Quartic Integral in Rigid Body-Gyrostat Dynamics

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.


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. e Liouville integrability concept is defined as the Hamiltonian system with n 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]. e 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]. e problem of integrability is split into two categories. e 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][4][5][6][7][8][9][10][11][12][13]). e second one deals with obtaining the necessary conditions of the integrability (see, e.g., [14][15][16][17][18][19][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][22][23]). So, it is a beneficial model for research from different points of view [24][25][26][27][28][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. e gyroscopic forces are specified by l � (0, 0, l 3 ), while the potential forces are characterized by V(γ). As it outlined in [30], this motion can be characterized by the Lagrangian: where ω � (p, q, r) is the angular velocity, and I � diag(A, B, C) is the inertia matrix of the body. e equations of motion corresponding Lagrangian (1) are [30][31][32] ω three classical integrals of motion. ey are as follows: (1) Area integral: where the arbitrary constant f denotes the value of area integral. is integral is sometimes named as a cyclic integral due to it correspondences the cyclic variable ψ, the angle of precession. (2) Jacobi integral: where the arbitrary constant h identifies the value of the Jacobi integral. (3) Geometric integral: 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). ence, 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 I 1 or valid on a single level of cyclic integral I 1 which is usually zero. e 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. ose works include three types of problems. e 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 by V � r 0 · c, and μ � 0, where r 0 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]). e 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. e second problem regards as a generalization to the first problem, and it is determined by V � r 0 · c 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. e 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. e third problem studies the problem of the motion of a rigid body in an incompressible ideal fluid, infinitely extending and at rest at infinity. e 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]). e 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.
e 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 � r 0 · c + (1/2)γJ · c and μ � k − 2cK, where J and K are the constant 3 × 3 matrices. e integrable cases for the third problem have been introduced in [34,40].
To dodge the ambiguity, we summarize those problems in Table 1. Obviously, each problem is a generalization of the previous one by inserting some of the additional parameters, which represented terms having certain physical interpretations.
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. ence, the current problem can be characterized by Routhian (see, [41]).
One can do the transformation, 2 Advances in Astronomy to Routhian (7), and we get where τ is the fictitious time, and dash refers to the derivative with respect to τ.

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]. is 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][5][6][44][45][46][47][48][49]). is method is restrictively employed for two mechanical systems. ere 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]. e two-dimensional mechanical systems are described by Lagrangian equation.
where the functions b ij , b i , and V rely on the generalized coordinates q 1 andq 2 , 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 where Λ, l 1 , l 2 , and V are the functions in the two variables xandy. e 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 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), to Lagrangian (11), we get where U � Λ(h − V), and ′ refers to the derivative with respect to τ. e Lagrangian equations corresponding to Lagrangian (14) are where Ω � zl 1 /zη − zl 2 /zξ. is system has a Jacobi integral in the form 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 where the functions P j , Q j , 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: where X � (P 1 P 2 P 3 Q 1 Q 2 Q 3 R U Ω) T is the vector of the unknown functions. D ξ X and D η X are partial derivatives according to the variables ξ and η of the vector X. M and N are the matrices and given as follows:   (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. e sixth equation in (18), which is zP 3 /zξ+ zQ 3 /zη � 4Ω, and the definition of Ω, allow us to write Lagrangian (14) in the form

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 ds 2 � dφ 2 + g(c 3 )dc 2 3 , it is more suitable to use the variable p instead of η through the relation where where p 1 , p 2 , andp 3 are the arbitrary parameters. We consider a certain class of problems in a rigid body dynamics in which the gyroscopic forces are determined by and the potential forces are characterized by where a 1 , a 2 , anda 3 are the arbitrary constants. e 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. is happens if a 1 � 0anda 2 � 0, e potential function (25) reduces to U � u(p) + a 3 v(p)sin ξ which is a Kowalevski-type potential. Furthermore, if we change ξ ⟶ 2ξ, the potential function takes the form U � u(p) + a 3 v(p)sin 2 ξ which is a Chaplygin-type potential. e 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][52][53]).
is 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.
is problem has been studied in several works such as [51,52].
ese studies lead to a generalization of a Kowalevski case and Chaplygin case by adding a constant gyrostatic moment. (b) When a 1 a 2 ≠ 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.

Two New Integrable Problems.
e metric corresponding to the Lagrangian (9) that describes the metric of a rigid body matches with the Lagrangian (26) if we set 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 a 1 a 2 ≠ 0 and a 2 � 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: Advances in Astronomy where a, b, λ, andρ are the arbitrary parameters, while k is a constant characterizing the gyrostatic moment. ese two cases were previously introduced in [53,54], respectively. It is worth noticing that the singular term ρ((1/c 4 3 ) − (1/c 6 3 )) 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.
e new cases will be directly announced without any details due to most of the calculations cannot be displayed in a suitable size.

First New Integrable Case.
We first consider the case in which a 2 a 1 ≠ 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. erefore, 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 condition A � B � 2C, and the potential and gyroscopic forces characterized by Vandμ, respectively, are given by or, equivalently,

Data Availability
No data were used to support this study.

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