The asymptotic solutions for the motion of a charged symmetric gyrostat in the irrational frequency case

The primary objective of this study is to explore the spatial rotary movements of a symmetrically charged rigid body (RB) that is rotating around a fixed point, akin to Lagrange’s scenario as a novel scenario where its center of mass experiences a slight displacement from the symmetry dynamic axis. The body’s movement is presumed to be affected by a gyrostatic moment and a force from an electromagnetic field, attributed to the presence of a located point charge on this axis. The regulating equations of motion that are pertaining to the equations Euler–Poisson are solved through the utilization of Poincaré’s small parameter method along with its adaptations when the scenario of irrational frequencies is considered. The three angles of Euler are derived and graphed to ascertain the body’s position at any point throughout the motion. The temporal evolutions of the achieved outcomes are drawn to showcase the significant impact of the selected parameters on the motion. The phase plane diagrams have been generated to illustrate the stability of the body during the motion. The novelty of studying the rotatory motion of a charged RB under these specific conditions lies in the intricate interplay of gyrostatic effects, magnetic interactions, and nonlinear dynamics. This research can push the boundaries of theoretical mechanics and provide valuable insights and tools for both theoretical advancements and practical applications. Moreover, the achieved results from this analysis can be utilized to improve the dynamic performance of diverse engineering applications, particularly those dependent on gyroscopic theory. This includes enhancing the functionality of satellites, compasses, submarines, and automatic pilots used in aircraft. Essentially, the findings have practical implications for optimizing the performance and stability of these systems.

One of the most significant problems that we deal with in our lives is the rotational motion of the RB.The significance of delving into this problem comes from its broad applicability across both the realms of physics and mathematics.Such problems are governed by both Euler and Poisson Equations 1 to determine the body's angular velocity and its configuration at any time.A lot of research has studied this problem from different perspectives [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] .In 2 , an influenced motion of a RB by a combination of gyroscopic and potential forces with axisymmetric characteristics was investigated, in which two integrals of this problem were presented.In 3 , a new class of two-dimension integrable systems characterized by an additional third-degree integral in velocities was presented.The utility of this system extends to addressing the motion challenges of both a particle and a RB about a fixed point.Furthermore, four new integrable scenarios that capture the dynamics of a particle navigating in various settings, such as the plane, surfaces, and pseudosphere with variable curvature, was presented.In 4 , the authors provided a fourth integral pertaining to the motion of a RB around a stationary point, particularly when it is influenced by a GM.In 5 , the authors ascertain the comprehensive structure of the potential associated with the motion of a RB.This formulation enables the angular velocity to consistently reside within a principal plane of body inertia.The complexity of the problem's solution is greatly reduced, making the process more straightforward and manageable.
In 6 , the emphasis is placed on examining the dynamics of a RB connected with an elastic spring, conceptualized as a pendulum model.The author explores numerical solutions through the application of Runge-Kutta making it an essential resource for understanding the behavior of rigid bodies in various fields.Under the influence of forces produced by the Barnett-London effect, the motion of a dynamically symmetric body around a fixed point in a uniform magnetic field was analyzed in 31 .The body's steady rotations and regular precession were explained, and the problem has been simplified to a quadrature.The rotational evolution of heavy rigid bodies when subjected to varying restoring and perturbation torques was examined in 32 .This study provided insights into the nonlinear dynamics of rigid bodies, which expanded our understanding of how unsteady forces influence rotational motion.
The current work aims to investigate the spatial rotational movement of a symmetrically charged RB revolving about a fixed point analogous to Lagrange's configuration when the body's center of mass is slightly off the body's axis of dynamic symmetry.It is considered that the body's motion is influenced by both a force of the electromagnetic field and a GM.The fundamental EOM associated with Euler-Poisson's equations are solved by applying PSPM along with its modifications in the case of irrational frequencies.The body's Euler angles have also been calculated and graphically represented to determine the body's position at any given moment during the motion.Plotting the time evolution of the obtained results demonstrates the substantial influence of the chosen gyrostat parameters on the movement.To demonstrate the body's stability during motion, diagrams of phase plane are created and analyzed.The combination of nonlinear dynamics, magnetic interactions, and gyrostatic effects makes investigating the rotatory motion of a charged rigid body (RB) under these particular circumstances intriguing.The results of this research have the potential to expand the field of theoretical mechanics and offer useful tools and insights for both theoretical development and real-world applications.The applications of studying the rotatory motion of charged rigid bodies under the action of gyrostatic moments (GM) are vast and diverse, spanning from space technology and robotics to navigation, energy systems, and beyond.The insights gained from this research can lead to significant advancements in the design, control, and optimization of various systems that rely on precise rotational dynamics.

