# The Slow Spinning Motion of a Rigid Body in Newtonian Field and External Torque

Authors: A. I. Ismail
Journal: Advances in Astronomy
 (2020)

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## Abstract

In this paper, the problem of the slow spinning motion of a rigid body about a pointO, 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.


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## Body

## 1. Introduction

In [1], the problem of rigid body dynamics is considered. The 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). The 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. The authors in [7] studied new types of integrable two-variable systems with quartic second integrals. The 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. The 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. The 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 ro about the moving z-axis which moves with the body. The authors achieved a small parameter proportional to 1/r0 and used the small parameter technique to solve the considered problems in the domain (t, ro⟶∞, ε⟶0). The 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, ro⟶0, ε⟶∞).


## 2. Equations of Motion and Change of Variables

Consider a rigid body of massM [10], with arbitrary ellipsoid of inertia surface, rotating about a fixed point O in the presence of the Newtonian force field O1 under the influence of the external torque vector about the moving axes ℓ¯=ℓ1i^¯+ℓ2j^¯+ℓ3k^¯. Let the attracting center O1 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 O1. Equations of motion and their first integrals are achieved and solved with a sufficiently large parameter proportional to 1/r0, where r0 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.The 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^¯=γ,γ′,γ″ be the unit vector fixed in space in the direction of the downward Z-axis, so(1)h¯o=Ap+ℓ1i^¯+Bq+ℓ2j^¯+Cr+ℓ3k^¯,(2)ω¯=pi^¯+qj^¯+rk^¯,where A,B,andC are the body’s principal moments of inertia in the moving frame. The six nonlinear equations of motion for this case are obtained in the following form:(3)dh¯odt=A∂p∂t+C−Bqri^¯+B∂q∂t+A−Cprj^¯+C∂r∂t+B−Apqk^¯,(4)dK^¯dt=∂K^¯∂t+ω¯∧K^¯=0¯.These equations have three first integrals named as follows:(a)
The Jacobi-integral(5)T+V=const,whereT is the kinetic energy of the body and V is the potential one.(b)
The angular momentum integral(6)h¯o.K^¯=const.(c)
The geometric integral(7)K^¯.K^¯=1.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¯ℓ1,ℓ2,ℓ3, around the x-axis, the y-axis, and the z-axis, respectively.These equations are of first order in unknown variablesp,q,r,γ,γ′,andγ″. The quantities A,B,C,ℓ1,ℓ2,andℓ3 are constants. The integration of such equations gives the solutions p,q,r,γ,γ′,andγ″ as functions in time t and the rigid body parameters.The 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).Letx0,y0,z0 be the center of mass in the moving coordinate system Oxyz; R is the distance from the fixed point O to the attracting center O1; p0,q0,r0,γ0,γ0′, and γ0″ are the initial values of the corresponding variables. Initially, let the body rotate about the z-axis with a sufficiently small angular velocity component r0 such that the z-axis makes an angle θ0≠0.5nπn=0,1,2,… with Z-axis being fixed in space.Without a loss of generality, we choose the positivez-axis, and the x-axis does not make an obtuse angle with Z-axis. According to this restriction, we obtain [9](8)γ0≥0,0<γ0″<1.Assume the parameters as follows:(9)a=AC,ab,c2=MgℓC,ε=cγ0″r0,x0=ℓx0′,xoyozo,ℓ2=x02+y02+z02,where ε is large since r0 is small and symbols such as (abc) mean cyclic permutations and indicate equations which are omitted.Introducing new variables as follows:(10)p=cγ0″p1,r=r0r1,γ=γ0″γ1,pq,γγ′γ″,k=3gRc−2,t=τr0.Substituting equation (10) into equations (3) to (7) when ℓ1=ℓ2=0, we obtain(11)p˙1+A1q1r1+A−1r0−1q1ℓ3=ε−1a−1y0′γ1″−z0′γ1′+kaA1γ1′γ1″,(12)q˙1+B1p1r1−B−1r0−1p1ℓ3=ε−1b−1z0′γ1−x0′γ1″+kbB1γ1γ1″,(13)r˙1=ε−2−C1p1q1+x0′γ1′−y0′γ1+kC1γ1γ1′,(14)γ˙1=r1γ1′−ε−1q1γ1″,(15)γ˙1′=ε−1p1γ1″−r1γ1,(16)γ˙1″=ε−1q1γ1−p1γ1′,.≡ddτ,(17)r12=1+ε−2S1,(18)r1γ1″=1+ε−1S2,(19)γ12+γ1′2+γ1″2=γ0″−2,where(20)S1=ap102−p12+bq102−q12−2x0′γ10−γ1+y0′γ10′−γ1′+z0′1−γ1″+kaγ102−γ12+bγ10′2−γ1′2+1−γ1″2,S2=ap10γ10−p1γ1+bq10γ10′−q1γ1′+1−γ1″ℓ3Ccγ0″.


