# Quartic Integral in Rigid Body-Gyrostat Dynamics

Authors: C. Mnasri; A. A. Elmandouh
Journal: Advances in Astronomy
 (2020)

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## Abstract

In this work, we investigate the problem of constructing new integrable problems in the dynamics of the rigid body rotating about its fixed point as results of the effect of a combination of potential and gyroscopic forces possessing a common symmetry axis. We introduce two new integrable problems in a rigid body dynamics that generalize some integrable problems in this field, known by names of Chaplygin and Yehia–Elmandouh.


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## Body

## 1. Introduction

One of the classical problems manifesting in mathematical physics is the issue of determining whether a dynamical system, especially a mechanical one, is integrable or not. Integrability in this context often points out to Liouville integrability. The Liouville integrability concept is defined as the Hamiltonian system withn degrees of freedom that is completely integrable if it has n independent integrals of motion which are in involution, i.e., their Poisson brackets are zero [1]. The integrable systems possess miscellaneous properties such as their behavior that can be globally tested in an infinite interval of time, gratitude to the theories of perturbation, those systems that can be applied to give an appointed inference about the nonintegrable systems nearby them, and in general, the motion equations can be solved by quadratures [2]. The problem of integrability is split into two categories. The first one is finding the sufficient conditions for the integrability, and this requires the construction of a sufficient number of first integrals of motion. Numerous methods can be utilized to construct the first integrals of motion such as the direct method, Darboux method, and Yehia method (see, e.g., [3–13]). The second one deals with obtaining the necessary conditions of the integrability (see, e.g., [14–20]), but we must introduce the required number of the integrals to confirm the integrability.One of the significant issues in applications in assorted branches of science such as physics and astronomy is the problem of a rigid body and its extension to a gyrostat (see, e.g., [21–23]). So, it is a beneficial model for research from different points of view [24–29]. Consequently, the present work interested in analyzing the general motion of a rigid body about its fixed point that happens under the effect of a combination of potential (velocity-independent) forces and gyroscopic (velocity-dependent) forces. The gyroscopic forces are specified by l=0,0,l3, while the potential forces are characterized by V(γ). As it outlined in [30], this motion can be characterized by the Lagrangian:(1)L=12ω·ωI+l·ω−V,where ω=p,q,r is the angular velocity, and I=diagA,B,C is the inertia matrix of the body. The equations of motion corresponding Lagrangian (1) are [30–32](2)ω.I+ω×ωI+μ=γ×∂V∂γ,γ.+ω×γ=0,where γ=γ1,γ2,γ3 is the unit vector which is fixed upward in space, and μ is expressed as(3)μγ1,γ2,γ3=∂∂γl·γ−∂∂γ⋅lγ.It is well-known that the Euler-Poisson equation (2) has three classical integrals of motion. They are as follows:(1)
Area integral:(4)I1=(ωI+l)·γ=f,where the arbitrary constantf denotes the value of area integral. This integral is sometimes named as a cyclic integral due to it correspondences the cyclic variable ψ, the angle of precession.(2)
Jacobi integral:(5)I2=12ω·ωI+V=h,where the arbitrary constanth identifies the value of the Jacobi integral.(3)
Geometric integral:(6)I3=γ⋅γ=1.Taking into account the Jacobi theorem on the last integrating multiplier [33], four integrals of motion are needed to confirm the integrability of the equation of motion (2). Thence, the existence of a fourth integral independent of those (4), (5), and (6) is sufficient to prove the integrability. It is worth noticing that the integrable case either generally integrable or conditionally integrable according to the fourth integral is either valid on an arbitrary level of the cyclic integral I1 or valid on a single level of cyclic integral I1 which is usually zero.The problem of a rigid body which is described by the equation of motion (2) was studied in diverse posterior works from the point of view of the integrability. Those works include three types of problems.The first problem deals with the problem of the motion of a rigid body about a fixed point under the action of its weight. It is characterized byV=r0⋅γ, and μ=0, where r0 is a constant vector that represents the center of the mass vector. It attracted the attention of the researchers for a long time, nearly two and a half centuries, and thus, it has a great history. It includes three (no more) general integrable problems bearing the names of who discovered them, Euler, Lagrange, and Kowalevski, and one conditional integrable problem of Goriatchev–Chaplygin (see, e.g., [34]).The second problem concerns the motion of a rigid body about its fixed point under the effect of its weight, and moreover, there is a rotor spinning about its axis of symmetry which is fixed in the body with a constant angular velocity. It is worth mentioning that it is a simple multibody that consists of the main body and the rotor, and it is termed in literatures a gyrostat. The second problem regards as a generalization to the first problem, and it is determined byV=r0⋅γ and μ=k, where k is a constant vector characterizing the gyroscopic moment due to the existence of the rotor. It contains three general and one conditional integrable problems generalizing those in the first problem by adding the gyrostatic moment. The general cases are Lagrange, Joukovsky [34], and Yehia [35], while the conditional case is the Sretensky case which generalizes the Goriatchev–Chaplygin case in the first problem. In [36], the author proved that the equations of the motion for the current problem does not own more than the three mentioned cases.The third problem studies the problem of the motion of a rigid body in an incompressible ideal fluid, infinitely extending and at rest at infinity. The simple connected body is either described by the traditional Kirchoff equations [37] or by their Hamiltonian [38] form, while the body bounded by a multiconnected surface is described either by Lamb equations [39] or by its equivalent Hamiltonian form (see, e.g., [34]). The utilization of the equations of Kirchhoff and Lamb to describe this problem lacks to demonstrate the link between this problem and the other problems of rigid body dynamics. The link between both problems is proved by Yehia who introduced the equations of motion for a rigid body in a liquid by removing the translation motion that appears as cyclic variables (see, [40]), and the reduced problem is described by V=r0⋅γ+1/2γJ⋅γ and μ=k−2γK¯, where JandK¯ are the constant 3×3 matrices. The integrable cases for the third problem have been introduced in [34, 40].To dodge the ambiguity, we summarize those problems in Table1. Obviously, each problem is a generalization of the previous one by inserting some of the additional parameters, which represented terms having certain physical interpretations.Table 1
Different problems in a rigid body dynamics.
No.ProblemPotential functionVVector functionμ1Heavy rigid bodyV=r0⋅γμ=02Heavy gyrostatV=r0⋅γμ=k3Rigid body in a liquidV=r0⋅γ+1/2γJ⋅γμ=k−2γK¯According to the methodology used, this work deals only with two-dimensional mechanical systems, as it is outlined down in section two. It is obvious that this problem has three degrees of freedom in which one of them can be ignored due to the presence of a cyclic variable, precession angle, by utilizing the Routh procedure. Thence, the current problem can be characterized by Routhian (see, [41]).(7)R=12γ˙21−γ32+C1−γ32φ.2A−A−Cγ32+fCγ3+Al31−γ32AA−A−Cγ32φ.−1AV+f−l3γ322A−A−Cγ32.One can do the transformation,(8)dt=C1−γ32A−A−Cγ32dτ,to Routhian (7), and we get(9)R0=A2φ´2+A−A−Cγ32C1−γ322γ´32+fCγ3+Al31−γ32A−A−Cγ32φ´−C1−γ32A−A−Cγ32V+f−l3γ322A−A−Cγ32,where τ is the fictitious time, and dash refers to the derivative with respect to τ.


