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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
*L*
_{4} inferior mass (in comparison of the other primaries) is placed at the equilibrium point of the R3BP.

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 [^{th} century but nobody established the proper mathematical model of the R4BP. Recently Ceccaroni and Biggs [

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 [

Let

The fourth body at

Let the line joining

ments of

Let

where

The total gravitational force acting on

Let

The Equation of motion of the infinitesimal mass

Since the synodic frame are revolving with constant angular velocity

In cartesian form, the equations of motion of the infinitesimal mass

Also the linear velocity of the infinitesimal mass

If

If mass of the infinitesimal body is supposed to be unity, then the kinetic energy of the infinitesimal mass is given by

Let

Combination of Equations ((9) and (10)) yields

The gravitational potential of the body of mass

then, total gravitational potential at

The Hamiltonian of the infinitesimal body of unit mass is given by

Let

then

The coordinates of

Clearly

where

is the reduced Hamiltonian corresponding to canonically conjugate variables

In our Hamiltonian

with

where

Clearly,

Also,

Thus the Hamiltonian

Let us introduce pseudo time

Thus the regularized Hamilton-canonical equations of motion of the infinitesimal body corresponding to the Hamiltonian

where the regularized Hamiltonian

Let us write

For generating solutions, we shall choose

where

Since

Putting

It may be noted that this differential equation is exactly the same as in Giacaglia [

where

Let us introduce a new quantity

Combination of Equations (29) and (30) yields

where

where

From Equation (33), we conclude that for general solution; we need only two arbitrary constants as

Let us introduce the parameters

where

It may be noted that for

We introduce a parameter

From Equations (33), (35) and (36), we get

Again from Equation (25)

Thus the equations of motion associated with

where

Now from

Also

and

Thus from the above relations, we have

From Equation (32), we get

From Equation (30),

where

where

If we take

From the second equation of system (41), we get the argument as

Since

hence for the problem generated by Hamiltonian

The variables

where

When

where

The original synodic cartesian co-ordinates in a non-uniformly rotating system are obtained from Equations (18) and (20), when

The sidereal cartesian co-ordinates are obtained by considering the transformations

where

where

In terms of canonical variables introduced, the complete Hamiltonian may be written as

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

where

The periods of

Here we shall follow the method given by Chaudhary [

Integrating these equations with respect to

These are the generating solutions of two-body problems. The generating solution will be periodic with the period

when

Following Poincare [

where

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

Expanding

Rejecting the second order term

and

The Equation (55) gives

Equation (45) gives

By solving the Equations (54)-(57), we can find the values of

where

Now,

From Equation (43),

then

From the Equation (26)

where

Thus

Taking only zero order terms i.e., for

where

Now from equations of system (52)

and from Equation (63)

where

Here

But

Now choosing suitably

and

Thus,

Now,

As

Thus,

Using Equation (65), we get

Thus the conditions for the existence of periodic orbits given by Duboshin [

In this previous section, we have shown that Duboshin’s condition [

together with the Jacobi Integral

To study the motion of the infinitesimal body by PSS, it is necessary to know its position

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

Levi-Civita [

dies. Bhatnagar [

for sufficiently small

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.

from the origin and return to it after

where

Therefore, the condition of Equation (69) became,

But,

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.

In section 1 of this paper, historical background has been sketched with original and previous contributions. In section 2, the equations of motion of the infinitesimal mass moving under the gravitational field of the three primaries situated at the vertices of an equilateral triangle taken by Ceccaroni and Biggs [

Hassan, M.R., Hassan, Md. A., Singh, P., Kumar, V. and Thapa, R.R. (2017) Periodic Orbits of the First Kind in the Autonomous Four-body Problem with the Case of Collision. International Journal of Astronomy and Astrophysics, 7, 91-111. https://doi.org/10.4236/ijaa.2017.72008

Bary-Centre: It is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit.

Synodic Co-ordinate System: The co-ordinate system, in which the xy-plane rotates in the positive direction with an angular velocity equal to that of the common velocity of one primary with respect to the other keeping the origin fixed, is called synodic co-ordinate system.

Reduced Mass: Mass ratio of the smaller primary to the total mass of the primaries or the non-dimensional mass of the smaller primary is known as reduced mass of the smaller primary.

Regularization: The process of elimination of the singularity from the force function is known as regularization.

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