## 3. Reduction of the Equations of Motion to a Quasi-Linear Autonomous System

In this section, we reduce the equations of motion to a quasi-linear autonomous system [11]. From equations (17) and (18), we obtain(21)r1=1+0.5ε−2S1+2z0′1−γ1″−k1−γ1″2+⋯,γ1″=1+ε−1S2−0.5ε−2S1+2z0′1−γ1″−k1−γ1″2+⋯.Differentiating equations (11) and (14) and using (21), one obtains(22)p¨1+ω′2p1=ε−1z0′a−1−A1b−1γ1+A1b−1x0′+kω2−A1γ1+b−1x0′−z0′γ1−kB1γ1A−1r0−1ℓ3+ε−2−ω2p1S1+A1b−1x0′S2+A1C1p1q12−A1q1x0′γ1′−y0′γ1+a−1y0′q1γ1−p1γ1′−a−1z0′p1+A1kp11−γ1′2+q11−C1γ1γ1′−S21+B1γ1+0.5r0−1ℓ3p1A−1B1−A1B−1×S1+2z0′1−γ1″−k1−γ1″2+A−1r0−1ℓ3b−1x0′−kb1γ1S2+ε−30.5z0′a−1−A1b−1γ1S1+2z0′1−γ1″−k1−γ1″2+0.5A−1r0−1ℓ3kB1γ1−b−1x0′S1+2z0′1−γ1″−k1−γ1″2+p1S22kA1−a−1z0′+⋯,(23)γ¨1+γ1=ε−11+B1−B−1r0−1ℓ3p1+ε−2−S1γ1+1+B1p1S2+1−C1p1q1γ1′+x0′γ1′2+x0′b−1−γ1y0′γ1′+z0′b−1+q12+kC1γ1′2−B1γ1′+ε−32b−1x0′−γ1b−1z0′+2kB1S2+⋯,where(24)ω2=−A1B1=A−CB−CAB=a−1b−1ab,ω′2=ω2−A−1B1−A1B−1r0−1ℓ3.We note thatω2>0 when A < B < C or A > B > C but ω′2>0 when A < B < C only.In caseA > B > C, we find that the term A−1B1−A1B−1 is positive and since ro is sufficiently small; that is, the term A−1B1−A1B−1r0−1ℓ3 tends to infinity, and ω′2 is negative.Solving equation (11) for q1 and equation (14) for γ1′, we obtain(25)q1=A1−1r1−11−A−1A1−1r0−1ℓ3r1−1+⋯−p˙1+ε−1a−1y0′γ1″−z0′γ1′+kaA1γ1′γ″,(26)γ1′=r1−1γ˙1++ε−1q1γ1″.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 p1,p˙1,γ1,γ˙1,p10,p˙10,γ10, and γ˙10.Introducing the new variables as follows:(27)p2=p1−ε−1χ+χ1γ2,γ2=γ1−ε−1νp2,where(28)χ=x0′bω′2−1A1+A−1r0−1ℓ3,ν=1−ω′2−11+B1−B−1r0−1ℓ3,χ1=1−ω′2−1−z0′a−1−A1b−1+kA1−ω2+A−1r0−1ℓ3b−1z0′+kB1.Using equations (27), (21), and (26), we