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M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51 : BMHSS, p.44 (Online Publication: Dec., 2013) 

 
 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (online) 
Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

Periodic Solution of the restricted three body problem 

M.R. Hassan
1
,  R. R.Thapa

2*
 

1
Dept. of Mathematics, S.M. College, Bhagalpur, Under T.M.B.University, Bhagalpur  

2Dept. of Mathematics, P.G.Campus, Biratnagar, Morang, Nepal 

*Email: thaparajuram@yahoo.com    

Article history: Received 25 August, 2013; Accepted 5 October, 2013 

 

Abstract 

The effect of perturbation in centrifugal force on the periodic solution of the restricted three-body 

problem representing analytic continuation of Keplerian rectilinear periodic motion has been 

examined. However, we have taken the perturbation in the centrifugal force to be of the order of µ , 

the reduced mass of the smaller primary. We have calculated the first order perturbations also. 

 

Keywords: Elliptic restricted three body problem; Generating solution; Hamiltonian equations; 

Perturbation; Periodic solution; Variational equations. 

Introduction 

 

Considering the two body problem, Poincare [1] classified three kinds of periodic solutions of the 

restricted three body problem. Solutions of I and II kinds are related to the planar case of the three 

body problem. In the solution of I kind, the eccentricity reduces to zero together with small mass. In 

the solution of II kind, the eccentricity does not reduce to zero with the small mass. Poincare studied 

in details the I kind and proved its existence. For the solution of the II kind, he deduced conditions 

under which they may exist but regorious proof for the existence of the solution of this kind was not 

given by him. Poincare used the De-Launay elements in his study and so he excluded the case when 

one of the eccentricity of the generating solution is equal to unity. Kurcheeva [2] considered the last 

case. Under certain conditions she proved the existence of so called periodic solution of Poincare i.e. 

having equal periods for their perturbed as well as for the unperturbed motions. 
Ahmad [3] studied the elliptic restricted three body problem generalizing the work of Kurcheeva. 

Ahmad and Khan [4] generalized the work of Ahmad with the introduction of perturbation in the 

Coriolis force. 

In this paper we have studied the generalization of the restricted three body problem introducing 

perturbation in the centrifugal force. We have examined the existence of the periodic solution taking 

the perturbation to be of the order of µ , the reduced mass of the smaller primary. We have also 

calculated first order perturbation. 
 

Hamiltonian Equations of Motion 
 

Using non-dimensional variables and a synodic system of co-ordinates, the Hamiltonian equations of 

motion are 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.45 (Online Publication: Dec., 2013) 

 

 

dX H dy H
,

dt X dt Y

dX H dY H
,

dt x dt y

∂ ∂
= =

∂ ∂
∂ ∂

= − = −
∂ ∂

       (1) 

where 

 ( ) ( )2 2

1 2

1 1
H X Y yX xY

2 r r

−µ µ
= + + + − −      (2) 

 
( )

( )

22 2

1

22 2

2

r x y

r x 1 y

= −µ +

= −µ + +
        (3) 

 

In the above equations, the parameter µ  is the ratio of the mass of the smaller primary to the total 

mass of the primaries and 0 1/ 2.< µ ≤  

We consider the perturbation in the centrifugal force with the help of the parameter β , the 

unperturbed value of β  being unity. The corresponding Hamiltonian function takes the following 

Bhatnagar and Hallan form [5] 

 ( ) ( ) ( )( )2 2 2 2

1 2

1 1 1
H X Y Xy Yx 1 x y

2 2 r r

−µ µ
= + + + + −β + − −  

Here β  may be taken as 

  1 ', ' |β = +∈ ∈ <<  

where '∈ represents the perturbation in the centrifugal force. Upto first order, we have 

 ( ) ( ) ( )2 2 2 2

1 2

1 ' 1
H X Y Xy Yx x y

2 2 r r

∈ −µ µ
= + + − − + − −    (4) 

