Periodic Orbits of the First Kind in the Autonomous Four-body Problem with the Case of Collision

In this manuscript, the existence of periodic orbits of collision of the first kind has been discussed on the model of Autonomous Four-body Problem by the method of analytic continuation given by Giacaglia [1] and Bhatnagar [2] [3]. For the existence of periodic orbits, Duboshin’s criterion [4] has been satisfied and it has been confirmed by analyzing the Poincare surfaces of section (PSS) [5]. Also it has been shown that the case of collision given by Levi-Civita [6] [7] is conserved by the method analytic continuation. In all sections of this manuscript, equilateral triangular configuration given by Ceccaroni and Biggs [8] has been considered. In this model, third primary of inferior mass (in comparison of the other primaries) is placed at the equilibrium point 4 L of the R3BP.


Introduction
We know that the four most popular methods of proving the existence of periodic orbits are: (i) the method of analytic continuation, (ii) the process of equating Fourier coefficients of equal frequencies, (iii) the application of fixed point theorem given by Poincare, (iv) the method of power series.
Giacaglia [1] used the method of analytic continuation to examine the existence of periodic orbits of collision in the Restricted Three-body Problem (R3BP).Bhatnagar [2] generalized the problem in elliptic case.The problem of Giacaglia [1] was further extended by Bhatnagar [3] in the R4BP by taking the primaries at the vertices of an equilateral triangle.With different perturbations like oblateness, triaxiality, photogravitation, Pointing-Robertson drag effects of the primaries, the existence of periodic orbits of collision in the R3BP and in the R4BP, have been studied by different authors in two and three-dimensional co-ordinate system during the period of last three decades of the 20 th century but nobody established the proper mathematical model of the R4BP.Recently Ceccaroni and Biggs [8] has studied the autonomous coplanar CR4BP by taking the third primary of comparatively inferior mass at the triangular equilibrium point 4 L of R3BP and with an extension to low-thrust propulsion for application to the future science mission.
In present paper, we have proposed to study the existence of periodic orbits of first kind in the Autonomous Four-body Problem by the method of analytic continuation.By using Poincare surfaces of section (PSS), the conditions for the existence of periodic orbits given by Duboshin [4] have been confirmed.For collision case, we have applied the criterion given by Levi-Civitas [6] [7] and it is satisfied by our model.

Equations of Motion
Let 1 2 3 , , P P P be the three massive bodies of masses ( ) m m m ≥  and the fourth body of mass m be at P .These bodies are moving in the same plane under some restrictions as follows: The fourth body at P of mass m is assumed to be of infinitesimal mass not influencing the motion of 1 2 3 , , P P P but motions of ( ) 4 P is being influenced by the motions of 1 2 3 , , P P P .Further, we have assumed that the mass 3 m at 3 P is taken small enough, so that it can't influence the motion of the dominating primaries 1 P and 2 P and it is placed at any one of the triangular libration points (Lagrangian Points) of the classical restricted three body problem.Since the third primary can't influence the motions of 1 P and 2 P , so the centre of rota- tion of the system remains at the barycentre of two main primaries 1 P and 2 P .
Also, it is supposed, all the primaries are moving in the same plane in circular orbits around the bary-centre of massive primaries 1 P and 2 P with the same angular velocity ω and the fourth body 4 P is moving under the gravitational field and plane of motion of three primaries 1 2 3 , , P P P then to check the nature of motion of infinitesimal mass 4 m .
Let the line joining 1 P and 2 P be taken as the x-axis and their mass centre (bary-centre) O, as the origin.Let the line through O and perpendicular to 1 2 P P lying in the plane of motion of the primaries be taken as the y-axis.Let the positions of masses ( ) P m by the prima- ries respectively, then ( ) where γ is the gravitational constant.
The total gravitational force acting on ( ) P m by the three primaries is given by Let n be the magnitude of angular velocity ω and k be the unit vector normal to the plane of motion of the primaries, then nk = ω .The Equation of motion of the infinitesimal mass 4 m in synodic frame is ( ) Since the synodic frame are revolving with constant angular velocity ω about the bary-centre, hence 0 t ∂ = ∂ ω and thus Equation (4) reduces to ( ) In cartesian form, the equations of motion of the infinitesimal mass 4 m in the gravitational field of three primaries, are given by Also the linear velocity of the infinitesimal mass m on its orbit; is given by If 1 2 , v v are two components of v , then from Equation ( 7), If mass of the infinitesimal body is supposed to be unity, then the kinetic energy of the infinitesimal mass is given by Let 1 2 , p p be the momenta corresponding to the co-ordinates , x y respec- tively, then ( ) ( ) Combination of Equations (( 9) and (10)) yields ( ) The gravitational potential of the body of mass i m at any point of 4 P out- side it, is given by ( ) then, total gravitational potential at 4 P due to three primaries is given by .
The Hamiltonian of the infinitesimal body of unit mass is given by ( ) Let µ be the reduced mass of the second primary and ε be the reduced mass of the third primary, then from the definition of reduced mass, we have The coordinates of 1 2 3 , , P P P are given by ( ) ( ) ( ) Clearly 1 2 2 3 3 1 1 P P P P P P = = = , which implies that 1 2 3 P P P forms an equila- teral triangle of sides of unit length.We know that ε is very small in compari- son of masses of the other two primaries, so we can choose ε as the order of µ i.e., ) is the reduced Hamiltonian corresponding to canonically conjugate variables , , p p .