## 2. Basic Equations

A method for constructing the two-dimensional integrable mechanical systems in which the additional integral is a polynomial in velocities has been presented by Yehia in [42], and it has been developed in [43]. This method has been successfully applied to construct new integrable problems (not necessarily plane) whose complementary integral is a polynomial in velocities up to degree four (e.g., [4–6, 44–49]). This method is restrictively employed for two mechanical systems. There are a wide class of beforementioned systems such as the n-dimensional mechanical systems admitting n−2 cyclic variables and the particle motion on a smooth surface under the influence of distinct types of forces. A further example is a present problem which describes the rotation of a rigid body about a fixed point under the effect of potential and gyroscopic forces possessing a common axis of symmetry, so the motion has a cyclic variable, and this enables us to apply Routh procedure to lessen the degrees of freedom from three to two [32, 33].The two-dimensional mechanical systems are described by Lagrangian equation.(10)L=12b11q1.2+2b12q1.q2.+b22q2.2+b1q1.+b2q2.−V,where the functions bij,bi, and V rely on the generalized coordinates q1andq2, and dots refer to the derivatives with respect to the time t. Birkhoff theorem [50] guarantees the existence of a certain canonical transformation which is applied to turn Lagrangian (10) into(11)L=Λ2ξ˙2+η˙2+l1ξ˙+l2η˙−V,where Λ,l1,l2,andV are the functions in the two variables xandy. The usefulness of this step is to diminish the number of functions from six to four. It is obvious that the Lagrangian (10) has a Jacobi integral in the form(12)I1=Λ2ξ˙2+η˙2+V=h,where h is an arbitrary constant. According to Liouville theorem for the equivalent Hamiltonian system, system (11) is completely integrable if it has another first integral independent on the Jacobi integral (12).Executing the time transformation (see Appendix A for more details about time transformation),(13)dt=Λdτ,to Lagrangian (11), we get(14)L0=12ξ′2+η′2+l1ξ′+l2η′−U,where U=Λh−V, and ′ refers to the derivative with respect to τ. The Lagrangian equations corresponding to Lagrangian (14) are(15)ξ″+Ωη′=∂U∂ξ,η″+Ωξ′=∂U∂η,where Ω=∂l1/∂η−∂l2/∂ξ. This system has a Jacobi integral in the form(16)I1=12ξ′2+η′2+U=0.Now, we are going to find an additional first integral that is independent on the Jacobi integral (16). Based on [42], the complementary integral that is assumed to be quartic in velocities can be expressed as(17)I2=ξ′4+P3ξ′3+Q3ξ′2η′+P2ξ′2+Q2ξ′η′+P1ξ′+Q1η′+R,where the functions Pj,Qj,andR depend on the two variables ξ and η. Calculating the derivative of (17) with respect to τ and using the Jacobi integral (16) to remove all the even powers of η′ as in [42], we get the following nonlinear system of partial differential equations:(18)M.DξX+N.DηX=B,where X=P1P2P3Q1Q2Q3RUΩT is the vector of the unknown functions. DξX and DηX are partial derivatives according to the variables ξ and η of the vector X. M and N are the matrices and given as follows:(19)M=10000003P3010000002Q3001000000001000000000100000000100000010000012P101000001P101000001Q20,(20)N=000−102U0Q300001000000000−1000000001000000000−100000000100000002U01Q200002U001Q10000000100.The right-side vectorB is given by B=−2ΩQ22ΩP2−3ΩQ3−4U3ΩP304Ω−ΩQ1+4ΩUQ32ΩQ2UΩP1T.System (18) composed of nine nonlinear partial differential equations with nine unknown functions is not easy to solve exactly. Notice that the solution of this system determines a two-dimensional integrable system with an additional quartic integral in the velocities which is valid on a zero level of Jacobi integral.The sixth equation in (18), which is ∂P3/∂ξ+∂Q3/∂η=4Ω, and the definition of Ω, allow us to write Lagrangian (14) in the form(21)L0=12ξ′2+η′2+14P3ξ′−Q3η′−U.