Limiting case µ=0 

  
For the elimination of singularity at the bigger primary we shall use Levi-Civita's [6] variables as used 

by Kurcheeva: 

 

( )

( )

2
x iy p iq

P iQ
X i Y

2(P iq)

+µ + = +

−
− +µ =

+

       (5) 

 ( )2 2dt
4 p q

ds
= +         (6) 

The regularized canonical equations of motion became 

 

dp dq
,

dS P dS Q

dP dQ
,

dS p dS q

∂Ω ∂Ω
= =

∂ ∂

∂Ω ∂Ω
= =

∂ ∂

        (7) 

 
where 

 
( ) ( ) ( ) ( )

( )

2 2 4 4

1 1 2

3 4 4

1 1

1
P Q 2r qP pQ 4 p q 4 4 4 r / r

2

2Cr 2 'r 4 ' p q

Ω = + + − + µ − − + µ − µ

+ − ∈ − µ∈ −
  (8) 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.46 (Online Publication: Dec., 2013) 

 

 ( ) ( )22 2 2 2 2 2 2

1 2r p q , r 1 2 p q p q= + = − + + +      (9) 

The solution of the system (7) being the solution of the system (8), we must have 

 0Ω =  

We suppose that '∈  is or the order of µ  i.e. 0' '∈ =∈ µ . If we put 0µ =  in the system (7) may be 

integrated. The solutions of the simplified system, which in sideral system of co-ordinates 
corresponds to collinear motion, has Krasinski form [7]. 

 
O O

O O

p sin , q cos

P sin , Q cos

= ρ θ = ρ θ

= ρ θ = ρ θɺ ɺ
       (10) 

where 

 

( )
( )

( )

0

0

3

0

2 / Ccos 2 CS

1
4 CS sin 4 CS 2

2n

d
n C meanmotion,

dS

ρ = −φ

 θ = + − φ
 

ρ
= = ρ =ɺ

 

The period of such a solution is 

 

( )

( )

0

0

K
if k m isan even number

2 C
s

K
if k m isan odd number

C

π +
= 

π +


     (11) 

The relation between t and s for 0µ =  is given by 

 ( ) ( )0 0

1
t 2 4 CS sin 4 C S 2

n
 = θ−ω = + − φ
 

    (12) 

 

Solutions of the Variational Equations of the Generating System 
 

The solutions of the variational equations of the system (7) for 0µ =  are well known. The solutions 

given by Krasinski are 

 

0 0
1 0 2 0 3 0 0 4 0 0

0

0 0
1 0 2 0 3 0 0 0 4 0 0

0

0 0
1 0 2 0 3 0 0 0 4 0 0 0 0

t q1 2
p q , p p , p p tq , p Q p S

n n2 C

t p1 2
q p , q q , q q t p , q P q S

n n2 C

t Q2 2
P Q , P P , P P t Q , p 2 C q p S p

n n n

 
δ = δ = δ = + δ = − + −  

 

 
δ = − δ = δ = − δ = + −  

 

 
δ = δ = δ = + δ = + + − 

 

ɺ ɺ

ɺ ɺ

ɺ ɺ

  (13) 

0 0
1 0 2 0 3 0 0 0 4 0 0 0 0

t P2 2
Q P , Q Q , Q Q t P , Q 2 C p Q S Q

n n n

 
δ = − δ = δ = − δ = − + + + 

 
ɺ ɺ  

Proof of  Isoperiodic Solution 

 

We see that our Hamiltonian function (8) possesses the same properties as that of Kurcheeva. 
Therefore, under the conditions: 

 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.47 (Online Publication: Dec., 2013) 

 

 q(o) p(o) o,p(S*) Q(S*) 0,S* S/ 4= = = = =      (14) 

and q(o) p(o) o,q(S*) P(S*) 0, S* S / 2= = = = =      (15) 

 

the general solutions xi (s), (i = 1, 2, 3, 4) are periodic with period S as in Kurcheeva  where by  xi (i = 

1, 2, 3, 4), p,q,P,Q are respectively denoted. 

