A novel study on the fourth first integral for the rotatory motion of an impacted charged rigid body by external torques

Abstract

The main purpose of this study is to explore the feasibility of achieving a fourth integral for a charged rigid body (RGB) rotary movement, considering the influence of gyrostatic torque (GT) and a uniform electromagnetic field. Six nonlinear first-order differential equations (DEs) and their three first integrals, pertaining to geometry, area, and energy, regulate this problem. The analytical solution to the issue necessitates approaching a crucial fourth integral. This paper presents a pivotal criterion for a function $F$ , contingent on all RGB variables, to qualify such integral is outlined. The achieved outcomes are validated through comparing them with the renowned cases and then they can be considered a consolidation or mainstreaming of prior research efforts.

Introduction

The RGB problem has captured the attention of eminent scientists for over two centuries, owing to its significant applications in various aspects of life, particularly in fields related to gyroscopic theory, such as spacecraft, airplanes, submarines, and similar technologies. This problem is back to the pivotal work of Euler in 1758, ¹ where he lays the foundational aspects and studies the free motion of the RGB. In 1788, ² Lagrange explored the symmetric case of this problem, incorporating the influence of a uniform gravity field. Remarkably, the solutions to these cases are expressed as elliptic functions of the time variable. The equations governing this motion are derived from the equations of Euler-Poisson, which, intriguingly, feature three first integrals of motion: the integrals of area, energy, and geometry. However, the resolution of these equations relies on the inclusion of an additional fourth first integral, which is independent of the aforementioned three.^3–5 One century from Lagrange’s case, Kowalevski in 1889 ⁶ made her significant contribution to identifying a third integrable case (IC). An extraordinary aspect of Kowalevski’s discovery is that the general solution of the system governing a rotating heavy RGB about a fixed point can be formulated for all initial conditions exclusively in these three cases.

Several IC are found after the aforementioned cases such as Hess, ⁷ Steklov, ⁸ Chaplygin ⁹ , and others. These scenarios have succeeded in the first three IC, which investigate integrability for the problem. However, the illusive fourth first integral has yet to be obtained in its general form. The pursuit of this integral results in the invention of several IC, as shown in Ref. 10–13, especially when the RGB is subjected to gyroscopic and potential stresses in various circumstances. These attempts aim to untangle the intricacies of the situation by identifying IC and contributing to a more complete knowledge of the dynamics involved. Generalizations of the introduced IC by Kovalevskaya, Lagrange, and six other scenarios have been successfully derived. Furthermore, in Ref. 14, additional conditions are formulated to extend the generalization to cover scenarios such as Chaplygin and Yehia. Moreover,^15,16 delves into the exploration of determining the urgent fourth integral for the RGB’s motion, considering the influence of a Newtonian field (NF) and GT vector in a straightforward procedure. For a more in-depth understanding of various IC, researchers are encouraged to see Ref. 3–5. The question that comes to mind now is how long the search for integrable states will be carried out. The answer is quite straightforward and can be expressed as “Forever,” especially when we can apply and determine the moments of forces that result in integrable problems with convenient conditions. Achieving the fourth first integral in a general scenario proves to be a formidable challenge, obstructing progress in finding general solutions for the equations of motion (EOMs). Consequently, an alternative approach is needed to unearth these solutions. Perturbation methods emerge as crucial tools in determining approximate solutions for such problems, for example, Ref. 17–25. As detailed in Ref. 17–22, the averaging technique is employed to construct the averaged EOMs for a symmetric RGB. It is noteworthy that this system has been successfully solved for various cases, particularly when the body rotates in a uniform field,^16,17 in the presence of GT,^19–21 and in the case of charged RGB.^21,22 Highly accurate approximate solutions for the RGB problem are achieved in Ref. 23–25 using Poincaré’s small parameter technique. These solutions are obtained in various contexts, including uniform field, GT, NF, and scenarios that are close to Kowalewski’s gyroscope. In Refs. 26,27, for Lagrange case, the averaged system of equations is generated once circumstances are supplied for averaging EOMs in regard to the nutation angle. The movement of the RGB in a medium with linear dissipation is represented by a real physical model of perturbation and a numerical solution is approached. In Ref. 28, with the addition of a perturbation term, the author creates an approximate solution to the system of Euler’s EOMs for symmetric spinning RGBs, as studied in Ref. 27.

Recently, the effects of a GT on the movement of the RGB that is both charged and immersed in a viscous, incompressible liquid have been examined in Ref. 29. The three-dimensional movement of the RGB when it is subjected to unchanging torques that are fixed to the RGB itself and a GT is presented in Ref. 30–35. The authors focused their work on phase space paths to show the evolving states of a system in three different cases. The first two cases deal with applied constant torques along the RGB’s major and minor axes, while the third one investigates the effects of applying a constant torque along the middle axis of the RGB. In Ref. 36, it is demonstrated that the triaxial ellipsoid of inertia equations for gyrostat orientation movement is reversible with two fixed sets. A third fixed set is found for a spherical ellipsoid of inertia. In the scenario of a circular equatorial orbit, it is found that two of these three sets match the corresponding sets of the generating issue. The generating problem’s families of symmetric oscillations are shown to split and produce single oscillations. In Ref. 37, an axisymmetric gyrostat satellite’s rotational motion in a circular orbit while subjected to its gravitational torque is examined. Investigations are conducted on periodic motion of the satellite’s symmetrical axis with respect to the orbital coordinate system. These motions show up as a gradual procession around the orbital plane’s normal in absolute space. In Ref. 38, using control torques derived from Lorentzian forces, the stability of regular precessions of a satellite in a circular orbit is examined. A unique family of linear time-varying systems that can be reduced to time-invariant ones includes the linearized system of EOMs. Both the original time-varying systems and the reduced time-invariant systems were used to study controllability.