Regularization at the Singularity (
) In our Hamiltonian H given in Equation (15), there are three singularities .To examine the existence of periodic orbits of collision with the first primary, we have to eliminate the singularity 1 0 r = .For this, let us define an extended generating function S given by ( ) ( ) where i Q is the momenta associated with new co-ordinate ( ) Clearly, r q q r q q q q r q q q q q q Thus the Hamiltonian H given in Equation ( 15), can be written in terms of new variables , Let us introduce pseudo time τ by the differential equation ( ) Thus the regularized Hamilton-canonical equations of motion of the infinitesimal body corresponding to the Hamiltonian 0 K = , are given by ( ) where the regularized Hamiltonian K is given by ( ) Let us write ( )

Generating Solution (i.e., Solutions When
For generating solutions, we shall choose 0 K for our Hamiltonian function, so in order to solve the Hamilton-Jacobi equation associated with 0 K , let us write , 1, 2 and 1 0, where α is an arbitrary constant.Since t is not involved explicitly in 0 K : hence by using Equation (27) in Equation ( 25), the Hamilton-Jacobi equation may be written as It may be noted that this differential equation is exactly the same as in Giacaglia [1] and Bhatnagar [2] [3] and therefore the solution of Equation ( 29) can be written by the method of separation of variables, as where G is an arbitrary constant.
Let us introduce a new quantity z by Combination of Equations ( 29) and (30) yields [ ] ( ) where ( ) ( ) where 1 z is the smaller root of the roots of the equation From Equation (33), we conclude that for general solution; we need only two arbitrary constants as α and G .Therefore the solution of Equation (30) may be regarded as a general solution.
Let us introduce the parameters , , a e l by the relations where a is the semi-major axis, e is the eccentricity and l is the latus-rec- tum of the elliptic orbit of the infinitesimal body.
It may be noted that for 1 , 0 z z l = = and 2 z is the other root of the equa- tion We introduce a parameter L by the relation ( ) ( ) From Equations (33), ( 35) and (36), we get i.e., sin .f z a e l = (37) Again from Equation (25) Thus the equations of motion associated with 0 K are given as where ( ) Thus from the above relations, we have From Equation (32), we get From Equation (30), ( )

∫
If we take L and G as arbitrary constants, the solutions may be written as From the second equation of system (41), we get the argument as Since ( ) hence for the problem generated by Hamiltonian 0 K (regularized two-body problem in rotating co-ordinate system), we have ( ) ( ) where .
The variables ( ) q Q i = can now be expressed in terms of the canonical elements for ( ) where ϕ is given by the first equation of system (42).
When 1 e = and 0 G = , then where ϕ is given by the second equation of system (42).
The original synodic cartesian co-ordinates in a non-uniformly rotating system are obtained from Equations ( 18) and (20), when 0 µ = , as The sidereal cartesian co-ordinates are obtained by considering the transformations where t is given by ( ) ( ) where 0 t is a constant.In terms of canonical variables introduced, the complete Hamiltonian may be written as ( ) , where R can be obtained from Equation (26) after changing into canonical variables.
The equations of motion for the complete Hamiltonian are ( ) ( ) Equation ( 49) forms the basis of a general perturbation theory for the present problem.The solution described by Equations ((44) and (45)) and is periodic if l and g have commensurable frequencies, i.e., if where p and q are integers.The periods of , i i q Q are 4π l n and 4π g n respectively, so that in case of commensurability, the period of the solution is 4π l p n or 4π g q n .