The solutions (10) of the generating system satisfy the conditions (14) when k+m is odd and (15) 

when k+m is even under the following values of the parameters 0φ  and .ω  

 I. 0 0, / 2φ = ω=± π  

 II. 0 0 / 2, L, L 0,1,2,.......φ = =± π ω = π =  

We observe that the value of 0Ω is the same as in Kurcheeva. 

Hence one can follow the work of Kurcheeva, step by step, to examine the existence of isoperiodic 

solutions. The isoperiodic solution exist for 0 0, / 2,k even, m odd.φ = ω = ±π − −  

For arbitrary k and m, there exist periodic solutions reducing to the generating one for 0.µ =  The 

period of these solutions is an analytic function of µ , S=S*+4S(µ ), S* being the period of the 

generating solution. 

 

Perturbation of the First Order 
 

In the proceeding paragraph the existence of the periodic orbits analytic relative to µ  with the period 

S=S*+ Sδ , where S* is the period of the generating solution has been pointed out. Let us pass on to 

such time t the period in which does not depend on µ : 

 
S* S*

S
S* S S*

τ = =
+δ +α

 

 Where 
k

S* .
4 C

π
=  Then the equations of motion are written as 

 ( ) ( )i
i 1 4

dx S*
X x ,..........x ,C , i 1,........., 4

d S*

+α
= =

τ
    (16) 

The solution of the system (16) may be sought in the form of series in integral positive powers of µ . 

 ( )(0) (i) i

k k k

i 1

x x x , k 1,....4
∞

=

= + µ =∑       (17) 

For C and α , we have 

 
i i

0 i i

i 1 i 1

C C C ,
∞ ∞

= =

= + µ α = α µ∑ ∑       (18) 

As it is proved by Kurcheeva series (17) and (18) representing solutions of the system (16) converge 

for sufficiently small value of µ . 

Let us substitute (17), (18) in (16) for the determination of the functions ( )( j)

ix τ . We have a system 

of linear differential equations. 

 

(i) 4
(i) (0) (i)k

kj j k i k i k

j 1

dx
p x x h C F

d =

= + + φ +
τ ∑  

where by hi we denote i / S*α  

 
 

 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.48 (Online Publication: Dec., 2013) 

 

 

(0)
(0) (0) (0)k

kj k k 1 4 0

j

k
k

0

x
P , x x (x ,.......x ,C ,0)

x

x

C

∂
= =

∂

∂
φ =

∂

 

( )(i) (i) (0) (i 1)

K K i 4 0 i 1 1 i 1
F F x ,......., x ,C ,......,C ,h ,......, h−

− −=  

In particular for i=1, we have 

 

(i)

1

(i)

2

(i) 3 '1 1
3 0 13

2 2

F 0

F 0

8qr (1 r )8q
F 16q 12 r p.

r r

=

=

+
= − + + + ∈

 

(i) 3 ' 21 1
4 0 13

2 2

8pr (1 r )8p
F 16p 12 r q.

r r

−
= + − + ∈      (20) 

  1 2 3 0 4 00, 4p , 4qφ = φ = φ = − φ = −     (21) 

(0) (0) (0)

1 0 0 2 0 0 3 0 0 0 4 0 0 0x P 2rq , x Q 2rp , x 2rQ 4p C , x 2rP 4q C= + = − = − = − −  (22) 

All the linearly independent solutions of the system 
4

i
ik k

k 1

dy
p .y

d =

=
τ ∑         (23) 

are obtained from the expression (13). 

Let us seek particular solution of the system (19) with the help of the method of variation of arbitrary 

constant. 

Let  
4

(i)

j k ik

k 1

x L .y
=

=∑        (24) 

where { }jk
y  is a fundamental system of solution of the homogenous system   (23) 

Differentiating (24), we obtain 

 
4

(i) (i)k
jk j 1 j i j

k 1

dL
y c x h F

d=

= φ + +
τ∑       (25) 

Now, in order to get rid of yjk from the coefficients with Lk depending on the time τ . We shall 

multiply the expression (25) by Zji , where { }jiZ is a matrix of solution for conjugate system (23) we 

shall add then in j from 1 to 4.  

