Periodic orbits in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body


 In this paper, we have studied the equations of motion for the problem, which are regularised in the neighbourhood of one of the finite masses and the existence of periodic orbits in a three-dimensional coordinate system when μ = 0. Finally, it establishes the canonical set (l, L, g, G, h, H) and forms the basic general perturbation theory for the problem.


Introduction
In this paper, we wish to study the three-dimensional generalisation of the problem studied by Bhatnagar (12)(13)(14) for the circular case. Since the Hamilton-Jacobi equation for generating a solution takes an unmanageable form for any solution, we have assumed that the third coordinate ( 3 of the infinitesimal mass is of the 0(µ). It will be interesting to observe that various equations and results worked out by Bhatnagar can be deduced from our results. In Section 2 we have determined the canonical form of the equations of motion, and in Section 3 these equations are regularised by the generalised Levi-Civita's transformation for three dimensions. Eqs (20)- (22) establish the canonical set (l, L, g, G, h, H) and Eq (32) form the basis of the general perturbation theory for the problem under consideration. During the last few years, many mathematician and astronomers have studied different types of periodic orbits in the restricted problem. Some of them are Giacaglia (7), Mayer and Schmidt (17), Markellos (19), Hadjidemetriou (10,11), Bhatnagar and Taqvi (15), Gomez and Noguera (8), Kadrnoska and Hadrava (9), Peridios et al. (21), Ahmad (1), Elipe and Lara (4), Mathlouthi (23), Scuflaire (22), Caranicolas (20), Poddar et al. (5,6), Abouelmagd and Guirao (2) and Abouelmagd et al. (3). In this work, we have presented an analytical study of the existence of periodic orbits for µ = 0 in the restricted problem of three bodies in a three-dimensional coordinate system when the smaller primary is a triaxial rigid body.

Equations of Motion
The equations of motion in the canonical form of an infinitesimal mass under the gravitational field of two finite and unequal masses and moving in circles are given bẏ where the Hamiltonian function H and consequently the energy integral is given by and C is a function of µ = C(µ) = C 0 + µ(C 1 ).

Regularisation of the Solution
We regularise the solution by Levi-Civita's (18) transformation generated by Such that where π i is the momenta associated with the new coordinate ξ i . We have from Eqs (3) and (4) From these equations, we have The Hamiltonian Eq. (2) given in terms of these new variables is where The equations of motion (1) will be transformed into Where K is the new Hamiltonian given by K can be put in the form K o + µK 1 , where where c 0 = c 0 n and The form given to k 0 ensures that the orbits which are analytically continued from the two-body orbits will belong to the K = 0 manifold. These are the solution to the regularised equation of the restricted problem. Here we have assumed that k 0 is negative (5). Thus, the corresponding two-body problem will admit bounded orbits as a solution in rotating coordinates. We can easily show that ||<1.

Generating Solution
To write the Hamilton-Jacobi equation corresponding to the Hamilton k 0 , we take For generating a solution, we shall choose k 0 for our Hamiltonian function. Since τ is not involved in k explicitly, the Hamilton-Jacobi equation corresponding to k 0 may be written as where α = 1 − ε. We take ξ 3 of the order of µ, then we have Putting ξ 1 = ξ cosφ , ξ 2 = sinφ Equation (9) may be written as Whose solution of Eq. (10) may be written as where G is an arbitrary parameter and taking ξ 2 = z we have We suppose that G + c 0 < 0 then the equation f (z) = 0 has two positive roots z 1 and z 2 and is positive between them. Also The solution of Eq. (12) is Let us introduce the parameter a, e, l using the relation where 0≤e≤1. It may be noted that Z = Z 1 when l = 0. The equations of motion to K 0 are Here denotes differentiation with respect to τ Now 1 4 (ξ 1 π 1 + ξ 2 π 2 ) = ξ ξ Therefore dz dτ = H2 + 2 G + c 0 · f (z) Integrating, we haveˆZ where z = z 1 at τ = τ 0 . Introducing L by relation Now taking L and G for the arbitrary constants instead of α and G, the solution may be given by the relation where f = and for e = 1, we have G = 0, f = 0. Eqs (20)-(22) establish the canonical set (l, L, g, G, h,H) since k 0 = α − 1 .

It follows that
and therefore, for the problem generated by this Hamiltonian (regularised two-body problems in Rotating coordinates), we have where l 0 , g 0 , h 0 are the values of l, g, h respectively at τ = 0. The angle θ is obtained from the equation The variables ξ i , π i (i = 1, 2, 3) can be expressed by the canonical elements we have and ∂ w ∂ = 2G Therefore, where ( is given by the first of Eq. (24). When e = 1 (G = 0), where φ is given by the second of the Eq. (24) The original synodic Cartesian coordinates are obtained from equations (µ = 0), i.e.
The sidereal Cartesian coordinates are given by where t 0 is a constant. It is seen that l is the eccentric anomaly of the problem of two-body.
In terms of the canonical variables, the complete Hamiltonian may be written as and ξ 1 , ξ 2 , ξ 3 , π 1 , π 2 , π 3 are given by Eq. (25). The equations of motion for the complete Hamiltonian are The period of ξ i π i is 4π n l and 4π n g , and therefore, in the case of commensurability the period of solution is 4π p n l or 4πq n g

Conclusion
We have shown that the equations of motion for the problem are regularised by the generalised Levi-Civita's transformation for three dimensions in the neighbourhood of one of the finite masses and the existence of periodic orbits for µ = 0 in the three-dimensional coordinate systems.
Equations where p and q are integers. The period of ξ i π i is 4π n l and 4π n g , so that in case of commensurability the period of solution is 4π p n l or 4πq n g .
T h i s p a g e i s i n t e n t i o n a l l y l e f t b l a n k