## 3. Applications to Rigid Body Dynamics

In the present section, we investigate the construction of new integrable systems with a quartic integral in the dynamics of a rigid body movement. Seeing that the metric corresponding to Lagrangian (9) is ds2=dφ2+gγ3dγ32, it is more suitable to use the variable p instead of η through the relation(22)η=∫dpfp,where(23)fp=p−p1p−p2p−p3p+p1+p2+p3,where p1,p2,andp3 are the arbitrary parameters. We consider a certain class of problems in a rigid body dynamics in which the gyroscopic forces are determined by(24)Ωξ,p=a1Ω1pcosξ+a2Ω0p,and the potential forces are characterized by(25)Uξ,p=up+a3vpsinξ−a1a28f0pcosξ+a12mpcos2ξ,where a1,a2,anda3 are the arbitrary constants. The motivation for the choice of the two functions (24) and (25) is that they represent a large class of problems in the dynamics of a rigid body. Certain clarifications should be made for two particular cases.(1)
Time-reversible case: this case is characterized by the absence of gyroscopic forces, i.e.,Ω=0. This happens if a1=0anda2=0, The potential function (25) reduces to U=up+a3vpsinξ which is a Kowalevski-type potential. Furthermore, if we change ξ⟶2ξ, the potential function takes the form U=up+a3vpsin2ξ which is a Chaplygin-type potential. The previous studies concerning those types of potentials lead to several generalizations for integrable problems in the rigid body dynamics in which the additional integral is a quartic polynomial in the velocities (e.g., [51–53]). This type of problem is referred in literatures as time-reversible systems.(2)
Time-irreversible case: this case involves a gyroscopic forces acting on the motion besides the potential forces. We can split it into two subcases.(a)
When settinga1=0, the gyroscopic forces is characterized by Ω=a2Ω0p, and the potential function (25) becomes U=up+a3vpsinξ. The potential function is a type of Kowalevski-gyrostat type potential or Chaplygin-gyrostat type potential (if ξ⟶2ξ). This problem has been studied in several works such as [51, 52]. These studies lead to a generalization of a Kowalevski case and Chaplygin case by adding a constant gyrostatic moment.(b)
Whena1a2≠0, the full structure of Ω and U is considered in [49], but the authors solved the basic equations for a special cases leading to the Kowalevski case, and they introduced two integrable problems generalize Kowalevski case and Sokolov case.Inserting the expressions (22), (24), and (25) into the equations (18)–(21), we have(26)L=12η′2+p′2fp+a1J1pcosξ+a2J0pξ′+up+a3vpsinξ−a1a28f0pvpcosξ+a12mpcos2ξ,(27)I=ξ′4+P3ξ′3+Q¯3ξ′2p′+P2ξ′2+Q¯2ξ′p′+P1ξ′+Q¯1p′+R=ε0,where Q¯j=Qj/fp. In what follows, we inscribe Q instead of Q¯ for simplicity. Taking into account the transformation (22) and inserting the two expressions (24) and (25) into the basic equation (18), we get the following nonlinear system of partial differential equations in the following form:(28)Q3dfdp+2f∂Q3∂p−2∂P3∂ξ=0,(29)∂P3∂p+∂Q3∂ξ−4a2Ω0−4a1Ω1cosξ=0,(30)8a3v+6a1fQ3Ω1cosξ−16a12msin2ξ−2f∂Q2∂p+2∂P2∂ξ+a1a2f0vsinξ−Q2dfdp+6a2fQ3Ω0=0,(31)∂Q2∂ξ+∂P2∂p−3P3a1Ω1cosξ+a2Ω0=0,(32)8a12Q3fdmdp+2fm∂Q3∂p+Q3mdfdpcos2ξ−a1a2f0vdf0dpQ3−24a3vP3+2a1a2ff0v∂Q3∂p+a1a2ff0Q3dvdp−16a1fQ2Ω1+a1a2Q3vfdf0dpcosξ+3a1a2vf0Q3+8a3vQ3dfdp+16a3fv∂Q3∂p+8a3Q3fdvdpsinξ+8Q3f×dudp−48a12P3msin2ξ−8f∂Q1∂p−4Q1dfdp+8∂P1∂ξ+8Q3udfdp+16fu∂Q3∂p+16fa2Q2Ω0=0,(33)a1a2f0vQ3sinξ+8a3vQ3−a1Ω1P2cosξ+4∂Q1∂ξ+4∂P1∂p−8a2P2Ω0−16a12mQ3sin2ξ=0,(34)∂R∂ξ=2a33fQ3mΩ1−a22a12f0fvQ3Ω0−a1fQ1Ω1+4a1Q3uΩ1f+a1a28Q2vfdf0dp+a1a24ff0v∂Q2∂p−2a3vP2+a1a28Q2ff0dvdp+a1a28f0vQ2dfdpcosξ+−a12mQ2dfdp−a2a124ff0vQ3Ω1−2a12mf∂Q2∂p+4a2a12fQ3m×Ω0+2a13fQ3mΩ1cos3ξ−a12fQ2dmdpcos2ξ+−2a3fv∂Q2∂p−a1a24f0vP2×Q2+4a2a3fQ3Ω0−a3fQ2dvdpsinξ+4a12P2m+2a1a3fvQ3Ω1sin2ξ−a2fQ1Ω0−Q2fdudp−Q2udfdp−a3vdfdp−Q2udfdp−a12a24ff0Q3Ω1+4a2fQ3uΩ0−2fu∂Q2∂p,(35)∂R∂p=2a12mQ2sin2ξ−a1a28Q2f0vsinξ+a1P1Ω1cosξ−a1a28Q2f0vsin3ξ+a2P1Ω0a1216fm∂Q1∂p+a2ff0vQ2Ω1−16a2fmQ2Ω0+8Q1mdfdp+8Q1fdmdpcos2ξ−8a13fQ2mΩ1cos3ξ+−a1a2f0vQ1dfdp+2a1a22ff0Q2Ω0−a1a2Q1ff0dvdp−16a1fuQ2Ω1−8a13fmQ2Ω1−a1a2vfQ1df0dp+8a13vP1−2a1a2f0fv∂Q1∂pcosξ+16a3fv∂Q1∂p+8a3vQ1dfdp−16a2a3fQ2vΩ0+a1a2P1f0v+8a3fQ1dvdpsinξ−8a12a1mP1+a3vfQ2Ω1sin2ξ+8Q1udfdp+a12a2fQ2f0v−16a2fQ2uΩ0+16fu∂Q1∂p+8Q1fdudp=0,where(36)Ω0=dJ0dp,Ω1=dJ1dp.It is obvious that systems (28)–(35) are a nonlinear system of partial differential equations, and so in general, their solution is somewhat difficult. For simplicity, we turn these equations to the system of ordinary differential equations by setting the integral coefficients in a more suitable form. After some trials, the integral coefficients can be expressed as follows:(37)P3ξ,p=a1f0p+a1f1pcosξ,Q3ξ,p=8a1sinξ,P2ξ,p=P0p+16a3psinξ+a1a2Gpcosξ+a12Tpcos2ξ,Q2ξ,p=a3q1pcosξ+a1a2q2psinξ+a12q3psin2ξ,P1ξ,p=f3pcosξ+a12a2f4pcos2ξ+a13f5pcos3ξ+a1a3f6psin2ξ+a2a3f7psinξ+f8p,Q1ξ,p=a2a3f9pcosξ+a1a3f10pcos2ξ+1+a1f11psinξ+a2a12f12psin2ξ+a13f13psin3ξ.Inserting the expressions of the integral’s coefficients (37) into the equations (28)–(35) and equating the coefficients of trigonometric functions to zero, we obtain the following system of ordinary differential equations:(38)a1f1+4dfdp=0,(39)a2fdf0dp−4Ω0=0,(40)a14Ω1−df1dp−8=0,(41)a3q1dfdp+2fdq1dp−32p−8v=0,(42)a1a2q2dfdp+2fdq2dp+2G−f0v−48fΩ0=0,(43)a12q3dfdp+2fdq3dp+4T+16m−24fΩ1=0,(44)a1a2dGdp−3f0Ω1−3f1Ω0−q2=0,(45)a122dTdp−3f1Ω1+4q3=0,(46)a3q1−16=0,(47)dP0dp−3a22f0Ω0−32a12f1Ω1=0,(48)a2a3f9dfdp+2fdf9dp−2f7−6f0v−4fq1Ω0=0,(49)a1a3f10dfdp+2fdf10dp+8ddpvp−4f6−3f1v−2q1fΩ1=0,(50)f3+a1fdf11dp+12f11dfdp−8fdudp−8udfdp+a134mdfdp+3mf1+a1a2238f0v−2fq2Ω0=0,(51)a2a128f12dfdp+16fdf12dp+8ddpf0vf+32f4+96f0m−32fq3Ω0−32fq3Ω0−16q2fΩ1=0,(52)a132fdf13dp+f13dfdp+6f5−8ddpmf−2q3fΩ1+6mf1=0,(53)a1a3f10dfdp+2fdf10dp−8ddpvf−2q1fΩ1−3f1v=0,(54)a12P0Ω1−f11+2a1a22GΩ0+a1316m+TΩ1−df3dp=0,(55)a12a22TΩ0−2f12+f0v+GΩ1−df4dp=0,(56)a13TΩ1−3f13−16m−df5dp=0,(57)a2a3df7dp−f9−32pΩ0=0,(58)a1a3df6dp−2f10−16pΩ1+8v=0,(59)a12a2GΩ1−f0v+2a2P0Ω0−df8dp=0,(60)∂R∂ξ=ℱ0p+ℱ1pcosξ+ℱ2pcos2ξ+ℱ3pcos3ξ+ℱ4psinξ+ℱ5psin2ξ+ℱ6psin3ξ,(61)∂R∂p=ℱ7pcosξ+ℱ8pcos2ξ+ℱ9pcos3ξ−a14q3m−Ω1f52cos4ξ+a3a22f7Ω0+a12f6Ω12+mq1−12vq3sinξ−a1a2a316vq1f0+8q2−8f7Ω1−16f6Ω0sin2ξ+a3a122f6Ω1+2mq1−vq3sin3ξ+a12f3Ω1−a322q1v+a2f8Ω0+a14q3m−a12a2216q2f0v,(62)a14a2f0f5v−8fq2mΩ1−vff13df0dp−f0f13vdfdp−16fq3mΩ0−16f4m+fq3Ω1f0v+vf0f5−2ff0vdf13dp+8ff12dmdp+8f12mdfdp−f13f0fdvdp+16fmdf12dp=0,(63)a3a13vf13dfdp+ff13dvdp−ff10dmdp−2fmdf10dp−2f6m−f10mdfdp−vf5−fq3Ω1v+2fvdf13dp+fq1Ω1m=0,(64)a1−16a1mf3+a1216f13fdudp+16fmdf11dp+8f11fdmdp+8f11mdfdp−16fq3uΩ1+16uf13dfdp+32fudf13dp+a12a22−ff0f12dvdp+fq2f0Ω1v+f4vf0−2ff0vdf12dp+2fq3f0Ω0v−vff12df0dp−16fq2Ω0m−f12f0vdfdp+a328ff10dvdp+8f10vdfdp+16fvdf10dp+8f6v−8fq1Ω1v−8a14q3fΩ1m=0,(65)a12a2a3f10f0vdfdp+16fq1mΩ0−16mf7+f0f6v+f0f10fdvdp−16fq3vΩ0+8ff12dvdp+8vf12dfdp−8fq2vΩ1−8f4v+ff10vdf10dp−16fmdf9dp−8ff9dmdp+16fvdf12dp−8f9mdfdp−fq1Ω1f0v+2ff0vdf10dp=0,(66)−2a12f8m+a1a216f0f3v+a2a1416−ff13vdvdp−ff13dvdp−2ff0vdf13dp−f0vf13dfdp−f0f5v+2fq3Ω1f0v+a2a12ff12dudp−116ff11vdfdp+2fudf12dp−fq2uΩ1+f12udfdp−116ff11f0dvdp−116ff11vdf0dp−2fq3uΩ0−18ff0vdf11dp+a2a322f7v+2fvdf9dp−2fq1Ω0v+f9fdvdp+f9vdfdp+a12a238fq2Ω0f0v=0,(67)a38f3v+a132fudf10dp−16fq1Ω1u−8f11vdfdp−8ff11dvdp+16f10udfdp+16ff10dudp−16vfdf11dp+a138f13vdfdp+16vfdf13dp+32fmdf10dp+16mf10dfdp+8f5v+16ff10dmdp−16fq1mΩ1+8f13fdvdp+a1a222ff0q1Ω0v−ff0f9dvdp−f0f7v−f9fvdf0dp−2f0fvdf9dp+16fq2Ω0v=0,(68)a316f8v+a216f9udfdp+16f9fdudp−32fq1Ω0u+32fudf9dp+a2a12−8fq2vΩ1−3f10vfdf0dp+8f9fdmdp+8f4v−6ff0vdf10dp−16fq1Ω0m−3f0f10fdvdp+16fvdf12dp−3f10f0vdfdp+3fq1Ω1f0v+8f12fdvdp+f6f0v+8f9mdfdp−16fq3Ω0v+8f12vdfdp+16fmdf9dp−16f7m=0,(69)a116f11udfdp+16f11fdudp+32fudf11dp−16a1f3m+2a2f0f8v+a12−8f11mdfdp−16fq3uΩ1−16fmdf11dp−8f11fdmdp+a148ff13dfdp+8f13mdfdp+16fmdf13dp−32a22fq2Ω0u+a32−8fq1Ω1v+8f6v+16fvdf10dp+8f10vdfdp+8ff10dvdp+a12a22−f0f12fdvdp+fq2Ω1f0v+16fq2Ω0m−f0vf12dfdp−f4f0v−vff12df0dp−2ff0vdf12dp+2fq3Ω0f0v=0,(70)a3a13f10mdfdp+8f10fdmdp−8fq1Ω1m+16fmdf10dp−8fq3Ω1v−16f6m+a1a22f7f0v−2ff0vdf9dp−f0ff9dvdp+2fq1f0vΩ0−16fq2Ω0v−f9fvdf0dp−f9f0vdfdp+32fudf10dp+8f11vdfdp+16fvdf11dp+16f10×udfdp−16fq1Ω1u+8f11fdvdp+16ff10dudp+8vf3=0,whereℱip,i=0,1,…,9 is given in Appendix B. Systems (38)–(68) consist of thirty-two nonlinear ordinary differential equations in nineteen unknown functions. The solution to this system is somewhat intricate, and we cannot generally solve it as in the reversible case for arbitrary values of the parameters. But we solve it for certain values of the parameters leading to rigid body dynamics.