obtain(29)Si=Si1+22−iε−1Si2+⋯,i=1,2,where(30)S11=ap202−p22+bχ32p˙202−p˙22−2x0′γ20−γ2−2y0′γ˙20−γ˙2+kaγ202−γ22+bγ˙202−γ˙22,S12=aχp20−p2+χ1p20γ20−p2γ2−bχ32a−1y0′p˙20−p˙2−χ2p˙20γ˙20−p˙2γ˙2−νx0′p20−p2−y0′ν1p˙20−p˙2+z0′−kS21+kνap20γ20−p2γ2+ν1bp˙20γ˙20−p˙2γ˙2,S21=ap20γ20−p2γ2−bχ3p˙20γ˙20−p˙2γ˙2,S22=aνp202−p22+χγ20−γ2+χ1γ202−γ22+bχ3−ν1p˙202−p˙22+a−1y0′γ˙20−γ˙2−χ2γ˙202−γ˙22−S21ℓ3Ccγ0″,where(31)χ3=A1−11−A−1A1−1r0−1ℓ3,χ2=χ1+a−1z0′−kA1,ν1=ν−χ3.Formulas (21) and (29) lead to(32)r1=1+0.5ε−2S11+ε−3S12−z0′S21+kS21+⋯,γ1″=1+ε−1S21+ε−2S22−0.5S11−ε−3S12−z0′S21+kS21+⋯.In terms ofp2·γ2, and the rigid body parameters, we find that(33)q1=−χ3p˙2+ε−1χ3a−1y0′−χ2γ˙2+ε−2χ3ν1p˙2kA1−a−1z0′+S11p˙2χ3−0.5A1−1+χ3S21kA1γ˙2+a−1y0′+⋯,γ1′=γ˙2+ε−1ν1p˙2+ε−2χ3a−1y0′−χ2γ˙2−S21p˙2−0.5S11γ˙2+⋯.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:(34)p¨2+ω′2p2=ε−2Fp2,p˙2,γ2,γ˙2,ε−1,γ¨2+γ2=ε−2ϕp2,p˙2,γ2,γ˙2,ε−1,where(35)F=F2+ε−1F3+…,ϕ=ϕ2+ε−1ϕ3+…,F2=f2−νχ11−ω′2p2,ϕ2=φ2+ν1−ω′2χ+χ1γ2,F3=f2−χ1φ2−νχ11−ω′2χ+χ1γ2,ϕ3=φ3−νf2+ν2χ11−ω′2p2,f2=−ω2S11p2+A1x0′b−1S21+χ3γ˙2p˙2+A1C1χ32p2p˙22−y0′χ3γ2p˙2A1+a−1−a−1p2z0′+y0′γ˙2+A1k1−γ˙22p2+C1−1χ3γ2γ˙2p˙2−1+B1S21γ2+0.5r0−1ℓ32p2A−1B1−A1B−1S11+A−1b−1x0′−kB1γ2S21,f3=−2ω2p2S12+χ+χ1γ2−ω2S11−a−1z0′+y0′γ˙2+A1C1χ32p˙22+k1−γ˙22+A1b−1x0′S22+A1χ3p˙2x0′ν1p˙2−y0′νp2−p2p˙2a−1y0′ν1+νχ+2A1kν1γ˙2+χ3p˙2ν1γ2p˙2+νγ˙2p2C1−1−1−B1νS21p2+S22γ2+0.5z0′a−1−A1b−1γ2S11+2kA1−a−1z0′p2S21+χ3a−1y0′−χ2γ˙2×−A12C1χ3p2p˙2+x0′γ˙2+y0′γ2A1+a−1+γ2γ˙221−C1+0.5r0−1ℓ3A−1B1−A1B−12p2S12−z0′S21+kS21+χ+χ1γ2S11+2A−1b−1x0′−kB1γ2S22−kB1νS21p2+A−1kB1γ2−b−1x0′S11,φ2=1+B1S21−1−C1χ3γ˙2p˙2p2+x0′b−1+γ˙22+kC1γ˙22−B1−y0′γ˙2−z0′b−1−χ32p˙22−S11γ2,φ3=1+B1p2S22+χ+χ1γ2S21+χ31−C1×a−1y0′−χ2γ˙2γ˙2p2−p˙0ν1p2p˙2+χ+χ1γ2γ˙2−2γ2S12−νp2S11+2x0′ν1γ˙2p˙2−y0′ν1γ2p˙2+νγ˙2p2−νp2b−1z0′+χ32p˙22+2χ32a−1y0′−χ2γ˙2×γ2p˙2+k2C1ν1γ2γ˙2p˙2+νC1γ˙22−B1p2+2b−1x0′−b−1z0′+2kB1γ2S21.System (34) has the first integral obtained from equations (17)–(19) as follows:(36)γ22+γ˙22+2ε−1νγ2p2+ν1γ˙2p˙2+S21+ε−2ν2p22+2χ3γ˙2a−1y0′−χ2γ˙2−S21p˙2−1+γ˙22S11+2S22+⋯=γ0″−2−1.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 r0.


