∂y20+α∂F0∂γ30lnr2−a2=φ4r.Based on the relation (30) and the assumptions of the function F0y20,γ30,r, the bracketed expressions will be algebraic functions of r. The function lnr2−a2 cannot be expressed as a function of r, and lnr−a/r+a. Then, the left side in (37) can be considered as a function of r if the bracketed expressions equal to zero. Therefore, the condition must be fulfilled for each r=±a.(38)∂F0∂y20=∂F0∂γ30=0,S≠0.According to the property of the expressionF0y20,γ30,r, condition (38) is not satisfied. This establishes that for the case under examination of the presence of a general fourth algebraic integral of (11) which is the same, for system (1) is an unattainable when ℓ1=ℓ2=0. As presented in [44], in this case, the issue does not have a single-valued generic solution on the entire plane. As a result, the necessary and sufficient requirement of the availability of the fourth independent general algebraic integral of system (1) at ℓ1=ℓ2=0 for A=B is the condition x0=y0=0.A comparison of the obtained manuscript’s results using the large parameter with that obtained for similar previous problems using the small parameter is summarized in Table1.Table 1
Distinctions between the prior issues and one being considered.
Ser.The preceding problemsThe current problem1The oscillations of the body are strongThe oscillations of the body are weak2The initial velocity is high sufficientlyThe initial velocity is low sufficiently3The parameterλ is smallThe parameterλ is large4The started required energy for the motion is highThe started required energy for the motion is low5The integrals domain isDt,r0⟶∞,λ⟶0The integrals domain isDt,r0⟶0,λ⟶∞