Based on the above survey of the fourth integral required to solve the RGB’s problem in its generality and the aforementioned regarding the use of perturbation methods to obtain the alternative approximate solutions of this problem, we present in this study a very simplified method for obtaining this integral. Henceforth, we have considered the charged RGB’s motion, taking into account, the influence of GT. In the present context for the existence of a fourth first integral, we aim to establish a crucial condition that is essential to function F, which is dependent on all body parameters. This will be accomplished through a straightforward procedure. The validity of this procedure is substantiated by its successful application to renowned cases, affirming its effectiveness and underscoring the significance of this criterion in the study. Analysis of rotational dynamics can be used in aircraft, satellites, and spacecraft for attitude control and stability. Moreover, the results may be utilized in understanding rotational motion for tires, braking systems, and suspension design. Furthermore, it utilized in helicopters, drones, and airplanes for generating lift and thrust.

A detailed exposition of the problem

For an RGB with mass $M$ rotating around a fixed point $O$ , two coordinate systems are considered: a stationary $OXYZ$ system and a mobile $Oxyz$ system, attached to the RGB. The mobile system rotates with angular velocity $\underset{\_}{\omega }=\left(p,q,r\right)$ about the principal inertia axes $Ox,Oy,$ and $Oz$ . The RGB undergoes the impact of the GT $\underset{\_}{\mathcal{l}}=\left({\mathcal{l}}_1,{\mathcal{l}}_2,{\mathcal{l}}_3\right)$ , which is aligned with the system’s principal inertia axes. ⁵ It is considered that the RGB rotates under the influence of a uniform electromagnetic field of strength $H$ due to a charge $e$ . Therefore, the resulted torque can be estimated in light of Lorentz force $e\left(\underset{\_}{V}\wedge \underset{\_}{H}\right)$ ³⁹ where $\underset{¯}{V}=\left(\underset{¯}{\omega }\wedge \underset{¯}{l\prime }\right)$ , $\underset{¯}{l\prime }\equiv \left(0,0,l\prime \right)$ is the position vector of $e$ with respect the origin $O$ . Hence, the corresponding torque has the form $e\left(\underset{\_}{H}.\underset{¯}{l\prime }\right)\left(\underset{\_}{\omega }\wedge \underset{¯}{l\prime }\right)$ with projections $\left(eH{l}^{\prime 2}q\cos \left(\hspace{0.17em}\theta \right),-eH{l}^{\prime 2}p\cos \left(\theta \right),0\right)$ on the same principal axes of inertia, in which $\theta$ is the angle $l^{\prime }$ make with the fixed axis $OZ$ . ⁴⁰

Consequently, the controlling system of motion takes the form^3–5

$$\begin{gathered} A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\dot{p}=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]r-\left(Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\hspace{0.17em}\right)q+d_2\gamma ″-d_3\gamma \prime , \\ {A}^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\dot{q}=\left(Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right)p-\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]r+d_3\gamma -d_1\gamma ″, \\ C\dot{r}=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]q-\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]p+d_1\gamma \prime -d_2\gamma ,\dot{\gamma }=r\gamma \prime -q\gamma ″, \dot{\gamma }\prime =p\gamma ″-r\gamma , \dot{\gamma }″=q\gamma -p\gamma \prime , d_1=Mg{x}_0, d_2=Mg{y}_0, d_3=Mg{z}_0, \end{gathered}$$

(1)

[See Figure 1]

Figure 1.

Description of the problem.

Here, $\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right),A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right),C\right]$ denote to the RGB’s inertia torques, $\left(\gamma ,\gamma \prime ,\gamma ″\right)$ refer to the unit vector components along the $OZ$ -axis. The over dot is used to signify the derivative with respect to time $t$ , $g$ is the gravitational acceleration, $\left(x_c,y_c,z_c\right)$ are the mass center position components that different from the origin $O$ .

Upon further examination of the system (1), it becomes evident that it is a self-governing system comprised of six nonlinear 1^st-order DEs. Notably, this system possesses three well-known first integrals: energy, area, and geometric integrals, all of which can be expressed as

$$\begin{gathered} {{\rm K}}_1=\frac{1}{2}\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p^2+A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q^2+C{r}^2\right]-Mg\left(x_0\gamma +y_0\gamma \prime +z_0\gamma ″\right)=C_1, \\ {{\rm K}}_2=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]\gamma +\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]\gamma \prime +\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \hspace{0.17em}\theta \right]\gamma ″=C_2, \\ {{\rm K}}_3={\gamma }^2+{\gamma }^{\prime 2}+{\gamma }^{″2}=1, \end{gathered}$$

(2)

where the energy constant is $C_1$ and the area constant is $C_2$ .