We have 

 
4

ik k i

k 1

A L f
=

=∑ ɺ  

where 
4

ik ji jk

j 1

A Z y
=

=∑ , do not depend on time (the property of the conjugate system) 

 ( )
4

(o) (i)

i k 1 k 1 k ki

k 1

f x h C x Z
=

= + φ +∑       (26) 

As the expression (23) is a system in variation of a cononic sytem, so for the determination of Zij we 

have the following relations 

 

 1i 3i 2i 4i 3i 1i 4i 2iZ y , Z y ,Z y ,Z y= − = − = − = −  



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.49 (Online Publication: Dec., 2013) 

 

As the coefficients Aij do not depend on time,  so for the determination we shall put 0τ = . Using the 

formula (13) for the calculation of Yij (0), we find 

 

 
ij

4
0 0 0

C

16
0 0 8

A n

0 8 0 0

4 16
0 0

nC

 
 
 
 
   =   

− 
 − −
 
 

 

 Thus we have the following system 

 

 

4 1 3 2 1

0

2 3 1 4 3

0

C 1 1
L f , L f f

4 8 2C

1 C 1
L f , L f f

8 4 2C

= = −

= − = +

ɺ ɺ

ɺ ɺ

 

 

 Integrating we get, 

 

 

2

0
1 1 4 3

0 0 0

3t C 1
L C b b

C 4C 4 2C

 τρ
= − + − + 
 

ɺ  

 0
2 1 1 3

t 1
L b C b

8 8
= τ + −  

22
(0)

3 1 2 1

0

1 1
L C b b

4 4 8 2C

 ρρ
= − + + −  
 

      (27) 

( )

4 1

1 1 3

20

2

4 4 ' 31
3 1 0 13 3 3

2 2 20 0 0

C
L b

4

1
b 16pqr 1 d

r

b

0, 0

8r1 1
b 8 p q 2 d 16 tpqr 1 d 12C r d

r r r

τ

τ τ τ

=

 
= − τ 

 

   ∂Ω ∂Ω
= − +   ∂µ ∂µ    τ = µ =

     
= − − + + τ + − τ + τ    

     

∫

∫ ∫ ∫

 

( ) 1 2
4 0

0

db dbt
b tan 2 C 2 / n d

n d d

τ   = τ −φ − + τ τ  τ τ  
∫     (28) 

 
In general, solution of the system (19) 

 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.50 (Online Publication: Dec., 2013) 

 

 ( ) ( )
4

(1)

k i i ki

i 1

x L a y , k 1,.....4
=

= + =∑       (29) 

There will enter six constraints h,C,ai (i=1,……4) (ai=constants of integration) which must be chosen 

in such a manner that the following relations are satisfied 

(1) (1) (1)

2 3 0 4

0

d k
x (0) x (0) x (S*) 0,S*

d 4 C
µ=

 Ω π
= = = = = µ 

 , if (k+m) is odd (30) 

and 
(1) (1) (1) (1)

2 3 1 4
0

d k
x (0) x (0) x (S*) x (S*) 0,S*

d 4 Cµ=

 Ω π
= = = = = = µ 

, if k+m is even (31) 

 

   

From the relations 

 

 
(1) (1)

2 3x (0) x (0) 0= =  

we find immediately that a1=a2=0 

  

we shall write the equations (30) and (31) in the following manner 

 

 

(1)

i i1 1 i2 1 i3 3 i4 4 i

(1)

j j1 1 j2 1 j3 3 j4 4 j

x (s*) F* h F* C F* a F* a F* 0

x (s*) F* h F* C F* a F* a F* 0

= + + + + =

= + + + + =
   (32) 

 