## 7. Numerical Solution

This section’s goal is to obtain the numerical solutions of the controlling system (1) utilizing the Runge-Kutta method from fourth order in the presence of all applied forces and moments. Therefore, the data listed below are utilized(39)A=10kg.m2,C=5kg.m2,B=6kg.m2,M=300kg,g=9.81ms−2,R=1000m,x0=0.0001m,y0=0.0002m,z0=0.0003m,ℓ1,ℓ2,ℓ3=5,10,15kg.m2.s−1.Figures2–4 are calculated when ℓ1,ℓ2, and ℓ3 have different values, respectively, with the constant values of the other used parameters. These figures reveal the variation of the numerical results of the governing system (1) with time.Figure 2
Variation of the numerical solutions ofp,q,r,γ1,γ2, and γ3 with time when ℓ1=5,10,15  kg.m2.s−1.
(a)(b)(c)(d)(e)(f)Figure 3
Time histories of the numerical solutions ofp,q,r,γ1,γ2, and γ3 with time when ℓ2=5,10,15  kg.m2.s−1.
(a)(b)(c)(d)(e)(f)Figure 4
Time histories of the numerical solutions ofp,q,r,γ1,γ2, and γ3 with time when ℓ3=5,10,15  kg.m2.s−1.
(a)(b)(c)(d)(e)(f)A closer inspection of the parts of Figure2 demonstrates that the related solutions with p,q,r, and γ1 have periodic behaviors with the increasing of time. Moreover, the waves’ amplitudes increase with the increasing of ℓ1 as in Figures 2(a) and 2(d) in which the oscillations number decreases, and the wavelengths increase. On the other hand, the amplitudes and the oscillation’s numbers of the waveforms that describe the solutions q and r decrease with the increasing of ℓ1 values. At all, increasing or decreasing of the angular velocity components yield increasing or decreasing of the total energy of the considered body which can be employed through various applications in life, especially for the applications that used the gyroscopic theory. An inspection of the plotted curves of parts (e) and (f) of the same figure reveals that the behavior of γ2 decreases gradually while the γ3 behavior increases gradually with time with slight vibrations of these curves.The variation ofℓ2 values on the behavior of the solutions has a distinct manner as drawn in parts of Figure 2, in which decaying of the waves is observed in parts (a), (c), and (e). The amplitudes of the periodic waves describing the solution q decrease with the increasing of ℓ2 values. On the other hand, oscillations of γ1 and γ3 become slightly with the increase of ℓ2 values, in which an increasing and decreasing manners are observed from Figures 3(d) and 3(f) for results of γ1 and γ3, respectively.The plotted curves in parts of Figure4 explore the behavior of the waves illustrating the results p,q,r, and γ3 has periodically formed, in which the time histories of the solutions p,q, and γ3 increase when ℓ3 increases as drawn in Figures 4(a), 4(b), and 4(f). For this case, the behavior of γ1 and γ2 increases and decreases with small vibrations with the increasing of ℓ3 values, see Figures 4(d) and 4(e).The numerical results of the controlling system (1) whenℓ1=ℓ2=0 and ℓ3=5,10,15 are graphed in parts of Figure 5 to observe the time histories of these results. It is noted that the plotted curves in parts (a), (b), and (e) of this figure have periodic forms. The oscillation numbers increase with the increasing of ℓ3 values, the wavelength decreases, and the amplitudes of these oscillations become steady to some extent as seen in parts (a) and (b). Whereas the amplitudes of the waves which describe γ3 and their wavelengths decrease when ℓ3 increases. On the other hand, the behavior of the γ1 and γ2 increases and decreases, gradually with time as graphed in Figures 5(d) and 5(f), respectively.Figure 5
Time histories of the numerical solutions ofp,q,r,γ1,γ2, and γ3 with time when ℓ1=ℓ2=0 at ℓ3=5,10,15  kg.m2.s−1.
(a)(b)(c)(d)(e)(f)Curves of Figure6 are plotted when ℓ1=ℓ2=0 and x0=y0=z0=0, which means that the body rotates under the impact of the third projections of the gyro vector and the center of mass coincides with the origin point of the coordinates of the used Cartesian systems. Periodic waves are noted for the curves of the solutions p and q as explored in parts (a) and (b), respectively. The number of oscillations increases with the increasing of ℓ3 values, while their wavelengths decrease, and their amplitudes become stationary. The waves describing the axial angular velocity r and the solutions γ1,γ2 have periodic decay manners with the increasing of ℓ3 values as plotted in parts (c), (d), and (e) of Figure 6. The significant impact of ℓ3 on the behavior of γ3 is drawn in Figures 6(f), in which the increasing of ℓ3 values, yield periodic waves with decreasing of their amplitudes and wavelengths.Figure 6
Time histories of the numerical solutions ofp,q,r,γ1,γ2, and γ3 with time when ℓ1=ℓ2=0 and x0=y0=z0=0 at ℓ3=5,10,15  kg.m2.s−1.
(a)(b)(c)(d)(e)(f)


## 8. Conclusions

(i)
The angular momentum principle is used to derive the EOM of an RB rotating, about a certain fixed point, under the action of an NFF exerted by an attractive center, and a rotating coupleℓ1,ℓ2, and ℓ3 about the principal axes of inertia.(ii)
The principles of the total energy and angular momentum conservation besides a geometric constraint are used to derive the famous three first integrals of the problem.(iii)
The existing conditions for the algebraic integrals are considered. In addition, in a special case, the fourth algebraic integral for the problem of the motion of an RB under the existence of an NFF with orbiting a coupleℓ3 about the z- axis is presented.(iv)
The necessary and sufficient requirements for the presence of the fourth algebraic integrals are considered using the large parameter for Lagrange’s case and Kovalevskaya's one.(v)
The large parameter is used for satisfying the existing conditions of the fourth algebraic integral for the system of the mentioned cases.(vi)
The necessary and sufficient requirement for the presence of the fourth independent algebraic integral of system (1) at ℓ1=ℓ2=0 for A=B is the condition x0=y0=0 is deduced.(vii)
The numerical results of the controlling system are achieved using the Runge-Kutta method from fourth-order and plotted in different figures when all of the applied forces and torques are taken into consideration to reveal the significance of these forces and moments on the behavior of the body at any time.



---

*Source: 9393658-2022-04-22.xml*