In order to approach the precise solution for equations in the system (1), a fourth first integral must be estimated. Despite significant efforts by scientists over the past two centuries, this integral has only been obtained for three specific famous cases: when the RGB moves according to Lagrange and Kovalevskaya cases. However, even in these cases, few restrictions must be considered on the mass center position, and the main inertia torques. ²⁵

Identification of the first integrals formulae

To fulfill this section’s objective, we aim to create a function $F\left(p,q,r,\gamma ,\gamma \prime ,\gamma ″\right)$ that is always constant and can be regarded as the first integral of the initial values of the system (1). Hence

$\frac{dF}{dt}=\frac{\partial F}{\partial p}\frac{dp}{dt}+\frac{\partial F}{\partial q}\frac{dq}{dt}+\frac{\partial F}{\partial r}\frac{dr}{dt}+\frac{\partial F}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial F}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial F}{\partial \gamma ″}\frac{d\gamma ″}{dt}=0.$

(3)

The third first integral in (2) can be rewritten as

${\gamma }^2+{\gamma }^{\prime 2}+{\gamma }^{″2}=C_3,$

(4)

It is evident that this assumption of an arbitrary non-negative constant $C_3$ makes the derivative of equation (4) zero, which eliminates any ambiguity in the preceding equations.

The above equation allows us to take the initial circumstances $\left({\gamma }_0,{\gamma }_0^{\prime },{\gamma }_0^{″}\right)$ into account to be arbitrary and is going to be utilized subsequently. An examination of equation (3) reveals that it can be regarded as an identity for all of the RGB parameters $p,q,r,\gamma ,\gamma \prime ,\gamma ″$ . ¹³ This clarifies that equation (3) needs to be fulfilled both at the beginning and at any point in time $t_0$ for the solution of (4).

The parameters $p,q,r,\gamma ,\gamma \prime ,\gamma ″$ initial values can be freely selected in this framework. As a result, (3) serves as an identity. This means that (3)’s partial derivative of $F$ in relation to the RGB’s angular velocity components and to the unit vector components along the $OZ$ -axis, is valid. It must be noted that three of the five independent solutions to this equation are expressed by functions (2). Obtaining the fourth integral refers to approaching (3)’s solution without relying on (2)’s solutions. This section’s purpose is to determine whether it is feasible to exclude the solutions from (2) when considering (3). For the intended fourth first integral, we obtain new DEs by the following theorem. The three first integrals are not generated from (3) since the (2)’s functions don’t reflect the outcomes of those equations.

The obtained theorem

The construction of a theorem for the function $F$ for a fourth first integral to exist is the main topic of this section. For this scope, we produce the following theorem. Therefore, the following theorem is derived:

Theorem (1)

The necessary and sufficient condition for the function $F$ , which depends on $p,q,r,\gamma ,\gamma \prime$ and $\gamma ″$ , to be a fourth first integral for system (1) is to satisfy the linear system:

$$\begin{gathered} U_1={\Omega }_1-A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)D\left[G_1\left(p{\mu }_1+\gamma {\delta }_1\right)+G_2\left(p{\mu }_2+\gamma {\delta }_2\right)+G_3\left(p{\mu }_3+\gamma {\delta }_3\right)\right], \\ {U}_2={\Omega }_2-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)D\left[G_1\left(q{\mu }_1+\gamma \prime {\delta }_1\right)+G_2\left(q{\mu }_2+\gamma \prime {\delta }_2\right)+G_3\left(q{\mu }_3+\gamma \prime {\delta }_3\right)\right], \\ {U}_3={\Omega }_3-CD\left[G_1\left(r{\mu }_1+\gamma ″{\delta }_1\right)+G_2\left(r{\mu }_2+\gamma ″{\delta }_2\right)+G_3\left(r{\mu }_3+\gamma ″{\delta }_3\right)\right], \end{gathered}$$

(5)

where

${\Omega }_1=\frac{\partial F}{\partial p},{\Omega }_2=\frac{\partial F}{\partial q},{\Omega }_3=\frac{\partial F}{\partial r},G_1=\frac{\partial F}{\partial \gamma },G_2=\frac{\partial F}{\partial {\gamma }^{\prime }},G_3=\frac{\partial F}{\partial {\gamma }^{″}},$

$D=-{\left({\displaystyle \sum }_{i=1}^3{d}_i{\mu }_i\right)}^{-1},{\delta }_1=d_2{\gamma }^{″}-d_3{\gamma }^{\prime },{\delta }_2=d_3\gamma -d_1{\gamma }^{″},{\delta }_3=d_1{\gamma }^{\prime }-d_2\gamma ,$

${\mu }_1=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]{\gamma }^{″}-\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\mathit{cos}\hspace{0.17em}\theta \right]{\gamma }^{\prime },$

${\mu }_2=\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\mathit{cos}\hspace{0.17em}\theta \right]\gamma -\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]{\gamma }^{″}{\mu }_3=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]{\gamma }^{\prime }-\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]\gamma .$

(6)