## 4. Formal Construction of the Periodic Solutions

Without a loss of generality, since the system (34) is autonomous, we assume that [14](37)p20,0=0,p˙20,0=0,γ˙20,ε−1=0.The generating system of equation (34) is(38)p¨20+ω′2p20=0,γ¨20+γ20=0,which has periodic solutions as follows:(39)p20=M1cosω′τ+M2sinω′τ,γ20=M3cosτ,with a period T0=2πn, where Mi, i=1,2,3, are constants to be determined. Consider the required periodic solutions of system (34) in the following form:(40)p2τ,ε−1=M1+β1cosω′τ+M2+β2sinω′τ+∑k=2∞ε−kGkτ,γ2τ,ε−1=M3+β3cosτ+∑k=2∞ε−kHkτ,with a period Tε−1=T0+αε−1. The quantities β1,ω′β2,andβ3 represent the deviations of solutions p2,p˙2,andγ2 at any ε from their initial values when ε⟶∞. Let the initial condition of system (40) be of the following form:(41)p20,ε−1=M1+β1,p˙20,ε−1=ω′M2+β2,γ20,ε−1=M3+β3,γ˙20,ε−1=0.Consider the new function as follows:(42)U=u+∂u∂M1β1+∂u∂M2β2+∂u∂M3β3+0.5∂2u∂M12β12+⋯,U=Gk,Hku=gk,hk,such that(43)gkτ=1ω′∫0τFk′t1sinω′τ−t1dt1,hkτ=∫0τϕk′t1sinτ−t1dt1k=2,3,where(44)Fk′τ=1k−2!dk−2Fdε2−kβ=ε−1=0,ϕk′τ=1k−2!dk−2ϕdε2−kβ=ε−1=0.We note that the right-hand sides of (34) begin with terms of order ε−2 and so(45)Fk′τ=Fkp20,p˙20,γ20,γ˙20≡Fk0,ϕk′τ=ϕkp20,p˙20,γ20,γ˙20≡ϕk0,k=2,3.Now, we find the expressions ofϕk0 and Fk0. Periodic solutions (39) are reformulated in the following form:(46)p20=Ecosω′τ−η,γ20=M3cosτ,where E=M12+M22 and η=tan−1M2/M1.Using equations (29) and (39), we obtain(47)Sij0=Sij0p20,p˙20,γ20,γ˙20i,j=1,2,S110=E2acos2η−0.5+bχ32ω′2sin2η−0.5+0.5bχ32ω′2−acos2ω′τ−η−2M3x0′1−cosτ+y0′sinτ−0.5kM32C11−cos2τ,S210=M3Eacos2η+0.5bω′χ3−acosω′−1τ−η−0.5bω′χ3+acosω′+1τ−η,S120=aEχ3cosη−cosω′τ−η+χ1M3cosη−cosτcosω′τ−η−bχ32Ecosη−cosω′τ−η+χ2M3sinτsinω′τ−η−νx0′Ecosη−cosω′τ−η+kEM3νacosη−cosτcosω′τ−η−ν1bsinτsinω′τ−η+z0′−kS210,S220=aνE2cos2η−cos2ω′τ−η+χM31−cosτ+χ1M32sin2τ+bχ3a−1y0′M3sinτ−ν1E2ω′2sin2η−sin2ω′τ−η+χ2M32sin2τ−S210ℓ3/Ccγ0″.Substituting equations (46) and (47) into equation (35), we