 
1 3

00 0

d 4 16 d
C a

d C n d
µ= τ=

   Ω Ω
= + +   µ µ   

 

 

Where 

 
i 1, j 4, k m isodd,

i 2, j 3, k m is even

= = +

= = +
 

 

 

k1 k2 k3 k3 k4 k4

2 2 2

0 0
k2 k1 k2 k3

0 0

F y , F y , F y

3t t (0)
F y y y

C 4C 8 4 4

= τ = =

   τρ ρ ρ
= − + +   

  

  

0

k 4 3 k1 3 k2 2 1 k3 1 k4

0 0

CC 1 1 1 1
F b b y b y b b y b y

4 2C 8 8 2C 4

   
= − + − + + +       

 (33) 

where h1 are defined by the formula (28). 

Thus we have three equations for the determination of four constants (h1, c1, a3, a4). One of them is 

chosen arbitrary when k is even m is odd, 0 0, / 2φ = ω = ± π  one may require in order that in 

perturbed as well as in unperturbed motion the period in independent variable t may coincide; then to 

these equations we shall supplement another. 

  ( )
0

k
t *, m 0, *

4 C

π
τ µ − π = τ =        (34) 

where t may be put in the form of the series inµ . 

 

 



 

 

M.R. Hassan and  R.R.Thapa / BIBECHANA 10 (2014) 44-51: BMHSS, p.51 (Online Publication: Dec., 2013) 

 

 
i

0 i i

i 1

t t t , t (0) 0
∞

=

= + µ =∑        (35) 

 

Let us pass in the expression (6) to new independent variation 

 
S*

S
S*

τ =
+α

 

  

 ( )2 2

1 2

dt S*
4 x x

d S*

+α
= +

τ
       (36) 

And we shall substitute expression (35) in (36) then for the determination of t1, we have the following 

equations. 

 ( ) ( )2 2
(0) (0) (0) (1) (0) (1)

1 2 1 1 1 1 1

dt
4 x x h 8 x x x x

d
= + + +

τ
 

Integrating we obtain 

 1 51 1 52 1 53 3 54 4 5t F h F C F a F a F= + + + + . 

 

where 

 

( )
( )

2 2

51 0 0

2 2 2
0 0 0 00

52

0 0 0

F 4 p q

t p q t 4 C sin 2 C 4sin 2 C
F

52 C C 3C

= +

+
= + − +

 

( )2 2

0 0 0
53 0 54

8 p q 2t
F 2t , F

n n

+
= = +       (37) 

( ) ( )2 2

5 2 4 3

0

2
F L L pp qq L p q d

n

τ
 = + + + τ 
 ∫ ɺ ɺ  

Then to the equation (32) when k is even, m is odd, 0, / 2,φ = ω = ±π one may add also another 

equation. 

 ( ) * * * * *

1 51 1 52 1 53 3 54 4 5t * F h F C F a F a F 0τ = + + + + =      (38) 

This is the required equation for calculating the first order perturbation in centrifugal forces. 

 

Acknowledgements 
 
We are thankful to Prof. Dr. Mrityunjay Jha, Prof. and Head, University Department of Mathematics, 

T.M.B., University; Bhagalpur, India for giving us encouragement and his valuable suggestions. 

 

References 
 

1. H. Poincare,  Les Mathod nouvels dela mecanique cilest., Paris, 1892. 
2. I.V. Kurcheeva,  Bulletin, Inst. Of Theo. Astro. (ITA) (1968)149. 

3. A. Ahmad, Ph.D. thesis, Bhagalpur University, Bhagalpur, 1981. 

4. A. Ahmad, and A.A. Khan, JBMS, 10(86) 199-128. 

5. K.B. Bhatnagar, and P.P. Hallen,  Celestial Mechanics 18 (1978) 105. 

6. T. Levi-Civita, Periodic orbits, Washington, 1906. 

7. G.A. Krasinski, Bulletin, Inst. of Theo. Astro. (ITA) 9 (1963) 105.