### 3.1. Two New Integrable Problems

The metric corresponding to the Lagrangian (9) that describes the metric of a rigid body matches with the Lagrangian (26) if we set(71)ξ=2φ−φ0,p1=p2=p3=1,p=1+γ341−γ32.We are going to solve equation (70) taking into account condition (71). As a result of the complexity of those equations, we utilize the Maple program. We consider separately the two cases that are a1a2≠0 and a2=0. Let us illustrate the causes of choosing those cases. In our works for constructing integrable systems with quartic integrals, it seems that some potentials are appropriate, with the attendance of a constant gyrostatic moment, while others are not. For more elucidation, we admit the Chaplygin case describing the motion of a rigid body in an incompressible ideal fluid and its generalization as an example:(72)V1=aγ12−γ22+2bγ1γ2+λ2γ32,μ=0,0,k,(73)V2=aγ12−γ22+2bγ1γ2+λ2γ32+ρ1γ34−1γ36,μ=0,0,0,where a,b,λ,andρ are the arbitrary parameters, while k is a constant characterizing the gyrostatic moment. These two cases were previously introduced in [53, 54], respectively. It is worth noticing that the singular term ρ1/γ34−1/γ36 is not compatible with the existence of the gyrostatic moment as it is outlined in (72), but in the absence of a gyrostatic moment, this term appears. As we see later, this situation appears, and it is followed by the discovery of two new cases. The new cases will be directly announced without any details due to most of the calculations cannot be displayed in a suitable size.


