The Slow SpinningMotion of a Rigid Body inNewtonian Field and External Torque

In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r0⟶ 0, define a large parameter proportional to 1/r0, and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.


Introduction
In [1], the problem of rigid body dynamics is considered. e author in [2] gave important space applications to this problem. In [3], the authors presented valuable and important studies for the evolution of motions of a rigid body about its mass center. In [4], the authors introduced a new procedure for solving Euler-Poisson equations (of a rotatory rigid body over a fixed point). e author in [5] constructed periodic solutions for Euler-Poisson equations utilizing power series expansion containing a small parameter proportional to the inverse of sufficiently high angular velocity component. In [6], the author studied many perturbation techniques for solving the linear and nonlinear systems of ordinary and partial differential equations such as Poincare's method, KBM method, Poincaré-Lindstedt method, and multiple scales method. e authors in [7] studied new types of integrable twovariable systems with quartic second integrals. e study in [8] presented the motion for the rigid body in the presence of a gyrostatic momentum in cases of external effects and without external effects. e author considered the fast spin motion of a rigid body and achieved a small parameter proportional to the inverse of high angular components about the z-axis. e author applies the small parameter of Poincare's method for solving this problem. In [9], the author investigated the motion over the fixed point O of a fast spinning heavy solid in a uniform gravity field (the classical problem). He assumed fast spinning of the body, achieved a small parameter, and used Poincare's method for the solution. In all previous works, the rotary motion for a fast-spinning body with gyro moments was studied. Initially, the authors assumed that the body rotates with a sufficiently large angular velocity component r o about the moving z-axis which moves with the body. e authors achieved a small parameter proportional to 1/r 0 and used the small parameter technique to solve the considered problems in the domain (t, r o ⟶ ∞, ε ⟶ 0). e fact of slow motion of that body which must be achieved on a new parameter named the large parameter and must be solved using a new procedure named the large parameter technique was not considered, although this motion saves high energy given at the initial moment of the body and can solve the problem in a new domain (t, r o ⟶ 0, ε ⟶ ∞).

Equations of Motion and Change of Variables
Consider a rigid body of mass M [10], with arbitrary ellipsoid of inertia surface, rotating about a fixed point O in the presence of the Newtonian force field O 1 under the influence of the external torque vector about the moving axes ℓ � ℓ 1 i + ℓ 2 j + ℓ 3 k . Let the attracting center O 1 lie on the Z-axis which is fixed in space. Let the element dm lie on the body at the point p (x, y, z) and have a position vector ρ from O and a position vector r from O 1 . Equations of motion and their first integrals are achieved and solved with a sufficiently large parameter proportional to 1/r 0 , where r 0 is sufficiently small. We deduce the system of equations of motion and their first integrals of the considered problem and use the large parameter method for solving it. e differential equations of motion and their first integrals are obtained [10]. Let h o be the angular momentum vector which rotates in space at the same angular velocity ω of the rigid body and k � (c, c ′ , c ″ ) be the unit vector fixed in space in the direction of the downward Z-axis, so where A, B, and C are the body's principal moments of inertia in the moving frame. e six nonlinear equations of motion for this case are obtained in the following form: ese equations have three first integrals named as follows: (a) e Jacobi-integral where T is the kinetic energy of the body and V is the potential one. (b) e angular momentum integral (c) e geometric integral Equations (3) and (4) are nonlinear differential equations for the motion of a rigid body around a fixed point in the field of Newtonian force with the presence of rotary torque vector ℓ(ℓ 1 , ℓ 2 , ℓ 3 ), around the x-axis, the y-axis, and the z-axis, respectively. ese equations are of first order in unknown variables p, q, r, c, c ′ , and c ″ . e quantities A, B, C, ℓ 1 , ℓ 2 , and ℓ 3 are constants. e integration of such equations gives the solutions p, q, r, c, c ′ , and c ″ as functions in time t and the rigid body parameters. e equations of motion for a coherent object around a fixed point in the asymmetric attraction field [5,9] and their three initial integrals result as special cases from equations (3), (4), (5), (6), and (7).
Let (x 0 , y 0 , z 0 ) be the center of mass in the moving coordinate system (Oxyz); R is the distance from the fixed point O to the attracting center O 1 ; p 0 , q 0 , r 0 , c 0 , c 0 ′ , and c 0 ″ are the initial values of the corresponding variables. Initially, let the body rotate about the z-axis with a sufficiently small angular velocity component r 0 such that the z-axis makes an angle θ 0 ≠ 0.5nπ(n � 0, 1, 2, . . .) with Z-axis being fixed in space.
Without a loss of generality, we choose the positive zaxis, and the x-axis does not make an obtuse angle with Zaxis. According to this restriction, we obtain [9] c 0 ≥ 0, 0 < c 0 ″ < 1.
Assume the parameters as follows: a � A C , (ab), where ε is large since r 0 is small and symbols such as (abc) mean cyclic permutations and indicate equations which are omitted.
Introducing new variables as follows: 2