obtain(48)F20=M1Lω′cosω′τ+M2Lω′sinω′τ+⋯,ϕ20=M3Nω′cosτ+⋯,(49)Lω′=ω2−aM12+bω′2χ32M22+bω′2χ32M12+M22+A1C1ω′2χ32M12+M22+2M3x0′ω2+kA1+0.5M32ω2C1−z0′a−1+νχ11−ω′2+0.5r0−1ℓ3A−1B1−A1B−1aM12+bω′2χ32M22−bω′2χ32M12+M22−2M3x0′−0.5kM32C1,(50)Nω′=−aM12+bω′2χ32M22−M12+M22aB1+ω′2χ321−b+2M3x0′−z0′b−1−νχ11−ω′2+kM32C1−B1.From equations (33), (49), and (50), we obtain(51)g2T0=−πnω′−1M2Lω′,g˙2T0=πnM1Lω′,h2T0=0,h˙2T0=πnM3Nω′,where the constants M1,ω′M2,andM3, the deviations β1ε−1,ω′β2ε−1,andβ3ε−1, and the correction of the period α are determined from the periodicity conditions and their derivatives:(52)ψ1=p2T0+α,ε−1−p20,ε−1=0,ψ2=p˙2T0+α,ε−1−p˙20,ε−1=0,ψ3=γ2T0+α,ε−1−γ20,ε−1=0,ψ4=γ˙2T0+α,ε−1−γ˙20,ε−1=0.Due to the existence of first integral (36) for system (34), the condition ψ3=0 is not independent [15]; then, integral (36) becomes(53)γ22T0+α,ε−1+γ˙22T0+α,ε−1+2ε−1νγ2T0+α,ε−1p2T0+α,ε−1+ν1γ˙2T0+α,ε−1×p˙2T0+α,ε−1+S21T0+α,ε−1+ε−2ν2p22T0+α,ε−1+2χ3γ˙2T0+α,ε−1a−1y0′−χ2γ˙2T0+α,ε−1−S21T0+α,ε−1p˙2T0+α,ε−1−S11T0+α,ε−11+γ˙22T0+α,ε−1+2S22T0+α,ε−1+…=γ220,ε−1+γ˙220,ε−1+2ε−1νγ20,ε−1p20,ε−1+ν1γ˙20,ε−1p˙20,ε−1+S210,ε−1+ε−2ν2p220,ε−1+2χ3γ˙20,ε−1a−1y0′−χ2γ˙20,ε−1−S210,ε−1p˙20,ε−1−S110,ε−11+γ˙220,ε−1+2S220,ε−1+….Using condition (41) and equation (52), we obtain(54)ψ32+2M3+β3ψ3+ε−1φ1ψ1,ψ2,ψ4,ε−1=0,where φ1 is an entire function in their variables and φ10,0,0,ε−1=0; then if M3≠0, form (54) gives(55)ψ3=f1ψ1,ψ2,ψ4,ε−1,where f1 is an entire function in all their arguments and f10,0,0,ε−1=0; then, the condition ψ3=0 in (52) is satisfied with the following condition:(56)ψ1=ψ2=ψ4=0.Substituting initial conditions (41) into equation (56) with τ=0, we obtain(57)M32+2M3β3+β32+2ε−1νM3M1+β1+⋯=γ0″−2−1.Assume thatγ0″ does not depend on ε; we obtain that(58)M32=γ0″−2−1,β32+2M3β3+2ε−1νM3M1+β1+⋯=0.From (58) and (8), we obtain(59)M3=1−γ0″2γ0″,0<M3<∞,β3=−ε−1νM1+β1+⋯,where γ0″ is an arbitrary parameter and M3 is the arbitrary constant.This means that periodic solutions (40) depend on arbitrary constant M3 and the function β3ε−1 which is equal to 0 when ε⟶∞. Independent periodic solutions (52) are expanded in a power series of α (neglecting terms of ε−2α); then, we