Description of the problem
The rotary motion of a symmetrical RB of mass M according to the criteria of Lagrange is investigated in this section.Through a fixed point O inside the body, two frames have been taken into consideration.The first frame OXYZ is the fixed in space, whereas the second one Oxyz is taken to be fixed in the body and moving along its configuration (see Fig. 1).The body experiences a GM ℓ , with its components denoted as ℓ j (j = 1, 2, 3) , operating along the body's inertia main axes Ox, Oy, and Oz , in which ℓ 1 = 0 .Furthermore, assuming that the body rotates under the action of a uniform electromagnetic field with the strength H , generated by a charge e positioned on the axis Oz at a distance ℓ * from O which makes an angle ν with the fixed axis OZ.
According to the aforementioned problem's formulation, the controlling EOM of the RB can be stated as follows 1,27,28 Here, h O is the total angular momentum of the body and L O is the total external moments affecting the body, A, B, and C are the body's inertia moments along the main axes, respectively.According to the examined (1) www.nature.com/scientificreports/Lagrange's case A = B .Within the definition of symbols in the preceding system (1), we find that g is the gravi- tational acceleration, r c = (x c , y c , z c ) refers to the position of the center of mass, ω = (p, q, r) signifies the body's angular velocity, K = (α, β, γ ) denotes the unit vector along the downward direction of the fixed axis OZ , and dots over the parameters denote their derivatives regarding the time t.Therefore, Eq. (1) can be represented as follows The three first integrals respecting to system (2) are Here, p 0 , q 0 , r 0 , α 0 , β 0 , and γ 0 are, respectively, the values of p, q, r, α, β, and γ at t = 0.
To apply the PSPM, let us define the following parameters Substituting (4) into systems ( 2) and (3) to get

Procedure of the PSPM
In this section, we are going to investigate the procedure of the used method trying to obtain the problem's solutions.
The first and third equations of system (6) allow us to write where (2) (5) www.nature.com/scientificreports/ Here, F d0 (d = 1, 2) denotes the initial values of F d0 .Substituting ( 7) into ( 5), one can reduce system (5) to the below first-order differential equation regarding the frequencies d where For d (d = 1, 2) , if we consider that 1 / 2 is a rational number by a suitable selection of r 0 , then the solution of the generating system of Eq. ( 9) is periodic with period However, we proceed by reformulating the problem to estimate the solution of system ( 9) with period τ 0 (ε) according to an extremely small value of ε .At ε = 0 , the periodic solution of the generated system of (8) exhibits a period T 0 .So, keep in mind the following substitution where ρ depends on ε , which can be determined subsequently.
Applying the aforementioned transformation of (11) to system (9), we have where ( 8) To achieve the desired solutions, it is imperative that system (12) receives considerable attention.Hence, by further differentiating of system (12) again, we can derive the below forms of nonlinear differential equations from second-order Consequently, we may assume the solutions of the aforementioned system as follows where Furthermore, considering the subsequent initial criteria where m j (j = 1, 2, 3, 4) and M (0) i (i = 1, 2, 3) represent the perturbed and unperturbed terms of M i , respectively.Observably, the solutions of system (9) exhibiting a period T 0 match the periodic solutions of the equations in (14) with a period T 1 = (1 + ερ)T 0 .Assuming ρ = (ρ 0 + m 4 ) to derive the solutions (15) in their periodic forms, and considering m j = m j (ε) where m j at ε = 0.
Following the procedure outlined in the SPMP, the initial criteria can be adjusted to align with arbitrary constants for the system's generating solutions.Making use of ( 10) and ( 8) to obtain ( 13) www.nature.com/scientificreports/Functions G j (j = 1, 2, 3, 4) in system (10) may have the forms where L s (s = 1, 2, 3, ..., 8) can be given in terms of f d (d = 1, 2) as in (Appendix 1).Therefore, one can write the non-perturbation terms of L s in the following way where S kη are the elements of the square matrix S kη (k, η = 1, 2, 3, ..., 8) .To obtain these elements, one can substitute d .So, S 1η , S 2η , ..., S 8η can be easily calculated (see Appendix 2).Referring to the solutions (15), the complete determination of these solutions is achievable, provided when the parameters C (n) j (T)(j = 1, 2, 3, 4) are obtained.As a result, we can simply create the system that specifies these parameters in (15) according to the substitution of ( 15) into (12) and then equating the like coefficients of ε in each side to get In addition to the following initial circumstances where H (n) j (T) are functions that can be found once C (χ) j have been estimated at χ < n.When n = 1 : Upon substituting the expression of H j from Eq. ( 12) into Eq. ( 20) one can then proceed to compare the coefficients of the distinct powers of ε on either side.Differentiating the resulting equations with respect to T will allow us to obtain equations that determine C (1) j (T) in the form where (1) (1) 4 (1) www.nature.com/scientificreports/Based on the Q j1 (u) expressions, the solutions of the aforementioned system ( 22) may be formulated as follows where −1 k Q j1 (T) can be obtained in terms of trigonometric functions, see (Appendix 2).Substituting of Q j1 (u) into ( 23), yields where It goes without saying that we may express the necessary and sufficient criteria for T periodic solutions of (15) as follows 33 Here, ψ j are expressed regarding to M i (i = 1, 2, 3), ρ, and ε .These criteria determine M (0) i , ρ 0 , and m j , and they are not independent according to system (11) 25 .In the scenario where M 3 = 0 , we may expect the third condition to directly result from the preceding conditions.Therefore, it is convenient to consider M (0) i or ρ 0 as a constant, and one of m j as a function of ε 26 .
The below necessary criteria for achieving the periodicity of C (1) j can be derived by dividing Eq. ( 26) by ε To enhance our understanding of the aforementioned criteria, one can utilize Eq. ( 23) to get It is clear that when the quotient of 1 divided by 2 equals 2 , 1/2 , 1 or −1 , we get the following forms nonzero mathematical formulae of R 11 , R 21 , R 31 and R 41 dT .

