Existential Properties of Algebraic Integrals of a Rigid Body

Ismail, A. I.; Amer, W. S.

doi:https://doi.org/10.1155/2022/9393658

Advances in Astronomy

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9393658 | https://doi.org/10.1155/2022/9393658

Existential Properties of Algebraic Integrals of a Rigid Body

A. I. Ismail^1,2and W. S. Amer³

Academic Editor: Zdzislaw E. Musielak

Received16 Jan 2022

Revised12 Feb 2022

Accepted18 Feb 2022

Published22 Apr 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.

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 mass rotates about a certain fixed point . Let be the fixed frame in space and be the rotating one fixed in the body and rotates with it. Assume that the body lies in a uniform field of gravity and a Newtonian one which emerges from the attracting center at a distance from . Suppose a couple that effects on the body, in which it concerns with the moving coordinate system, see Figure 1. Consider an object at a given point of the body, exerted by a Newtonian attraction force , with positions and concerning the points and, respectively.

The EOM is derived in the form [2]here and whatever the symbols indicate canceled equations.

This system represents the nonlinear differential EOM of the RB about a fixed point in an NFF with gyrostatic couple components and effect on the axes and , respectively. This system is of the first order of and which represents the angular velocity components and the projections of the fixed unit vector on the principal axes, respectively. The quantities and represent the body’s moments of inertia and the mass center’s coordinates which are constants. The integration of this system yields the variables and which are functions of time t.

System (1) permits us to write the following integrals, which are related to energy, angular momentum, and geometry:

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 as the acceleration of gravity, , and 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 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 component 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 . It must be noted that these integrals do not explicitly include the time , 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 thatwhich 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., and if are satisfied, then the system of EOM (1) will turn to the forms of [37, 39] as follows:

Based on the above system, one gets

This case cannot be integrated due to the existence of the term unless . Therefore, for the case of Euler or besides , 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 case , and , the fourth algebraic integral is possible only if , 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 variables and in place of the variables and as followsand replace by .

Based on the abovementioned statement and the substitution of and , we can rewrite the system of equations (1) in the form

This system admits the following first integrals:

Now, we introduce an arbitrary large parameter where is assumed to be sufficiently small utilizing the substitution of.

in place of , thus system (7) and its first integrals (8) are rewritten in the formswhere and are certain arbitrary constants.

It must be noted that the zero values of and 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 and Kovalevskaya 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 which is small when 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 variables , and (where 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, is independent of or even , in addition to the right-hand sides of system (9), and also the relations (10), are polynomials for and ; 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 formwith their the three first algebraic integrals

Replacing in system (11), the quantities and by and , where is an arbitrary parameter which is considered to be large, to obtain the following system of equationswhich is met with the following first algebraic integrals

Here, and are certain arbitrary constants depending on . At , the above first integrals (14) will be identical to the integrals (2) when .

Substituting (14) about the quantities into (13), we obtain

Based on [41], the fourth algebraic integral is found for system (11), then the previous system produces an algebraic integral in the formwhich can be extended as a power series of ( is an integer) in a very close region of the parameter to getwith coefficients , which are algebraic functions of all their arguments.

At least one of the quantities or must be considered while expanding . Furthermore, if is large enough [34], the solutions and of system (15) may be expanded according to a series of integral power of as follows:where the coefficients and will be determined from the next equations

The expansions of the functions and take the forms

These expansions besides the above ones (18) must satisfy the integrals (14).

Substituting (18) into (17) to produce

According to the above equation, the first integrals of systems (19) and (20) must arise, which they can be formulated as follows:

It must be mentioned that the left-hand side of (23) will depend necessarily upon at least one of the variables or .

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:where is independent of . Making use of the expansions (18) and (21) into the first integrals (14), then we can write the solution under evaluation in the form

Therefore, the following expressions are obtained

In addition to the relations of the quantities and as follows

Then, equation (19), which employs (27), can be represented as follows:in which they are satisfied by the solutions

Now, the quantities and must be determined. Therefore, substituting relations (27), (28), and (30) into equations (20) and transforming the variables to new ones and according to the formulasto obtain the following system:

Integration of the last of equations yields

The substitution from the previous expression into the first equation of (32) yieldswhere

Here, and in the following equations represent algebraic functions of . Returning to the previous variables, we get

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

Based on the relation (30) and the assumptions of the function , the bracketed expressions will be algebraic functions of . The function cannot be expressed as a function of , and . Then, the left side in (37) can be considered as a function of if the bracketed expressions equal to zero. Therefore, the condition must be fulfilled for each .

According to the property of the expression , 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 . 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 for is the condition .

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 Table 1.

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

Figures 2–4 are calculated when and 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.

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

(a)

(b)

(c)

(d)

(e)

(f)

A closer inspection of the parts of Figure 2 demonstrates that the related solutions with and have periodic behaviors with the increasing of time. Moreover, the waves’ amplitudes increase with the increasing of 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 and decrease with the increasing of 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 decreases gradually while the behavior increases gradually with time with slight vibrations of these curves.

The variation of 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 decrease with the increasing of values. On the other hand, oscillations of and become slightly with the increase of values, in which an increasing and decreasing manners are observed from Figures 3(d) and 3(f) for results of and , respectively.

The plotted curves in parts of Figure 4 explore the behavior of the waves illustrating the results and has periodically formed, in which the time histories of the solutions , and increase when increases as drawn in Figures 4(a), 4(b), and 4(f). For this case, the behavior of and increases and decreases with small vibrations with the increasing of values, see Figures 4(d) and 4(e).

The numerical results of the controlling system (1) when and 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 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 and their wavelengths decrease when increases. On the other hand, the behavior of the and increases and decreases, gradually with time as graphed in Figures 5(d) and 5(f), respectively.

(a)

(b)

(c)

(d)

(e)

(f)

Curves of Figure 6 are plotted when and , 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 and as explored in parts (a) and (b), respectively. The number of oscillations increases with the increasing of values, while their wavelengths decrease, and their amplitudes become stationary. The waves describing the axial angular velocity and the solutions have periodic decay manners with the increasing of values as plotted in parts (c), (d), and (e) of Figure 6. The significant impact of on the behavior of is drawn in Figures 6(f), in which the increasing of values, yield periodic waves with decreasing of their amplitudes and wavelengths.

(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 couple and 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 couple about the - 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 for is the condition 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.

Data Availability

Data sharing is not applicable to this publication because no datasets were collected or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4240002DSR03.

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Copyright © 2022 A. I. Ismail and W. S. Amer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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