The Stability of Certain Motion of a Charged Gyrostat in Newtonian Force Field

)is work aims to study the stability of certain motions of a rigid body rotating about its fixed point and carrying a rotor that rotates with constant angular velocity about an axis parallel to one of the principal axes.)is motion is presumed to take place due to the combined influence of the magnetic field and the Newtonian force field. )e equations of motion are deduced, and moreover, they are expressed as a Lie–Poisson Hamilton system. )e permanent rotations are calculated and interpreted mechanically. )e sufficient conditions for instability are presented employing the linear approximation method. )e energyCasimir method is applied to gain sufficient conditions for stability. )e regions of linear stability and Lyapunov stability are illustrated graphically for certain values of the parameters.


Introduction
A gyrostat is a simple multibody which consists of a rigid body and other bodies which are usually called rotors moving in such a way that their motion does not change the distribution of mass for the gyrostat [1]. e gyrostat is also well known in the literature as a dual-spin body due to the motion of the two bodies which compose the gyrostat. Volterra [2] had first introduced the notions of gyrostat when he strived to study the motion of the Earth's polar axis and interpret variations in the Earth's latitude by means of the internal motion which keeps the mass distribution of the planet fixed. is model is used in a variety of numerous applications in different branches of physics besides their classical applications in astronomy and mechanics. For example, the gyrostat was utilized as a model of the Earth that takes into account some stationary transport processes on it [2], as a model of the atmosphere and of rotating fluid (e.g., [3]) and as a controlling device in satellite dynamics (e.g., [4]).
Most of the works related to the rigid body and its extension to gyrostat can be assorted into three categories. e first is the integrability problem and the searching for the complete set of the first integrals of the motions. Borisov and Mamaev [5] contain most of those integrable problems up to 2001, and some cases were presented by several authors (see, e.g., [6][7][8][9][10]). e second category regards the problem of study periodic solutions, bifurcation, and chaos in some problems of rigid body-gyrostat (see, e.g., [11][12][13][14]). e third one is the stability problem of the equilibria in the dynamics of a rigid body-gyrostat moving in an orbit or about its fixed point (see, e.g., [15][16][17][18][19][20][21][22][23]). e current work is interested in studying the stability of permanent rotations for the motion of a charged gyrostat moving due to the combined influence of the magnetic field and Newtonian force field. is work is regarded as an extension of some previous works. In [17], Iñarrea et al. examined the stability of permanent rotations of a heavy rigid body carrying a rotor that rotates about one of the principal axes by a constant angular velocity. is study was followed by Elmandouh who studied this problem in the case of a charged heavy gyrostat [20]. Vera studied the stability of relative equilibrium for a gyrostat in Newtonian force field [19].
is work is organized as follows: in Section 2, we deduce equations of the motion and rewrite them as a Lie-Poisson Hamilton system. Section 3 contains the permanent rotations and their interpretation mechanically. In Section 4, we study the stability of those permanent rotations by applying both methods linear approximation method and energy-Casimir method. Section 5 involves the results found.

Equations of Motion
We consider the rotation of a charged rigid body about its fixed point and assume this body carries an axisymmetric rotor aligning along one of the principal axes of the body and rotating with a constant angular velocity.
is motion is assumed to happen due to the combined influence of a homogeneous magnetic field H �→ and Newtonian forces field. For motion description, we choose two frames Oξ 1 ζ 1 η 1 and Oξ 2 ζ 2 η 2 fixed in the space and in the body, respectively (see Figure 1). Furthermore, the body frame Oξ 2 ζ 2 η 2 is assumed to be the principal axes of the inertia at the fixed point O, and consequently, the principal inertia matrix of the gyrostat is are the unit vector along Oη 1 -axis and the angular velocity of the gyrostat, respectively. e two vectors ω → and c → are referred to the body frame. e vector c → is written in terms of Eulerian angle as illustrated in [1]: where θ, ψ, and φ indicate the angle of nutation between the two axes Oη 2 and Oη 1 , the precession angle about the Oη 2 axis, and the angle of proper rotation, respectively. e homogeneous magnetic field N �→ is presumed to be a constant and influences in the direction of the Oη 1 -axis, and therefore, it can be written as where β denotes the magnitude of the magnetic field. Now, we are going to introduce the equations of the motion. e whole angular momentum of the gyrostat is where π → � I ω → is the angular momentum of the gyrostat when the rotor is relatively at rest and K �→ � (0, 0, k) is the gyrostatic momentum, that is, the relative angular momentum of the rotor with respect to the body. Following the theorem of angular momentum about the point O, fixed point, we obtain where M �→ 0 indicates the total torque of the external forces about the fixed point O.
According to [1], the potential function for the Newtonian forces field takes the following form: where r 0 → � (x 0 , y 0 , z 0 ) is the center of mass vector, and for simplicity, we assume the center of mass lies on Oη 2 and n � (3g/R), where R is the distance between the center of the attraction and the fixed point O (R is assumed very large compared with the dimensions of the body). e torque due to the potential forces derived from the potential function (5) is given as follows [24]: Now, we calculate the torque appearing due to the magnetic field about the fixed point O. Let p be any point from the body which moves with velocity u → (p), carry a charge dq, and its position vector r → with respect to the fixed where ρ is the charge density and dV is the element's volume from the body. Hence, the torque arising due to the magnetic field takes the following form: where A is the 3 × 3 constant matrix which is assumed to be A � diag(a, b, d) for simplicity. us, the total torque about the fixed point O is 2 Advances in Astronomy Taking into account the two equations (4) and (8), the equations of motion in the body frame take the following form: where Notably, the expression (10) represents the torque of the gyroscopic forces (forces rely on the velocity). Because c → is a unit vector fixed in the space, we obtain Despite the variables utilized in two equations (9) and (11) are not canonical, we can describe this motion by means of a Hamiltonian function in the framework of Lie-Poisson systems. e Hamiltonian function takes the following form: (12) According to [25], we write the equations of the motion (9) and (11) as a Hamiltonian-Poisson system that is spanned by the matrix Π μ →: as long as the Jacobi identity is verified, i.e., Alternatively, the condition (14) is equivalently verified if the equation is satisfied. By direct calculations, we can prove that vector μ → , shown in (10), satisfied the condition (15), and consequently, the equations of motion (9) and (11) are rewritten as where Y � (π 1 , π 2 , π 3 , c 1 , c 2 , c 3 ) and ∇ → H is the naive gradient of H. In addition to the Hamilton (12), this system admits two Casimirs: where α 0 is a constant.