obtain(60)p2T0,ε−1+αp˙2T0,ε−1+…=p20,ε−1,p˙2T0,ε−1+αp¨2T0,ε−1+…=p˙20,ε−1,γ˙2T0,ε−1+αγ¨2T0,ε−1+…=γ˙20,ε−1.Using initial values (41) in the above relations, we put independent periodicity conditions (56) in the following form:(61)p2T0,ε−1+αω′M2+β2=M1+β1,p˙2T0,ε−1−ω′M2+β2=αω′M1+β1,γ˙2T0,ε−1=αM3+β3.Using equations (40), (59), and the last equation of (61), we obtain the following function:(62)αε−1=ε−2M3+β3−1H˙2T0+εH˙3T0+….Thus, neglecting the terms of orderα2 and ε−2α in (61), we find that the terms of the order ε−4 are neglected. Using equations (37) and (41), we obtain the periodic solutions with basic amplitudes equal to zero, that is [16],(63)M1=M2=0.Substituting equations (62), (63), and (40) into the first two equations from (61), we obtain the following system for determining β1 and β2:(64)G2T0+ε−1G3T0+ω′β2M3+β3−1H˙2T0+ε−1H˙3T0+ε−2…=0,G˙2T0+ε−1G˙3T0−ω′2β1M3+β3−1H˙2T0+ε−1H˙3T0+ε−2…=0.Due to (51), the above system becomes(65)−πnβ2ω′−1L1ω′−ω′2N1ω′+ε−1G3T0+…=0,πnβ1L1ω′−ω′2N1ω′+ε−1G˙3T0+…=0,where L1ω′ and N1ω′ are obtained from (50) replacing M1,M2,M3 by β1,β2,andM3+β3. Making use of equations (24), (28), (31), and (49), we obtain(66)L1ω′−ω′2N1ω′=β12+β22W1ω′+z0′W2ω′+kW3ω′+W4ω′,where(67)W1ω′=d1+d2+d3r0−1ℓ3,W2ω′=d4−d5d6d7+r0−1ℓ3d5d6d8+d9+B−1d7−d101+a−1d6d7,W3ω′=d5d6d11+d12+r0−1ℓ3d5d13−d14−B−1d11+b−1d10a−1d6d11+d15r0−1ℓ3,W4ω′=−0.5ad10β12+a−1b−1β22r0−1ℓ3,d1=b−1a−12a−b−1,d2=b−2ba−b+a−1aA−1a−11−b−1+bB−1,d3=0.5A−11−Aab−11−a1−b−1+AB−1,d4=a−11−b−2a−1b−1,d5=ab−2ab+a−1b−1,d6=b−1a+b−1,d7=ab−12b−1ab+a−1b−1,d8=Ab−1ab+a−1b−1,d9=ab−12b−1A−1aa−1+B−1bb−1,d10=Ab−1a−1+aB−1b−1,d11=ab−11−ba+b−1ab+a−1b−1,d12=ab2−11−bb2−a−12+0.05ba−1b−aM32,d13=Ab−1a−1ab+a−1b−1,(68)d14=ab−11−ba+b−1aA−1a−1+bB−1b−1,d15=0.75bb−aM32−a−1.Since thez-axis is oriented towards the minor axis of the ellipsoid of inertia for the body, then W1ω′>0 for all ω′ under consideration. Assume that [17](69)z0′W2ω′+kW3ω′≠0.Using (65), the expressions for β1 and β2 are obtained in the form of power series expansions beginning with terms of order greater than ε−2. So, we obtain