Proof

Since $U_i;i=1,2,3$ generate a nonzero solution, then

$\frac{d{U}_1}{dt}=U_2\left\{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)-C\right],r,-,{\mathcal{l}}_3,+,eH{l}^{\prime 2},\hspace{0.17em},\cos ,\left(\theta \right)\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}+C^{-1}{U}_3\left[A^0\epsilon \left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)q+{\mathcal{l}}_2\right]-A^{0}D\left(1+\epsilon {\mathrm{\Delta }}_1\right)\left[F_1\left(p{\mu }_1+\gamma {\delta }_1\right)+F_2\left(p{\mu }_2+\gamma {\delta }_2\right)+F_3\left(p{\mu }_3+\gamma {\delta }_3\right)\right],\frac{d{U}_2}{dt}=C^{-1}{U}_3\left[A^0\epsilon \left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)p-{\mathcal{l}}_1\right]+U_1\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)r+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right]{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}-\left[A^{0}D\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]\left[F_1\left(q{\mu }_1+\gamma \prime {\delta }_1\right)+F_2\left(q{\mu }_2+\gamma \prime {\delta }_2\right)+F_3\left(q{\mu }_3+\gamma \prime {\delta }_3\right)\right],\frac{d{U}_3}{dt}=U_1\left\{\left[\mathrm{C}-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q-{\mathcal{l}}_2\right]{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}\right\}+U_2\left\{{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)-C\right]\left[p+{\mathcal{l}}_1\right]\right\}-CD\left[F_1\left(r{\mu }_1+\gamma ″{\delta }_1\right)+F_2\left(r{\mu }_2+\gamma ″{\delta }_2\right)+F_3\left(r{\mu }_3+\gamma ″{\delta }_3\right)\right],$

(7)

then

$U_1\frac{dp}{dt}+U_2\frac{dq}{dt}+U_3\frac{dr}{dt}=0,$

(8)

where

$F_1=S{\dot{d}}_1+S^{*}\left[\left({\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right)q-{\mathcal{l}}_{2}r\right]+d_2{C}^{-1}{U}_3-d_3{U}_2{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1},$

$F_2=S{\dot{d}}_2+S^{*}[({\mathcal{l}}_{1}r-\left({\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right)p]+d_3{U}_1{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}-d_1{C}^{-1}{U}_3,$

$F_3=S{\dot{d}}_3+S^{*}\left({\mathcal{l}}_{2}p-{\mathcal{l}}_{1}q\right)+d_1{U}_2{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}-d_2{U}_1{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1},$

${\sigma }_1=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]d_3-\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right]d_2,$

${\sigma }_2=\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right]d_1-\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]d_3,{\sigma }_3=\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]d_2-\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]d_1.$

whereas, $S$ and $S^{*}$ represent functions of time.

Assuming that the function $F\left(p,q,r,\gamma ,\gamma \prime ,\gamma ″\right)$ is a necessary first integral of the initial system (1), then equation (3) can be used to get

${\Omega }_1\frac{dp}{dt}+{\Omega }_2\frac{dq}{dt}+{\Omega }_3\frac{dr}{dt}+G_1\frac{d\gamma }{dt}+G_2\frac{d\gamma \prime }{dt}+G_3\frac{d\gamma ″}{dt}=0.$

(9)

Given that the aforementioned formula may be viewed as a unique identity with regard to every RGB parameter, it can be differentiated with respect to each of these parameters. The partial derivative of (9) with respect to $p$ results in

$\frac{\partial {\Omega }_1}{\partial p}\frac{dp}{dt}+\frac{\partial {\Omega }_2}{\partial p}\frac{dq}{dt}+\frac{\partial {\Omega }_3}{\partial p}\frac{dr}{dt}+\frac{\partial G_1}{\partial p}\frac{d\gamma }{dt}+\frac{\partial G_2}{\partial p}\frac{d\gamma \prime }{dt}+\frac{\partial G_3}{\partial p}\frac{d\gamma ″}{dt}=\left\{[,A^0,(1+\epsilon {\mathrm{\Delta }}_1),-,C,],r,-,{\mathcal{l}}_3,+,e,H,l^{\prime 2},\hspace{0.17em},\cos ,\left(\theta \right)\right\}[A^0(1+\epsilon {\mathrm{\Delta }}_2){]}^{-1}{\Omega }_2+[A^0\epsilon ({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1)q+{\mathcal{l}}_2]C^{-1}{\Omega }_3-G_2\gamma ″+G_3\gamma \prime .$

(10)

Given the relationship that exists between the 2^nd-order partial derivatives of $F$ , equation (10) may be expressed as

$\frac{\partial {\Omega }_1}{\partial p}\frac{dp}{dt}+\frac{\partial {\Omega }_1}{\partial q}\frac{dq}{dt}+\frac{\partial {\Omega }_1}{\partial r}\frac{dr}{dt}+\frac{\partial {\Omega }_1}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial {\Omega }_1}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial {\Omega }_1}{\partial \gamma ″}\frac{d\gamma ″}{dt}=\left\{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)-C\right],r,-,{\mathcal{l}}_3,+eH{l}^{\prime 2}\hspace{0.17em}\hspace{0.17em}\cos \left(\theta \right)\right\}[A^0(1+\epsilon {\mathrm{\Delta }}_2){]}^{-1}{\Omega }_2+[A^0\epsilon ({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1)q+{\mathcal{l}}_2]C^{-1}{\Omega }_3-G_2{\gamma }^{″}+G_3{\gamma }^{\prime },$