(27) C
(1) i and ρ 0 meet Eq. ( 28), then it is assumed that the Jacobi matrices of C (1) i , ρ = ρ 0 , and ψ j must be determined in terms of m j , under the condition m j = ε = 0. Since the calculation of the second matrix does not depend on ε, we can safely set ε = 0.As both M j , ρ and m j are evident in the solutions as linked units, these matrices can be denoted by J.This suggests that these variables play a role in the outcomes or results being discussed, indicating their interconnectedness or dependence within the context of the problem or analysis.Hence, the solutions of Eq. ( 26) enable us to explore the necessary periodic solutions based on the following scenario.

Irrational frequency scenario
This section elucidates the desired solutions pertaining to the scenario where 1 −1 2 is irrational.It is worth noting that one may derive these solutions when M (0) 3 S 53 = 0 (for any quantity of M 3 ) 28 , J is from a third-rank, and are met.
ii) for 1 iii) for 1 iv) for 1 In the current context, it is foreseeable that the solutions of Eq. ( 26) take the form of a power series in terms of the small parameter, involving m 1 , m 2 , and m 4 .Within this framework, we may assume that m 3 is negligible.Hence, these solutions become null at ε = 0 .Building upon the outlined procedure, we can express the perio- dicity criteria as follows Based on the Eq. ( 30), we observe that the solutions p 2 (T, ε) , q 2 (T, ε) , α 2 (T, ε), and β 2 (T, ε) of the unper- turbed scenario ( ε = 0 ) have the following forms Using (32), one obtains directly m 1 and m 2 as follows In terms of power series of ε and based on the aforementioned formulas in ( 4), ( 7), ( 10), ( 15), (23), and (24), one may achieve the required solutions in the form (34) 2 sin 1 T, 3 cos 1 T, (36) However, the self-rotation angle ϕ increases progressively as time goes on with the increase of ℓ 2 , ℓ 3 , and e values, as seen in Figs.8b, 9b and 10b.This behavior can be attributed to the third term in the third equation of the system (39), wherein the magnitude of the first two terms is lower than that of the third term.
Conversely, the precession angle ψ exhibits a decreasing fluctuation pattern over the analyzed time span, as depicted in Figs.8c, 9c and 10c.On may state that, the reason of various initial points of these fluctuations is going back to the mathematical form of ψ 0 in Eq. (39).

Conclusion
The dynamical rotatory movement of a charged symmetric RB, induced by a point charge positioned on its dynamic symmetry axis, has been examined, considering a slight displacement of the body's center of mass from this axis.The analysis also incorporates the effects of two components of the GM about two of the body's main axes of inertia, along with the impact of an electromagnetic field.The governing EOM associated with the equations of Euler-Poisson have been solved using PSPM, along with its modifications, particularly in a case involving irrational frequencies.The derived mathematical representations of Euler's angles are utilized to determine the orientation or positioning of the body at different points in time.The newly acquired results have been plotted to better understand how the body moves over time, considering the defined parameter values.Phase plane graphs have been presented to illustrate the body's stability throughout its motion.The novelty of examining the rotational dynamics of a charged RB in these specific conditions lies in the intricate interaction of gyrostatic effects, magnetic forces, and nonlinear dynamics.This research can extend the frontiers of theoretical mechanics and provide essential insights and tools for both academic progress and practical implementation.The obtained