the first terms of the required periodic solutions and the correction of the period αε−1 in the following forms:(70)p1=ε−1−x0′a−1−11+bB−1a−1−1r0−1ℓ3+χ1M3cosτ+⋯,(71)q1=ε−1a1−b−1y0′a−1+χ2M3sinτ−A−11−b−1r0−1ℓ3y0′+z0′−kaA1M3sinτ+ad5kb1−bd6−z0′2b−1+⋯,(72)r1=1−ε−2M3x0′1−cosτ+y0′sinτ+0.25M3C11−cos2τ+⋯,(73)γ1=M3cosτ+⋯,(74)γ1′=−M3sinτ+⋯,(75)γ1″=1+ε−21−b−1M3y0′sinτ+1−a−1M3x0′1−cosτ−0.5b−11−b−1d7M32z0′1−cos2τ+0.25M32k2abd5d6+C11−cos2τ+r0−1ℓ3−abA−11−B−2M3y0′sinτ+abB−1a−1−2M3x0′1−cosτ+0.5b−11−b−1×z0′M321−cos2τA−1a2bd51−b−12b2−2b+1+d9+0.5k1−b−1M321−cosτb−1d13−aA−1d111−b−1−1−b2b−1−1d6d9+⋯,(76)αε−1=ε−2πnM3x0′−z0′b−1+ab−1kd11−z0′d7d6+kM32C1−B1+ab−1r0−1ℓ3z0′d6d8+d9+d7B−1+kd6d13−d14−d11B−1+….New solutions (70)–(76) are obtained in terms of the large parameter ε and a sufficiently small angular velocity component ro about the minor axis of the ellipsoid of inertia. The case of the motion of the body with a sufficiently small angular velocity component ro about the major axis of the ellipsoid of inertia is considered in a separate paper since ω′2 is negative in this case. The motion considered here is a generalization of many problems studied in a previous work [18]. That is, the obtained solutions give many special cases for gyroscopic problems with new treatment by the large parameter technique [19] which saves high energy given for the body at the initial motion. The correction terms in our solutions in terms of the parameter ε are(77)△p1=ε−1x0′b−1B1−11−ω2ω′−2+ω′−2A−1r0−1ℓ3+χ−χ1∗M3cosτ+⋯,△q1=ε−1−y0′aAA12−1r0−1ℓ3+A1−1M3sinτχ1−χ1∗−χ2A−1A1−1r0−1ℓ3+⋯,△r1=ε−30+…,△γ1=ε−10+…,△γ1′=ε−10+⋯,△γ1″=ε−2aM3χ−χ∗1−cosτ−bM3y0′aAA12−1r0−1ℓ3sinτ+0.5M321−cos2τa1−b−1χ1−χ1∗−χ2bAA12−1r0−1ℓ3+⋯,Δαε−1=ε−2πn1−B1χ1−χ1∗−B−1χ1r0−1ℓ3+⋯.Also,(78)Δp11=Δp1+ε−1χ1∗−χ1∗∗M3cosτ+⋯,Δq11=Δq1+ε−1A1−1M3χ1∗−χ1∗∗−kA1sinτ+⋯,Δr11=−0.25ε−2M32C11−cos2τ+⋯,Δγ11=ε−10+…,Δγ11′=ε−10+⋯,Δγ11″=Δγ1″+ε−20.25kM32C11−cos2τ+0.5M321−cosτa1−b−1χ1∗−χ1∗∗−kb+⋯,Δα1ε−1=Δα+ε−2πnz0′2−b−1+kM32C1−B1+χ1∗1+B1+⋯,where(79)χ∗=A1x0′bω2−1,χ1∗=1−ω2−1kA1−ω2−z0′a−1−A1b−1,χ1∗∗=−z0′1−ω2−1a−1−A1b−1.