Reduction of the Equations of Motion to a Quasi-Linear Autonomous System
In this section, we reduce the equations of motion to a quasilinear autonomous system [11]. From equations (17) and (18), we obtain Differentiating equations (11) and (14) and using (21), one obtains where We note that In case A > B > C, we find that the term ( is positive and since r o is sufficiently small; that is, the term 0 ℓ 3 tends to infinity, and ω ′ 2 is negative. Solving equation (11) for q 1 and equation (14) for c 1 ′ , we obtain Advances in Astronomy Making use of equations (21) and (26) into equations (22) and (23), we obtain a quasi-linear autonomous system with two degrees of freedom and depend on 10 , and _ c 10 . Introducing the new variables as follows: where Using equations (27), (21), and (26), we obtain where where Formulas (21) and (29) lead to In terms of p 2 · c 2 , and the rigid body parameters, we find that Substituting equations (27), (29), (30), (32), and (33) into equations (23) and (24), we obtain a quasi-linear autonomous system of two degrees of freedom in the following form: where 4 Advances in Astronomy System (34) has the first integral obtained from equations (17)- (19) as follows: We aim to find the periodic solutions for system (36) under the condition A < B < C (ω ′ 2 is positive) [12]. In this case, the body rotates about the minor axis of the ellipsoid of inertia surface [13] with initial sufficiently small angular velocity r 0 .

Comparison between the Previous Problems and the Considered Problem
In this section, we make a comparison between the previous works and the considered work through Table 1.

Conclusions
From this study, we treat the problem of the slow spinning motion about the minor axis of the ellipsoid of inertia of a rigid body to find the periodic solutions and the correction Poincare's method is used for solving the problems e large parameter method is used for solving the problem 5 High kinetic energy is required for the motions Low kinetic energy is required for the motion 6 e domain of the solutions 2 is positive for A < B < C and negative for A > B > C of the period of the equations of motion of it in the presence of Newtonian force field and an external torque. is problem is solved in a new domain of the angular velocity component r o ⟶ 0. e well-known Poincare's method [5] cannot solve this problem because we cannot achieve the small parameter which must be proportional to a sufficiently high angular velocity component r o ⟶ ∞. So we must search other techniques that come from the sufficiently small assumption of r o and depend on achieving large parameter instead of a small one.
is technique is named the large parameter method. e advantage of this method is as follows: assuming low energy at the initial instant instead of high energy, obtaining a slow periodic motion instead of a fast periodic one, and giving the solutions in a new domain of motion r o ⟶ 0 and ε ⟶ ∞. e case when A < B < C [21] cannot be solved here since ω ′ 2 is negative in this case. So we will treat this case separately in the future, in shaa Allah. e correction terms for our solutions are given in terms of r o and ε. e geometric interpretation of motions is given to describe the orientation of the motion at any instant of time.
e cases of gyroscopic motions and regular precession are obtained as special cases from this study when we apply the symmetry conditions. e practical importance of this work is very wide since it is used in many applications of life such as military life and civil one. e case of the gyro motion which is symmetric about the z-axis, i.e., A � B < C, is obtained as a special case from our work [22]. ere are many interesting space applications of these problems in [2].

Data Availability
e data used to support the findings of this study are available from the corresponding author upon reasonable request.

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