Permanent Rotations
To find the permanent rotations, we place _ π → � _ c → � 0 → into the two equations of motion (9) and (11); we obtain Equation (19) implies the two vectors ω → and c → are parallel, and consequently, ω where ω is the magnitude of angular velocity of the gyrostat in the body frame.

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O lies down the center of the mass of the gyrostat. In a similitude way, the permanent rotation E − 1 is explained as the rotation of the gyrostat in the down direction; i.e., the fixed point O lies above the center of mass of the gyrostat. (ii) When c 1 � 0 and c 2 ≠ 0, the geometric integral (2) reduces to c 2 2 + c 2 3 � 1, which has a parametric solution c 2 � sin θ and c 3 � cos θ. Taking into account the obtained results, equations (21)-(23) become which represents the existence condition for a family of the permanent rotations taking the form E 2 � (0, Bω sin θ, Cω cos θ, 0, sin θ, cos θ). It represents a rotation of a gyrostat with a constant angular velocity about an axis having a direction cosine (0, sin θ, cos θ).
Collecting the obtained results, we introduce down the following. Theorem 1.
e mechanical system (9) and (11) characterizing the rotations of a charged gyrostat in Newtonian field has four permanent rotations. ey are as follows:

Stability Analysis
is section aims to examine the stability of the permanent rotations introduced in eorem 1. We apply a linear approximation method to determine the sufficient conditions for instability that are also necessary conditions for the stability. We evaluate the tangent flow for the equations of the motion at the permanent rotation E, and we get where J(E) is the Jacobi matrix calculated at the permanent rotation E. To examine the linear stability, we find the eigenvalues of the Jacobi matrix and those eigenvalues are the roots of the characteristic equation where I 6 refers the 6 × 6 identity matrix. e energy-Casimir method is utilized to find sufficient conditions for stability.
is definite for W that is defined by en, E is stable and E is usually stable if W � 0 { }.

Stability of
is section aims to examine the stability of the permanent rotation E ± 1 which describes the rotation about the vertical axis with a constant angular velocity in two cases characterized by whether the fixed point O is above or down the gyrostat center of mass.
To obtain the necessary conditions for the stability, we compute the tangent flow of the equations of the motion (9) and (11) in the permanent rotation E ± 1 , and we obtain an equation in the form (26) and its characteristic equation (27) admits the following form: where 4 Advances in Astronomy Now, we are going to determine the sufficient conditions for the stability by employing the energy-Casimir method that is presented in eorem 2. We introduce the augmented Hamilton in the following form: where ] 1 and ] 2 are arbitrary constants which are determined by taking into account and E ± 1 is a critical point for the augmented Hamilton N, i.e., Equation (33) implies e subspace W is determined by where After some calculation, the basis of the subspace W is where e → i are the canonical basis of R 6 . e Hessian matrix for the augmented Hamilton (32) in the reduced subspace W is where We investigate the definiteness of the Hessian matrix (38) by applying the Sylvester criterion and so we evaluate its principal minors: It is obvious that Δ 1 > 0 and Δ 2 > 0 while Δ 3 > 0 if Advances in Astronomy and e two inequalities (41) and (42) verify together if us, we can formulate the theorem.