$\frac{\partial {\Omega }_2}{\partial p}\frac{dp}{dt}+\frac{\partial {\Omega }_2}{\partial q}\frac{dq}{dt}+\frac{\partial {\Omega }_2}{\partial r}\frac{dr}{dt}+\frac{\partial {\Omega }_2}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial {\Omega }_2}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial {\Omega }_2}{\partial \gamma ″}\frac{d\gamma ″}{dt}=\left[\epsilon A^0\left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)p-{\mathcal{l}}_1\right]\times C^{-1}{\Omega }_3+\left\{\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]r+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1-G_3\gamma +G_1\gamma ″,$

$\frac{\partial {\Omega }_3}{\partial p}\frac{dp}{dt}+\frac{\partial {\Omega }_3}{\partial q}\frac{dq}{dt}+\frac{\partial {\Omega }_3}{\partial r}\frac{dr}{dt}+\frac{\partial {\Omega }_3}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial {\Omega }_3}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial {\Omega }_3}{\partial \gamma ″}\frac{d\gamma ″}{dt}=\left\{\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right],q,-,{\mathcal{l}}_2\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1+\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2-C\right)p+{\mathcal{l}}_1\right]{\Omega }_2-G_1{\gamma }^{\prime }+G_2\gamma ,$

$\frac{\partial G_1}{\partial p}\frac{dp}{dt}+\frac{\partial G_1}{\partial q}\frac{dq}{dt}+\frac{\partial G_1}{\partial r}\frac{dr}{dt}+\frac{\partial G_1}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial G_1}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial G_1}{\partial \gamma ″}\frac{d\gamma ″}{dt}=d_2{C}^{-1}{\Omega }_3 -{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{d}_3{\Omega }_2+r{G}_2-q{G}_3,$

$\frac{\partial G_2}{\partial p}\frac{dp}{dt}+\frac{\partial G_2}{\partial q}\frac{dq}{dt}+\frac{\partial G_2}{\partial r}\frac{dr}{dt}+\frac{\partial G_2}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial G_2}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial G_2}{\partial \gamma ″}\frac{d\gamma ″}{dt}=-d_1{C}^{-1}{\Omega }_3+d_3{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1+p{G}_3-r{G}_1,$

$\frac{\partial G_3}{\partial p}\frac{dp}{dt}+\frac{\partial G_3}{\partial q}\frac{dq}{dt}+\frac{\partial G_3}{\partial r}\frac{dr}{dt}+\frac{\partial G_3}{\partial \gamma }\frac{d\gamma }{dt}+\frac{\partial G_3}{\partial \gamma \prime }\frac{d\gamma \prime }{dt}+\frac{\partial G_3}{\partial \gamma ″}\frac{d\gamma ″}{dt}=d_1{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{\Omega }_2-d_2{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1+q{G}_1-p{G}_2.$

(11)

It is worthwhile to notice that the left side of (11) is equal to $\frac{d{\Omega }_1}{dt}$ . The subsequent system is then generated with regard to the two functions ${\Omega }_i$ and $G_i;\left(i=1,2,3\right)$ by taking a partial derivative of (9) with respect to $p,q,r,\gamma ,\gamma \prime ,$ and $\gamma ″$

$\frac{d{\Omega }_1}{dt}=\left\{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)-C\right]r-{\mathcal{l}}_3+eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{\Omega }_2+\left[A^0\epsilon \left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)q+{\mathcal{l}}_2\right]C^{-1}{\Omega }_3-\gamma ″G_2+\gamma \prime G_3,\frac{d{\Omega }_2}{dt}=\left[A^0\epsilon \left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)p-{\mathcal{l}}_1\right]C^{-1}{\Omega }_3+\left\{\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]r+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1-\gamma G_3+\gamma ″G_1,\frac{d{\Omega }_3}{dt}=\left\{\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]q-{\mathcal{l}}_2\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1+\left\{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)-C\right]p+{\mathcal{l}}_1\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{\Omega }_2-\gamma \prime G_1+\gamma G_2,\frac{d{G}_1}{dt}=d_2{C}^{-1}{\Omega }_3-d_3{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{\Omega }_2+r{G}_2-q{G}_3,\frac{d{G}_2}{dt}=d_3{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1-d_1{C}^{-1}{\Omega }_3+p{G}_3-r{G}_1,\frac{d{G}_3}{dt}=d_1{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{\Omega }_2-d_2{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{\Omega }_1+q{G}_1-p{G}_2,$

(12)

Six partial nonlinear DEs make up the prior system. Thus, the vectors comprising the necessary partial derivatives of the integrals from (1) are fundamental to the solutions of (12). Now, consider the usage of the following substitution in (12)

${\Omega }_1=A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\left(pS+\gamma S^{*}\right)+U_1, G_1=-d_{1}S+\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)p+{\mathcal{l}}_1\right]S^{*}+\gamma S^{**},{\Omega }_2=A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\left(qS+\gamma \prime S^{*}\right)+U_2, G_2=-d_{2}S+\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)q+{\mathcal{l}}_2\right]S^{*}+\gamma \prime S^{**},{\Omega }_3=C\left(r{S}^{*}+\gamma ″S^{**}\right)+U_3, G_3=-d_{3}S+\left[Cr+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right]S^{*}+\gamma ″S^{**}.$