#### 3.1.1. First New Integrable Case

We first consider the case in whicha2a1≠0; taking into account the condition (71) and using the Maple program, we obtain a new integrable problem in a rigid body dynamic after tedious manipulations which are not writable in a suitable size in the generalized coordinates θ and φ. Therefore, we introduce it in the traditional Euler-Poisson variables for the sake of simplicity and to make the comparison clear with previous results.Theorem 1.
Let the principal inertia matrix for a rigid body satisfy the conditionA=B=2C, and the potential and gyroscopic forces characterized by Vandμ, respectively, are given by(74)V=κcγ22−γ12−2dγ1γ2+λγ32+K2cγ1γ2+dγ22−γ12k−νγ12+γ32γ12−kνγ32γ12−ν2γ32γ32+2γ222γ14+K2d22γ34+4γ12γ22−c2γ32γ12+γ22+2γ12γ222+cdγ1γ2γ12−γ22,(75)μ=2Kγ3cγ2−dγ1−2νγ31+γ22γ13,2Kγ3cγ1+dγ2+2νγ2γ3γ12,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,or, equivalently,(76)l=0,0,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,where k,κ,c,d,λ,K,andν are the arbitrary parameters. Then, the Euler-Poisson equation 2) with (74) and (76) is integrable on a zero level of the area integral:(77)I1=2pγ1+2qγ2+r+k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12γ3.The additional integral admits the form(78)I2=p2−q2+cκγ32+cK2γ32cγ12−γ22+2dγ1γ2−Kd2ν+2k+γ323k−3ν−r−λγ12−γ22γ322+2pq+dκγ32+dK2γ32γ12−γ22+2dγ1γ2+cKγ323k−3ν−r−2λγ1γ2γ322+k−νr−k+ν−K2cγ1γ2+dγ22−γ122p2+q2+2λ1+1γ32+2γ32γ32−1c2+d2K2−2dk−νK+cκ−4γ3k−ν2Kk−νcγ1+2dγ2+κdγ1−2cγ2q+γ2p2ck−νK+dκ−8k−νKc2k−ν×γ34−γ32−14K−κγ1γ2γ32+c2k−νdK2γ1γ2γ32+Kdκ2γ32+dκγ32γ12−12−2λγ1γ2−κk−ν2γ32−k−νd2K216γ12γ32+18+dk−ν2+λγ32−2dλ14−γ12K+K2νγ34γ12c2+d2νγ12γ322γ12+γ32−2γ12+γ22−2Kγ12+γ222cγ1γ2+dγ22−γ12+22γ32−1r+2ν24γ14−γ32γ14p2−q2−2kνγ32r2γ12+2νγ32p2+q2γ122Kcγ2−dγ1γ1−r−4ν4γ324γ18γ16−1+γ22γ44+2γ22+12γ12+γ22γ32γ22+2+νγ16r×−2ν2γ12γ12+3γ22+γ32γ12+γ22−2kνγ32γ12γ12−γ22+2−γ142λγ32−1+4kKγ322cγ1γ2+dγ12+γ22−2ν3γ32γ16kγ125γ12+4γ32−6+γ322−γ32+Kγ22cγ1γ12−γ22−dγ23γ12+γ32+dγ12γ12+2γ223γ12+γ32+2νλγ14νγ22−γ12+γ12k+K2cdγ1γ2+dγ22−γ12−2νγ32κγ14νγ2γ24γ14+γ32−2dγ12γ12+γ32+γ122γ3×cγ2−dγ1q−2γ3cγ1+dγ2p+k4dγ1γ2−4γ32+4γ22+2ν2−5Kdkγ36−42cq−12dpγ1+γ2cp+32dq×Kγ35+2k5−8γ12d+5cγ1γ2K−k2−2q2γ32−45cγ13q+γ2cp+4dqγ12+γ1dp2−2cq−γ2cp+3dq2×Kγ33+γ32k16γ12−8γ1−5d+2γ1γ26γ12−5cK+γ122p2−6q2−4k2+4γ1γ2pq+k2+8K−cq−dpγ13γ2cp−dqγ12+32cqγ1+γ22cp+2dqγ3γ12+4qγ122γ1γ2p+q+4νγ12−12kK2γ32c2+2d2+γ342ckγ1×cγ1+dγ2+d2kK2+3k2+p2−2q2d−3pqcK+2k3qγ2−pγ1d+c2qγ1+γ2qKγ33+2kγ14−cdγ1γ2−c2γ12γ22+γ32K2+γ322k2γ12−2−12p2+32q2d+2cpq−2k2γ12γ22K+k32−4Kkdqγ2+cγ12γ2p−γ1q+12γ2p+3γ1qγ3.Theorem1 characterizes a new integrable problem in a rigid body dynamics. The present case generalizes a special version of the case introduced by Yehia and Elmandouh in 2016 by adding a new parameter ν [48]. Also, it includes the case announced by Elmandouh in 2015 (K=0) [3]. Moreover, it generalizes the case presented by Yehia and Elmandouh in 2013 by inserting two arbitrary constants (k=ν=0) [6]. It generalizes the integrable case which was introduced by Goriatchev in 1916 by adding four arbitrary parameters ν,k,K,andρ [55]. It also contains five arbitrary parameters, ν,k,K,λ,andρ, more than the case introduced by Chaplygin in 1903 [56]. To avoid confusion, we summed up the comparisons between this case and the related earlier cases in Table 2.Regrettably, the physical interpretation for the whole system with the full set of all associated parameters is unknown. Disregarding the singular terms in both potential and vector functions, the problem describes physically the motion of an electrically charged heavy rigid body about a fixed point under the action of potential and gyroscopic forces admitting a common axis of symmetry [6].Table 2
Comparison the first integrable case with previous results.
No.AuthorsConditions of the parametersReferences1Yehia and Elmandouh [48]ν=0[48]2Elmandouh [3]K=0[3]3Yehia and Elmandouh [6]k=ν=0[6]4Goriatchev [55]ν=K=ρ=k=0[55]5Chaplygin [56]ν=K=ρ=k=λ=0[56]