## 5. Geometric Interpretation of Motion

In this section, we explain the geometric interpretation of motion using Euler’s anglesθ,ψ,andφ [20] which are determined from the obtained periodic solutions. Since the initial system is autonomous, the periodic solutions remain so, if the time t is replaced by t+t0, where t0 is the arbitrary interval time. So, Euler’s angles for this problem are given by(80)cosθ=γ″,dψdt=pγ+qγ′1−γ″2,tanφ0=γ0γ0′,dφdt=r−cosθdψdt.Substituting equations (70)–(76) into equation (80) in which t is replaced by t+t0 and using the relations (10), the following expressions for Euler’s angles θ,ψ,andφ are obtained:(81)φ0=π2+r0−1t0+⋯,θ0=tan−1M3,θ=θ0−ε−2θ1t+t0−θ1t0,ψ=ψ0+ε−1ccosecθ0cosθ0ψ1t+t0−ψ1t0,φ=φ0+r0−1t−ε−1ccotθ0cosθ0φ1t+t0−φ1t0−ε−2tanθ0φ2t+t0−φ2t0,where(82)θ1t=a1sinr0−1t−a2cosr0−1t−a3tanθ0cos2r0−1t,ψ1t=a4r0−1sinr0−1t+a5r0−1cosr0−1t+0.5χ1−a6tanθ0+0.25r0−1χ1−a6tanθ0sin2r0−1t,φ1t=ψ1t,a1=1−b−1y0′1−abA−11−b−1r0−1ℓ3,a2=1−a−1x0′1+abB−11−a−1r0−1ℓ3,a3=0.5z0′b−11−b−1r0−1ℓ3a2bd5A−11−b−12b2−2b+1+d9−d7+0.25k2abd5d6+C1+21−b−1r0−1ℓ3b−1d13−aA−11−b−1d11+b−12b−1−1d6d9,a4=−x0′a−1−11+bB−1a−1−1r0−1ℓ3,a5=1−by0′−aA−11−b−2r0−1ℓ3y0′+ad5kb1−bd6−z0′2b−1,a6=a1−b−1χ2−aA−1r0−1ℓ31−b−1χ2−χ1,a7=x0′+0.25kC1tanθ0.


## 6. 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 Table1.Table 1
The differences between the previous works and the considered work.
Ser.The previous problemsThe considered problem1The body rotates fastThe body rotates slow2ro is sufficiently highro is sufficiently small3ε⟶0ε⟶∞4Poincare’s method is used for solving the problemsThe large parameter method is used for solving the problem5High kinetic energy is required for the motionsLow kinetic energy is required for the motion6The domain of the solutionsF (t, ro⟶∞, ε⟶0)The domain of the solutionsF (t, ro⟶0, ε⟶∞)7θ,ψ,andφ have the domain G (t, ro⟶∞, ε⟶0)θ,ψ,andφ have the domain G (t, ro⟶0, ε⟶∞)8ω′2 is positive for A<B<C or A>B>Cω′2 is positive for A<B<C and negative for A>B>C


## 7. 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 of the period of the equations of motion of it in the presence of Newtonian force field and an external torque. This problem is solved in a new domain of the angular velocity componentro⟶0.The 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 ro⟶∞. So we must search other techniques that come from the sufficiently small assumption of ro and depend on achieving large parameter instead of a small one. This technique is named the large parameter method. The 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 ro⟶0 and ε⟶∞. The 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. The correction terms for our solutions are given in terms of ro and ε. The geometric interpretation of motions is given to describe the orientation of the motion at any instant of time. The cases of gyroscopic motions and regular precession are obtained as special cases from this study when we apply the symmetry conditions. The practical importance of this work is very wide since it is used in many applications of life such as military life and civil one. The 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]. There are many interesting space applications of these problems in [2].



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