Theorem 4.
e sufficient condition of the stability for the permanent rotation E ± 1 to be Lyapunov stable is (43). Figure 2 determines the regions of linear stability and Lyapunov stability for E ± 1 . In Figure 2(a), the regions in pink determine the linear stability, while the white zones represent the instability. Notably, the solid lines in Figure 2 are determined by P 1 0, specifies the regions of Lyapunov stability in yellow, and the dash lines are specified by mgz 0 � min(χ 1 , χ 2 ). Figure 3(c) clarifies the regions of Lyapunov stability appears as a portion from the regions of linear stability. Similar figures can be used to describe the zones of the stability and instability for the permanent rotation E − 1 .

4.2.
Stability of E 2 . We endeavor to find the necessary and sufficient conditions for a family of the permanent rotation E 2 by utilizing the linear approximation and energy-Casimir method, respectively. Calculating the tangent flow of the equations of the motion (9) and (11) in the equilibrium position E 2 , we get an equation in the form (26) and its characteristic equation (27) takes the following form: where (45) Notice that we use the existence condition (24) to eliminate the weight of the gyrostat mg. Now, we can write down the following.

Theorem 5.
e family of permanent rotation E 2 is linearly stable if the conditions P 2 ≥ 0, Q 2 ≥ 0, and P 2 2 − 4Q 2 ≥ 0 are met. Or, equivalently, it is Lyapunov instable if at least one of these conditions is not satisfied.
To complete our study about the stability of E 2 , we apply the energy-Casimir method to find the stability's sufficient conditions. We utilize the same augmented Hamilton (34) and determine the values the two constants ] 1 and ] 2 leading to E 2 becomes a critical point for this augmented Hamilton, i.e., Equation (46) leads to Following eorem 2, the subspace W is specified by After some manipulations, we show that the subspace W is spanned by the vectors  where e → i are the canonical basis of R 6 . e Hessian matrix corresponding to the Hamiltonian (32) with (47) takes the following form: where To investigate the definiteness of Hessian matrix (51), we utilize the Sylvester criterion and so we compute the principal minors. ey take the following form: It is clear that Δ 1 and Δ 2 are always positive, whereas Δ 3 and Δ 4 are positive if e regions of linear and nonlinear (Lyapunov) stability are clarified by Figure 3. Figure 3(a) delimits the regions of linear stability in pink color, while as the white regions display the sufficient conditions for the instability of the permanent rotation E 2 . Notably, the solid lines which are the boundary of those regions are characterized by the equations P 2 � 0, Q 2 � 0, P 2 2 − 4Q 2 � 0. e regions in yellow appear in Figure 3(b) represent the regions of Lyapunov stability for E 2 . Figure 3(c) illustrates that the regions of Lyapunov stability appear as a part of the regions of linear stability. Notably, on the boundary of linear stability regions (solid lines in Figure 3(a) and also in Figure 3(c)), the permanent rotation E 2 is unstable.

Stability of E 3 .
e stability analysis for the family of the permanent rotations E 3 is done by applying similar procedures to E 2 stability study. e necessary and sufficient conditions for the stability of E 3 can be obtained by replacing A↔B and a↔b in the eorems 5 and 6, respectively.
us, we can formulate the following two theorems.

Theorem 7.
e family of permanent rotations E 3 is linearly stable if Or, equivalently, it is unstable if at least one of the conditions (57) is verified.
e sufficient condition for the stability of the equilibrium position E 3 is where one of the two conditions B > A, ω ∈ ]− ∞,

Conclusions
In this work, we have considered the motion of a charged rigid body carrying a rotor rotating about an axis which is parallel to one of the principal axes with a constant velocity. We have assumed the motion happens due to the combination influence of a uniform constant magnetic field and Newtonian force field. e equations of the motion have been constructed and rewritten as a Lie-Poisson Hamilton system. e permanent rotations E 1 have been obtained and collected in eorem 1. e first two permanent rotations E ± 1 are mechanically interpreted as the rotation of the gyrostat about the vertical up or down with a constant angular velocity. e permanent rotation E 2 is explained as the rotation of a gyrostat with a constant angular velocity about an axis having a direction cosine (0, sin θ, cos θ). e permanent rotation E 3 is interpreted as the rotation of the gyrostat with a constant angular velocity about an axis having a direction cosine (sin θ, 0, cos θ). e linear stability of those equilibrium positions has been studied by applying the linear approximation method, and the obtained results have been collected in eorems 3, 5, and 7. However, the Lyapunov stability of those permanent rotations have been examined by utilizing the energy-Casimir method, and the finding results have been presented in eorems 4, 6, and 8. In the plane of the two variables k and ω, the regions of linear stability have been clarified in pink (see Figures 2(a), 3(a), and 4(a)), while the regions of Lyapunov stability have been illustrated in yellow (see Figures 2(b), 3(b), and 4(b)). Moreover, we illustrate the regions of Lyapunov stability appear as a part of the regions of linear stability.

Data Availability
No data were used to support the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Advances in Astronomy