### 3.2. Second New Integrable Problem

In this subsection, we solve the basic equations in the case in whicha2=0 taking into account the conditions.Theorem 2.
Let the inertia matrix of a rigid body beI=diag2C,2C,C, and assume this body be in motion under the action of a combination of following potential and gyroscopic forces which are characterized by Vandμ, respectively,(79)V=κ2dγ1γ2+cγ12−γ22+λγ32+ρ1γ34−1γ36−Kνγ32γ122cγ1γ2+dγ22−γ12+ν2γ32γ32−22γ12+K22dcγ1γ2γ12−γ22+d22γ32+4γ12γ22−c2γ32γ12+γ22+2γ12γ22,(80)μ=2γ3Kcγ2−dγ1−ν1+γ22g13,2γ3Kcγ1+dγ2+νγ22γ12,K2cγ1γ2+dγ22−γ12+ν2−γ32γ12,or, equivalently,(81)l=0,0,K2cγ1γ2+dγ22−γ12+ν2−γ32γ12,where c,d,κ,K,λ,ρ,andν are the arbitrary parameters. Then, the Euler-Poisson equation (2) with the two expressions (79) and (80) is completely integrable on a zero level of the cyclic integral:(82)I1=2pγ1+qγ2+r+K2cγ1γ2+dγ22−γ12+ν2−γ32γ12γ3.Its additional integral takes the form(83)I2=p2−q2+γ32Kdr+cκ+cK2γ32cγ12−γ22+2dγ1γ2−λγ12−γ22γ322+2pq+dκγ32−γ32Kcr−dK2cγ12−γ22+2dγ1γ2−2λγ12γ22γ32−dK2cγ12−γ22+2dγ1γ2−2λγ12γ22γ322−ν2γ12+γ222γ18γ342γ14ρ+λγ34−ν2γ32+4νpγ1+qγ2γ12×ν2γ33γ12+γ12γ14+γ32−1γ34λ+ργ35−2K2c2+d22γ32−1γ33+2K2cγ1γ2+dγ22−γ12×−νK2γ36c2+d2γ12−ρrγ34+νλγ34−ργ12γ32+2ν3γ34γ12+γ22γ16+2ρκγ34+ν2γ34γ142dγ1γ2+cγ12−γ22+2p2+q2ργ32−1γ36+2νγ3pγ1+qγ2γ12+4νKγ32cγ2−dγ1γ1+ρρ−2λγ341−γ322γ312+2K2γ34c21−2γ32−4γ1γ2cγ2−dγ1cγ1+dγ2+ν2γ32γ14K2c2+d2γ325γ32−8γ32+2+4Kγ3γ12+γ22×4cq−dpγ1+γ22cp+3dq−2γ124Kγ3cγ1+dγ2q+dγ1−cγ2p+2q2−p2−γ12+γ22×p2+3q2−4pqγ1γ2−2νγ32γ14Kγ326cpq+d5q2−p2+2cp+dqκγ1γ3+2Kqcp+dq−4κγ2γ3cq−dp.Theorem2 introduces a new integrable problem in the dynamics of a rigid body. Furthermore, it represents an extension for the related previous results. It adds to the case that was discovered by Yehia and Elmandouh in 2013, one arbitrary parameter ν [6]. It modifies the case introduced by Goriatchev in 1916 by inserting three parameters K,ν,andρ [55]. It generalizes the Chaplygin case that was found in 1903 by entering four arbitrary parameters ν,ρ,λ,andK [56]. The comparisons with previous results are summarized in Table 3.Table 3
Comparison the second new integrable case with previous results.
No.AuthorsConditions of the parametersReferences1Yehia and Elmandouh [6]ν=0[6]2Goriatchev [55]K=ν=ρ=0[55]3Chaplygin [56]K=ν=λ=ρ=0[56]


## 3.1. Two New Integrable Problems

The metric corresponding to the Lagrangian (9) that describes the metric of a rigid body matches with the Lagrangian (26) if we set(71)ξ=2φ−φ0,p1=p2=p3=1,p=1+γ341−γ32.We are going to solve equation (70) taking into account condition (71). As a result of the complexity of those equations, we utilize the Maple program. We consider separately the two cases that are a1a2≠0 and a2=0. Let us illustrate the causes of choosing those cases. In our works for constructing integrable systems with quartic integrals, it seems that some potentials are appropriate, with the attendance of a constant gyrostatic moment, while others are not. For more elucidation, we admit the Chaplygin case describing the motion of a rigid body in an incompressible ideal fluid and its generalization as an example:(72)V1=aγ12−γ22+2bγ1γ2+λ2γ32,μ=0,0,k,(73)V2=aγ12−γ22+2bγ1γ2+λ2γ32+ρ1γ34−1γ36,μ=0,0,0,where a,b,λ,andρ are the arbitrary parameters, while k is a constant characterizing the gyrostatic moment. These two cases were previously introduced in [53, 54], respectively. It is worth noticing that the singular term ρ1/γ34−1/γ36 is not compatible with the existence of the gyrostatic moment as it is outlined in (72), but in the absence of a gyrostatic moment, this term appears. As we see later, this situation appears, and it is followed by the discovery of two new cases. The new cases will be directly announced without any details due to most of the calculations cannot be displayed in a suitable size.