(13)

obtains

$\frac{d{U}_1}{dt}=\{[A^0(1+\epsilon {\mathrm{\Delta }}_1)-C]r-{\mathcal{l}}_3+eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\}[A^0(1+\epsilon {\mathrm{\Delta }}_2){]}^{-1}{U}_2-A^0(1+\epsilon {\mathrm{\Delta }}_1)\times \left[p,\frac{dS}{dt},+,\gamma ,\frac{d{S}^{*}}{dt}\right]+\left[A^0\epsilon \left({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1\right)q+{\mathcal{l}}_2\right]C^{-1}{U}_3,\frac{d{U}_2}{dt}=\{[C-A^0(1+\epsilon {\mathrm{\Delta }}_2)]r+{\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\}[A^0(1+\epsilon {\mathrm{\Delta }}_1){]}^{-1}{U}_1-A^0(1+\epsilon {\mathrm{\Delta }}_1)\times \left[q,\frac{dS}{dt},+,\gamma ,\prime ,\frac{d{S}^{*}}{dt}\right]+[A^0\epsilon ({\mathrm{\Delta }}_2-{\mathrm{\Delta }}_1)p-{\mathcal{l}}_1]C^{-1}{U}_3,\frac{d{U}_3}{dt}=\left\{\left[C-A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]q-{\mathcal{l}}_2\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)\right]}^{-1}{U}_1-C\left[r,\frac{dS}{dt},+,\gamma ,″,\frac{d{S}^{*}}{dt}\right]+\left\{[,A^0,(1+\epsilon {\mathrm{\Delta }}_1),-,C,],p,+,{\mathcal{l}}_1\right\}{\left[A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\right]}^{-1}{U}_2,$

(14)

and

$\frac{dS}{dt}=D\left(F_1{\mu }_1+F_2{\mu }_2+F_3{\mu }_3\right), \frac{d{S}^{*}}{dt}=D\left(F_1{\delta }_1+F_2{\delta }_2+F_3{\delta }_3\right), \frac{d{S}^{**}}{dt}=D\left(F_1{\sigma }_1+F_2{\sigma }_2+F_3{\sigma }_3\right),$

(15)

in which $S^{**}$ and $\left(U_i;i=1,2,3\right)$ are time functions. Although the right side of (14) is independent of $S,S^{*},S^{**}$ , three DEs make up the system, which corresponds to (7).

It is easy to demonstrate that equations (5) and (8) are fulfilled when (13) is taken into account. Furthermore, (5) can be obtained immediately by solving (13) for $U_i$ . Multiplying $\left({\Omega }_i;i=1,2,3\right)$ that are given in (13) by $\frac{dp}{dt},\frac{dq}{dt},\frac{dr}{dt}$ and the reminder three ones by $\frac{d\gamma }{dt},\frac{d\gamma \prime }{dt},\frac{d\gamma ″}{dt}$ , respectively, then adding the resulting equations to obtain

${\Omega }_1\frac{dp}{dt}+{\Omega }_2\frac{dq}{dt}+{\Omega }_3\frac{dr}{dt}+G_1\frac{d\gamma }{dt}+G_2\frac{d\gamma \prime }{dt}+G_3\frac{d\gamma ″}{dt}=S\frac{d{{\rm K}}_1}{dt}+S^{*}\frac{d{{\rm K}}_2}{dt}+S^{**}\frac{d{{\rm K}}_3}{dt}+U_1\frac{dp}{dt}+U_2\frac{dq}{dt}+U_3\frac{dr}{dt}.$

(16)

Carefully examining (9) leads us to the conclusion that the left side of (16) is zero. Additionally, we can determine that $\left(\frac{d{{\rm K}}_i}{dt}=0;i=1,2,3\right)$ based on (2), therefore (8) is also fulfilled.

One critical element of this theorem is establishing the necessity of the condition. For this, it is sufficient to prove that $U_i=0$ are ruled out. In the current situation, the contradiction method in the proof is employed. Considering that $U_i=0$ , system (5) becomes obvious when we examine it more closely. The resulting system will be homogenous in terms of the function $F$ , that is,

$X_1\left(F\right)={\Omega }_1-A^{0}D\left(1+\epsilon {\mathrm{\Delta }}_1\right)\left[G_1\left(p{\mu }_1+\gamma {\delta }_1\right)+G_2\left(p{\mu }_2+\gamma {\delta }_2\right)+G_3\left(p{\mu }_3+\gamma {\delta }_3\right)\right],X_2\left(F\right)={\Omega }_2-A^{0}D\left(1+\epsilon {\mathrm{\Delta }}_2\right)\left[G_1\left(p{\mu }_1+\gamma \prime {\delta }_1\right)+G_2\left(q{\mu }_2+\gamma \prime {\delta }_2\right)+G_3\left(q{\mu }_3+\gamma \prime {\delta }_3\right)\right],X_3\left(F\right)={\Omega }_3-CD[G_1(r{\mu }_1+\gamma ″{\delta }_1)+G_2\left(r{\mu }_2+\gamma ″{\delta }_2\right)+G_3\left(r{\mu }_3+\gamma ″{\delta }_3\right)],$