Existence of Periodic Orbits When 0 µ ≠
Here we shall follow the method given by Chaudhary [9] to prove the existence of periodic orbits.Let Integrating these equations with respect to τ , we get ( ) ( ) ( ) These are the generating solutions of two-body problems.The generating solution will be periodic with the period 0 when i κ are integers, so that ( ) Following Poincare [5], the general solution in the neighbourhood of the generating solution, may be given as where ς is the new independent variable given by . 1 The necessary and sufficient conditions for the existence of periodic solution are Restricting our solution only up to the first order infinitesimals, the equations of motion may be written as where Equation (54) may be written as , 0 .
By solving the Equations ( 54)-(57), we can find the values of 1 2 2 , , β β γ , as analytic function of µ , reducing to zero with µ , if the conditions for periodic orbits given by Duboshin [4] are satisfied i.e., (iii) , , 0 for 0 , , where [ ] 1 K is the zero degree terms of 1 K given in Equation ( 2 , and 0. 2 From the Equation (26) ( ) Taking only zero order terms i.e., for where ( ) Now from equations of system (52) x a q x a q y y and from Equation ( [ ] Here [ ] ϕ ω ∂ = ∂ so from Equation (63), we have ) Thus the conditions for the existence of periodic orbits given by Duboshin [4] are satisfied i.e., in the region of motion of the infinitesimal body, periodic orbits exist.

Poincare Surfaces of Section (PSS)
In this previous section, we have shown that Duboshin's condition [4] for the existence of periodic orbits when 0 µ ≠ , are satisfied.So to justify the mathe- matical model given in Equations ( 58)-(60), we have applied the method of Poincare surfaces of section (PSS) to the reduced equations of motion together with the Jacobi Integral ( ) To study the motion of the infinitesimal body by PSS, it is necessary to know its position ( ) , x y and velocity ( ) , x y   which correspond to a point in four- dimensional phase space.By defining a plane 0 y = , in the resulting three-di- mensional space, the values of x and x  can be plotted.Every time the par- ticle has 0 y = , whenever the trajectory intersects the plane in a particular di- rection say 0 y >  .
The techniques of PSS suggest to determine the regular or chaotic nature of the trajectories.If there are smooth, well-defined island then the trajectory is likely to be regular and the islands correspond to oscillation around a periodic orbit.As the curves shrink down to a point, the points represent a periodic orbit as per Kolmogorov-Arnold-Moser (KAM) theory.Any fuzzy distribution of points in surfaces of section, implies that trajectory is chaotic.In Figure 2 in which atleast nine points are visible towards which the regular trajectories shrink, so we can say that the periodic orbits exist in the region of motion of infinitesimal mass.Other than the neighbourhood of these points, the quasi-periodic and chaotic regions are seen in the PSS.In Figure 4, in PSS for 3.17     for sufficiently small µ and 0 ε .
Further, he has proved that, in particular, such relation is uniform integral of the differential equation of motion along any collision orbit.He has also proved this integral is a power series in terms of the distance from the origin and the series is convergent through the radius of convergence is generally small.In section (5), we have shown that periodicity is conserved by analytic continuation.
Let us show that the condition of collision is also conserved by analytic continuation.
Figure 7 shows the geometrical configuration of collision orbits.In order to show the validity of that continuation, we shall consider orbits corresponding to the case when ( ) . When 1 e = , the orbits starts as an ejection from the origin and return to it after 4 T .Bhatnagar [2] [3] and Levi-Civita [6] [7] finds the condition for collision as tan so tan , q q q q ϕ ϕ q q q q q q q q q q q q q ϕ τ ϕ τ ( ) Thus from Equations ((70) and ( 71)) ( ) Here the Equation (71) corresponds to the Equation (68), so it is easy to say that the collision orbits exist.
Hamilton canonical equations of motion of the infinitesimal body 4 P are given by

N
= is only the case for which [ ] Poincare surfaces of section have been plotted in which atleast seven points are visible towards which the regular trajectories shrink, hence by KAM theory, periodic orbits exists.Again Figure3represents a Poincare surfaces of section for 3