### 3.1.1. First New Integrable Case

We first consider the case in whicha2a1≠0; taking into account the condition (71) and using the Maple program, we obtain a new integrable problem in a rigid body dynamic after tedious manipulations which are not writable in a suitable size in the generalized coordinates θ and φ. Therefore, we introduce it in the traditional Euler-Poisson variables for the sake of simplicity and to make the comparison clear with previous results.Theorem 1.
Let the principal inertia matrix for a rigid body satisfy the conditionA=B=2C, and the potential and gyroscopic forces characterized by Vandμ, respectively, are given by(74)V=κcγ22−γ12−2dγ1γ2+λγ32+K2cγ1γ2+dγ22−γ12k−νγ12+γ32γ12−kνγ32γ12−ν2γ32γ32+2γ222γ14+K2d22γ34+4γ12γ22−c2γ32γ12+γ22+2γ12γ222+cdγ1γ2γ12−γ22,(75)μ=2Kγ3cγ2−dγ1−2νγ31+γ22γ13,2Kγ3cγ1+dγ2+2νγ2γ3γ12,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,or, equivalently,(76)l=0,0,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,where k,κ,c,d,λ,K,andν are the arbitrary parameters. Then, the Euler-Poisson equation 2) with (74) and (76) is integrable on a zero level of the area integral:(77)I1=2pγ1+2qγ2+r+k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12γ3.The additional integral admits the form(78)I2=p2−q2+cκγ32+cK2γ32cγ12−γ22+2dγ1γ2−Kd2ν+2k+γ323k−3ν−r−λγ12−γ22γ322+2pq+dκγ32+dK2γ32γ12−γ22+2dγ1γ2+cKγ323k−3ν−r−2λγ1γ2γ322+k−νr−k+ν−K2cγ1γ2+dγ22−γ122p2+q2+2λ1+1γ32+2γ32γ32−1c2+d2K2−2dk−νK+cκ−4γ3k−ν2Kk−νcγ1+2dγ2+κdγ1−2cγ2q+γ2p2ck−νK+dκ−8k−νKc2k−ν×γ34−γ32−14K−κγ1γ2γ32+c2k−νdK2γ1γ2γ32+Kdκ2γ32+dκγ32γ12−12−2λγ1γ2−κk−ν2γ32−k−νd2K216γ12γ32+18+dk−ν2+λγ32−2dλ14−γ12K+K2νγ34γ12c2+d2νγ12γ322γ12+γ32−2γ12+γ22−2Kγ12+γ222cγ1γ2+dγ22−γ12+22γ32−1r+2ν24γ14−γ32γ14p2−q2−2kνγ32r2γ12+2νγ32p2+q2γ122Kcγ2−dγ1γ1−r−4ν4γ324γ18γ16−1+γ22γ44+2γ22+12γ12+γ22γ32γ22+2+νγ16r×−2ν2γ12γ12+3γ22+γ32γ12+γ22−2kνγ32γ12γ12−γ22+2−γ142λγ32−1+4kKγ322cγ1γ2+dγ12+γ22−2ν3γ32γ16kγ125γ12+4γ32−6+γ322−γ32+Kγ22cγ1γ12−γ22−dγ23γ12+γ32+dγ12γ12+2γ223γ12+γ32+2νλγ14νγ22−γ12+γ12k+K2cdγ1γ2+dγ22−γ12−2νγ32κγ14νγ2γ24γ14+γ32−2dγ12γ12+γ32+γ122γ3×cγ2−dγ1q−2γ3cγ1+dγ2p+k4dγ1γ2−4γ32+4γ22+2ν2−5Kdkγ36−42cq−12dpγ1+γ2cp+32dq×Kγ35+2k5−8γ12d+5cγ1γ2K−k2−2q2γ32−45cγ13q+γ2cp+4dqγ12+γ1dp2−2cq−γ2cp+3dq2×Kγ33+γ32k16γ12−8γ1−5d+2γ1γ26γ12−5cK+γ122p2−6q2−4k2+4γ1γ2pq+k2+8K−cq−dpγ13γ2cp−dqγ12+32cqγ1+γ22cp+2dqγ3γ12+4qγ122γ1γ2p+q+4νγ12−12kK2γ32c2+2d2+γ342ckγ1×cγ1+dγ2+d2kK2+3k2+p2−2q2d−3pqcK+2k3qγ2−pγ1d+c2qγ1+γ2qKγ33+2kγ14−cdγ1γ2−c2γ12γ22+γ32K2+γ322k2γ12−2−12p2+32q2d+2cpq−2k2γ12γ22K+k32−4Kkdqγ2+cγ12γ2p−γ1q+12γ2p+3γ1qγ3.Theorem1 characterizes a new integrable problem in a rigid body dynamics. The present case generalizes a special version of the case introduced by Yehia and Elmandouh in 2016 by adding a new parameter ν [48]. Also, it includes the case announced by Elmandouh in 2015 (K=0) [3]. Moreover, it generalizes the case presented by Yehia and Elmandouh in 2013 by inserting two arbitrary constants (k=ν=0) [6]. It generalizes the integrable case which was introduced by Goriatchev in 1916 by adding four arbitrary parameters ν,k,K,andρ [55]. It also contains five arbitrary parameters, ν,k,K,λ,andρ, more than the case introduced by Chaplygin in 1903 [56]. To avoid confusion, we summed up the comparisons between this case and the related earlier cases in Table 2.Regrettably, the physical interpretation for the whole system with the full set of all associated parameters is unknown. Disregarding the singular terms in both potential and vector functions, the problem describes physically the motion of an electrically charged heavy rigid body about a fixed point under the action of potential and gyroscopic forces admitting a common axis of symmetry [6].Table 2
Comparison the first integrable case with previous results.
No.AuthorsConditions of the parametersReferences1Yehia and Elmandouh [48]ν=0[48]2Elmandouh [3]K=0[3]3Yehia and Elmandouh [6]k=ν=0[6]4Goriatchev [55]ν=K=ρ=k=0[55]5Chaplygin [56]ν=K=ρ=k=λ=0[56]


## 3.1.1. First New Integrable Case

We first consider the case in whicha2a1≠0; taking into account the condition (71) and using the Maple program, we obtain a new integrable problem in a rigid body dynamic after tedious manipulations which are not writable in a suitable size in the generalized coordinates θ and φ. Therefore, we introduce it in the traditional Euler-Poisson variables for the sake of simplicity and to make the comparison clear with previous results.Theorem 1.
Let the principal inertia matrix for a rigid body satisfy the conditionA=B=2C, and the potential and gyroscopic forces characterized by Vandμ, respectively, are given by(74)V=κcγ22−γ12−2dγ1γ2+λγ32+K2cγ1γ2+dγ22−γ12k−νγ12+γ32γ12−kνγ32γ12−ν2γ32γ32+2γ222γ14+K2d22γ34+4γ12γ22−c2γ32γ12+γ22+2γ12γ222+cdγ1γ2γ12−γ22,(75)μ=2Kγ3cγ2−dγ1−2νγ31+γ22γ13,2Kγ3cγ1+dγ2+2νγ2γ3γ12,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,or, equivalently,(76)l=0,0,k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12,where k,κ,c,d,λ,K,andν are the arbitrary parameters. Then, the Euler-Poisson equation 2) with (74) and (76) is integrable on a zero level of the area integral:(77)I1=2pγ1+2qγ2+r+k+K2cγ1γ2+dγ22−γ12+ν1+γ22γ12γ3.The additional integral admits the form(78)I2=p2−q2+cκγ32+cK2γ32cγ12−γ22+2dγ1γ2−Kd2ν+2k+γ323k−3ν−r−λγ12−γ22γ322+2pq+dκγ32+dK2γ32γ12−γ22+2dγ1γ2+cKγ323k−3ν−r−2λγ1γ2γ322+k−νr−k+ν−K2cγ1γ2+dγ22−γ122p2+q2+2λ1+1γ32+2γ32γ32−1c2+d2K2−2dk−νK+cκ−4γ3k−ν2Kk−νcγ1+2dγ2+κdγ1−2cγ2q+γ2p2ck−νK+dκ−8k−νKc2k−ν×γ34−γ32−14K−κγ1γ2γ32+c2k−νdK2γ1γ2γ32+Kdκ2γ32+dκγ32γ12−12−2λγ1γ2−κk−ν2γ32−k−νd2K216γ12γ32+18+dk−ν2+λγ32−2dλ14−γ12K+K2νγ34γ12c2+d2νγ12γ322γ12+γ32−2γ12+γ22−2Kγ12+γ222cγ1γ2+dγ22−γ12+22γ32−1r+2ν24γ14−γ32γ14p2−q2−2kνγ32r2γ12+2νγ32p2+q2γ122Kcγ2−dγ1γ1−r−4ν4γ324γ18γ16−1+γ22γ44+2γ22+12γ12+γ22γ32γ22+2+νγ16r×−2ν2γ12γ12+3γ22+γ32γ12+γ22−2kνγ32γ12γ12−γ22+2−γ142λγ32−1+4kKγ322cγ1γ2+dγ12+γ22−2ν3γ32γ16kγ125γ12+4γ32−6+γ322−γ32+Kγ22cγ1γ12−γ22−dγ23γ12+γ32+dγ12γ12+2γ223γ12+γ32+2νλγ14νγ22−γ12+γ12k+K2cdγ1γ2+dγ22−γ12−2νγ32κγ14νγ2γ24γ14+γ32−2dγ12γ12+γ32+γ122γ3×cγ2−dγ1q−2γ3cγ1+dγ2p+k4dγ1γ2−4γ32+4γ22+2ν2−5Kdkγ36−42cq−12dpγ1+γ2cp+32dq×Kγ35+2k5−8γ12d+5cγ1γ2K−k2−2q2γ32−45cγ13q+γ2cp+4dqγ12+γ1dp2−2cq−γ2cp+3dq2×Kγ33+γ32k16γ12−8γ1−5d+2γ1γ26γ12−5cK+γ122p2−6q2−4k2+4γ1γ2pq+k2+8K−cq−dpγ13γ2cp−dqγ12+32cqγ1+γ22cp+2dqγ3γ12+4qγ122γ1γ2p+q+4νγ12−12kK2γ32c2+2d2+γ342ckγ1×cγ1+dγ2+d2kK2+3k2+p2−2q2d−3pqcK+2k3qγ2−pγ1d+c2qγ1+γ2qKγ33+2kγ14−cdγ1γ2−c2γ12γ22+γ32K2+γ322k2γ12−2−12p2+32q2d+2cpq−2k2γ12γ22K+k32−4Kkdqγ2+cγ12γ2p−γ1q+12γ2p+3γ1qγ3.Theorem1 characterizes a new integrable problem in a rigid body dynamics. The present case generalizes a special version of the case introduced by Yehia and Elmandouh in 2016 by adding a new parameter ν [48]. Also, it includes the case announced by Elmandouh in 2015 (K=0) [3]. Moreover, it generalizes the case presented by Yehia and Elmandouh in 2013 by inserting two arbitrary constants (k=ν=0) [6]. It generalizes the integrable case which was introduced by Goriatchev in 1916 by adding four arbitrary parameters ν,k,K,andρ [55]. It also contains five arbitrary parameters, ν,k,K,λ,andρ, more than the case introduced by Chaplygin in 1903 [56]. To avoid confusion, we summed up the comparisons between this case and the related earlier cases in Table 2.Regrettably, the physical interpretation for the whole system with the full set of all associated parameters is unknown. Disregarding the singular terms in both potential and vector functions, the problem describes physically the motion of an electrically charged heavy rigid body about a fixed point under the action of potential and gyroscopic forces admitting a common axis of symmetry [6].Table 2
Comparison the first integrable case with previous results.
No.AuthorsConditions of the parametersReferences1Yehia and Elmandouh [48]ν=0[48]2Elmandouh [3]K=0[3]3Yehia and Elmandouh [6]k=ν=0[6]4Goriatchev [55]ν=K=ρ=k=0[55]5Chaplygin [56]ν=K=ρ=k=λ=0[56]