(17)

From a mathematical perspective, the expressions $X_j\left[X_k\left(F\right)\right]-X_k\left[X_j\left(F\right)\right]$ correspond to Poisson’s brackets for each of the operators $(X_k\left(F\right)$ and $X_j\left(F\right);j,k=1,2,3,j>k)$ . Through all types of Poisson’s brackets for the system (17), it is reinforced as a complete system. ¹⁰ It is important to observe that for any complete system of $k$ equations, expressed as a function of $n$ independent variables, the system possesses $\left(n-k\right)$ independent integrals. The integral of this system is treated as an arbitrary function of the $\left(n-k\right)$ particular integrals. ¹² Based on the present theorem, we conclude that the system of equation (17) has three independent solutions. Additionally, any other solution can be written as a function of these three. It is simple to verify that the first integrals (2) satisfy the system of equation (16). Hence, the general solution of (17) is represented as an arbitrary function of (2). The function $F$ , as described, represents the desired first integral of (1) and, consequently, cannot depend on the integrals in (2). Hence, it becomes evident that $U_i$ is invalid, meaning $U_i$ forms a nonzero solution of the system (7). This conclusion affirms the necessity of the aforementioned theorem. Assuming that $F=F\left(p,q,r,\gamma ,\gamma \prime ,\gamma ″\right)$ satisfies both (5) and (8). In actuality, the new integral of (1) is such a function, as has been previously investigated, and is the exact demonstration of the requirements. Adding the resulting equations after multiplying (5) by the time derivative of the angular velocity components, $p,q,$ and $r$ , respectively, yields

${\Omega }_1\frac{dp}{dt}+{\Omega }_2\frac{dq}{dt}+{\Omega }_3\frac{dr}{dt}+G_1\frac{d\gamma }{dt}+G_2\frac{d\gamma \prime }{dt}+G_3\frac{d\gamma ″}{dt}=0.$

(18)

Therefore, the function $F$ is the first integral of system (1), as demonstrated by the preceding in (18). Proving that the derived first integral reflects the fourth, or that the function $F$ is independent of first integrals (2), is one of the crucial elements. As a result, we use disparity to offer the proof. Given that $U_i$ and that the resulting first integral depends on the integrals (2), and then (5) being satisfied. The requirement of the hypothesis that $U_i$ forms a nonzero solution of (7) causes us to be in contradiction. This means that the proof is complete.

Applications for well-known cases

This section’s main purpose is to highlight the importance of the condition by presenting applications of the previous hypothesis to the two most renowned scenarios: Lagrange’s case and the kinetic symmetrical case.

Lagrange’s case

Here, the RGB’s mass center is located on one of its inertia primary axes, such as $Oz$ , which is symmetric about it, that is, $A^0\left(1+\epsilon {\mathrm{\Delta }}_1\right)=A^0\left(1+\epsilon {\mathrm{\Delta }}_2\right)\ne C,x_0=y_0=0$ (see Figure 2).

Figure 2.

Lagrange’s case.

The functions listed below, in turn, fulfill (14)

$U_1=0, U_2=0, U_3=1, S=S^{*}=S^{**}=0.$

(19)

Using (13) and (19), yields

${\Omega }_1=0, {\Omega }_2=0, {\Omega }_3=1, G_1=G_2=G_3=0.$

(20)

Therefore, the fourth integral is

$F=r.$

(21)

Consequently, this specific outcome is consistent with the established outcomes in Refs. 12,13, and 25 which confirm the significance of the previously indicated condition.

The case of kinetic symmetry

This specific scenario, in which every inertia torque is the same, is referred to as a spherical case for RGB’s motion. Thus, the aforementioned functions satisfy (14) in a manner similar to the earlier scenario

$U_1=x_0, U_2=y_0, U_3=z_0, S=S^{*}=S^{**}=0,$

(22)

By altering (22) with (13), yields

${\Omega }_1=x_0, {\Omega }_2=y_0, {\Omega }_3=z_0, G_1=G_2=G_3=0.$

(23)

As a result, the fourth first integral in this situation is

$F=x_{0}p+y_{0}q+z_{0}r.$

(24)

This integral matches the obtained one in Refs. 11 and 12 indicating efficiency of the condition given in the hypothesis.

Kovalevskaya’s case

In this scenario, when two inertia torques are added together, it equals double the third torque’s value, and the RGB’s mass center is situated in the plane that contains these equal torques [refer to Figure 3]. Hence, (14) is satisfied by the functions and gives