## 3.2. Second New Integrable Problem

In this subsection, we solve the basic equations in the case in whicha2=0 taking into account the conditions.Theorem 2.
Let the inertia matrix of a rigid body beI=diag2C,2C,C, and assume this body be in motion under the action of a combination of following potential and gyroscopic forces which are characterized by Vandμ, respectively,(79)V=κ2dγ1γ2+cγ12−γ22+λγ32+ρ1γ34−1γ36−Kνγ32γ122cγ1γ2+dγ22−γ12+ν2γ32γ32−22γ12+K22dcγ1γ2γ12−γ22+d22γ32+4γ12γ22−c2γ32γ12+γ22+2γ12γ22,(80)μ=2γ3Kcγ2−dγ1−ν1+γ22g13,2γ3Kcγ1+dγ2+νγ22γ12,K2cγ1γ2+dγ22−γ12+ν2−γ32γ12,or, equivalently,(81)l=0,0,K2cγ1γ2+dγ22−γ12+ν2−γ32γ12,where c,d,κ,K,λ,ρ,andν are the arbitrary parameters. Then, the Euler-Poisson equation (2) with the two expressions (79) and (80) is completely integrable on a zero level of the cyclic integral:(82)I1=2pγ1+qγ2+r+K2cγ1γ2+dγ22−γ12+ν2−γ32γ12γ3.Its additional integral takes the form(83)I2=p2−q2+γ32Kdr+cκ+cK2γ32cγ12−γ22+2dγ1γ2−λγ12−γ22γ322+2pq+dκγ32−γ32Kcr−dK2cγ12−γ22+2dγ1γ2−2λγ12γ22γ32−dK2cγ12−γ22+2dγ1γ2−2λγ12γ22γ322−ν2γ12+γ222γ18γ342γ14ρ+λγ34−ν2γ32+4νpγ1+qγ2γ12×ν2γ33γ12+γ12γ14+γ32−1γ34λ+ργ35−2K2c2+d22γ32−1γ33+2K2cγ1γ2+dγ22−γ12×−νK2γ36c2+d2γ12−ρrγ34+νλγ34−ργ12γ32+2ν3γ34γ12+γ22γ16+2ρκγ34+ν2γ34γ142dγ1γ2+cγ12−γ22+2p2+q2ργ32−1γ36+2νγ3pγ1+qγ2γ12+4νKγ32cγ2−dγ1γ1+ρρ−2λγ341−γ322γ312+2K2γ34c21−2γ32−4γ1γ2cγ2−dγ1cγ1+dγ2+ν2γ32γ14K2c2+d2γ325γ32−8γ32+2+4Kγ3γ12+γ22×4cq−dpγ1+γ22cp+3dq−2γ124Kγ3cγ1+dγ2q+dγ1−cγ2p+2q2−p2−γ12+γ22×p2+3q2−4pqγ1γ2−2νγ32γ14Kγ326cpq+d5q2−p2+2cp+dqκγ1γ3+2Kqcp+dq−4κγ2γ3cq−dp.Theorem2 introduces a new integrable problem in the dynamics of a rigid body. Furthermore, it represents an extension for the related previous results. It adds to the case that was discovered by Yehia and Elmandouh in 2013, one arbitrary parameter ν [6]. It modifies the case introduced by Goriatchev in 1916 by inserting three parameters K,ν,andρ [55]. It generalizes the Chaplygin case that was found in 1903 by entering four arbitrary parameters ν,ρ,λ,andK [56]. The comparisons with previous results are summarized in Table 3.Table 3
Comparison the second new integrable case with previous results.
No.AuthorsConditions of the parametersReferences1Yehia and Elmandouh [6]ν=0[6]2Goriatchev [55]K=ν=ρ=0[55]3Chaplygin [56]K=ν=λ=ρ=0[56]


## 4. Conclusion

In the current work, we had interest in studying the integrability issue of the motion of a rigid body about a fixed point under the action of potential and gyroscopic forces having a common axis of symmetry. We have assumed this problem has a complementary quartic integral in the velocities. We have applied the method by Yehia. The basic equations have been formulated and introduced in a general setting. But as it is outlined in the literature, in the case of the existence of gyroscopic forces, the basic equations have not been solved in a general setting, but it is usually solved for certain values of the parameters leading to the metric of a rigid body dynamics which are valuable and significant problems. We have announced two new integrable problems generalizing the Chaplygin case in a rigid body and its subsequent works by different authors such as Goriatchev, Yehia and Elmandouh, and Elmandouh. The comparison of new results with previous ones is summarized and collected in Tables2 and 3.



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