$U_1=4d_{1}D{C}^{-1}\{-{\mu }_1\left[2C{\left(p+{\mathcal{l}}_{13}\right)}^3+\left(q+{\mathcal{l}}_{23}\right){\delta }_3\right]-{\mu }_2\left[2C{\left(p+{\mathcal{l}}_{13}\right)}^2\left(q+{\mathcal{l}}_{23}\right)-\left(p+{\mathcal{l}}_{13}\right){\delta }_3\right]-{\delta }_3\left[\left(p+{\mathcal{l}}_{13}\right){\mu }_3+2C{\left(p+{\mathcal{l}}_{13}\right)}^2{\gamma }_3-{\gamma }_1{\delta }_2\right]\},U_2=4d_{1}D{C}^{-1}\{-{\mu }_1\left[2C{\left(p+{\mathcal{l}}_{13}\right)}^2\left(q+{\mathcal{l}}_{23}\right)+\left(p+{\mathcal{l}}_{13}\right){\delta }_3\right]-{\mu }_2[2C\left(p+{\mathcal{l}}_{13}\right){\left(q+{\mathcal{l}}_{23}\right)}^2-\left(q+{\mathcal{l}}_{23}\right){\delta }_3]+{\delta }_3\left[2C\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right){\gamma }_3-{\gamma }_2{\delta }_2\right]\},U_3=2d_{1}D{C}^{-1}\{{\mu }_1\left[2\left(p+{\mathcal{l}}_{13}\right){\delta }_2-C{\left(p+{\mathcal{l}}_{13}\right)}^2\left(q+{\mathcal{l}}_{23}\right)+C{\left(q+{\mathcal{l}}_{23}\right)}^2\left(r+{\mathcal{l}}_{33}\right)\right]-{\mu }_2\left(q+{\mathcal{l}}_{23}\right)\left[2C\left(p+{\mathcal{l}}_{13}\right)\left(r+{\mathcal{l}}_{33}\right)-2{\delta }_3\right]-{\gamma }_3{\delta }_3\left[2C\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)+{\delta }_1\right]\},S=2d_{1}D{C}^{-1}\left\{{{\mu }_1\left[{\left(p+{\mathcal{l}}_{13}\right)}^2-{\left(q+{\mathcal{l}}_{23}\right)}^2\right]+2\mu }_2\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)-2{\delta }_3\left[\left(q+{\mathcal{l}}_{23}\right){\gamma }_1-\left(p+{\mathcal{l}}_{13}\right){\gamma }_3\right]\right\}{S}^{*}=2d_{1}D{C}^{-1}{\delta }_2\left[2C\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)+{\delta }_3\right],S^{**}=-2d_{1}D{C}^{-1}\left(r+{\mathcal{l}}_{33}\right)\left[2C\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)+{\delta }_3\right].$

(25)

where

${\mathcal{l}}_{13}=C^{-1}{\mathcal{l}}_1, {\mathcal{l}}_{23}=C^{-1}{\mathcal{l}}_2, {\mathcal{l}}_{33}=C^{-1}\left({\mathcal{l}}_3-eH{l}^{\prime 2}\hspace{0.17em}\cos \left(\theta \right)\right).$

Figure 3.

Kovalevskaya’s case.

By utilizing (13) and (25), we can directly derive the expressions for the derivatives of the fourth integral as

${\Omega }_1=2\left(p+{\mathcal{l}}_{13}\right)\left[2{\left(p+{\mathcal{l}}_{13}\right)}^2+{\left(q+{\mathcal{l}}_{23}\right)}^2\right]-d_1{C}^{-1}\left[2\left(p+{\mathcal{l}}_{13}\right){\gamma }_1+\left(q+{\mathcal{l}}_{23}\right){\gamma }_2\right],$

${\Omega }_2=2\left(q+{\mathcal{l}}_{23}\right)\left[2{\left(q+{\mathcal{l}}_{23}\right)}^2+{\left(p+{\mathcal{l}}_{13}\right)}^2\right]-d_1{C}^{-1}\left[\left(p+{\mathcal{l}}_{13}\right){\gamma }_2-2\left(q+{\mathcal{l}}_{23}\right){\gamma }_1\right],$

${\Omega }_3=0, G_1=d_1{C}^{-1}\left[{\left(q+{\mathcal{l}}_{23}\right)}^2-{\left(p+{\mathcal{l}}_{13}\right)}^2+2d_1{C}^{-1}{\gamma }_1\right],$

$G_2=-2d_1{C}^{-1}\left[\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)-d_1{C}^{-1}{\gamma }_2\right], G_3=0.$

Thus, the fourth integral turns into

$F={\left[{\left(p+{\mathcal{l}}_{13}\right)}^2-{\left(q+{\mathcal{l}}_{23}\right)}^2-d_1{C}^{-1}{\gamma }_1\right]}^2+{\left[2\left(p+{\mathcal{l}}_{13}\right)\left(q+{\mathcal{l}}_{23}\right)-d_1{C}^{-1}{\gamma }_2\right]}^2.$

(26)

This result is in good agreement with the obtained in Ref. 12 and 15 when ${\mathcal{l}}_1={\mathcal{l}}_2={\mathcal{l}}_3=0$ which demonstrates the precision and significance of the condition under consideration in this paper.

Concluding remarks

In this study, the EOMs that control the rotation of the RGB around a stationary point are examined taking into account the impact of the GT. The expression for the hypothesis that is essential to identifying a function $F$ as the problem’s fourth first integral is approached based on a significant condition, enabling a straightforward calculation of this integral. The results obtained from this analysis align well with previous findings from studies ^13,24 and ²⁵ when the RGB is solely influenced by the charge and study ¹² when the RGB moves inside a uniform field. By reducing the GT vector, the fourth integrals for ICs are obtained, showcasing the importance and reliability of this condition. Therefore, this study can be considered an expansion of previous research conducted in studies ¹² and ¹³ offering a more comprehensive examination of the topic.