M.A.A. Khan et al./ BIBECHANA 13 (2016) 18-22 : RCOST p.18 (Online Publication: Dec., 2015) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (Print), 2382-5340 (0nline) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Research Council of Science and Technology, Biratnagar, Nepal Study of the stability of the perturbed solutions of the restricted three body problem M.A.A. Khan1, M.R. Hassan2, R.R.Thapa3* 1Teacher of Maths, Muslim Higher School, Bhagalpur-812002, India 2Dept. of Mathematics, S.M. College, Bhagalpur, T.M.B. University, Bhagalpur – 812007, India 3Dept. of Mathematics, P.G. Campus, Tribhuvan University, Biratnagar, Nepal *E-mail: thaparajuram@yahoo.com Article history: Received 14 August, 2015; Accepted 04 September, 2015 DOI: http://dx.doi.org/10.3126/bibechana.v13i0.13321 Abstract In this paper we have been examined the stability of the perturbed solutions of the restricted three body problem. We have been restricted ourselves only to the first order variational equations. Our variational equations depend on the periodic solutions. Here the applications of the method of Fuchs and Floquet Proves to be complicated and hence we have been preferred Poincare's Method of determination of the characteristic exponents. With the determination of the characteristic exponents we have been abled to conclude regarding the stability of the generating solution. We have obtained that the motions are unstable in all the cases. By Poincare's implicit function theorem we have concluded that the stability would remain the same for small value of the parameter  and in all types of motion of the restricted three-body problem. ©RCOST: All rights reserved. Keywords: Characteristic equations; Characteristic exponents; Complex plane; Poincare's implicit function. 1. Introduction According to poincare [1] there are three kinds of periodic solutions of the restricted three body problem. Planer case of three body problem depends on first and second kind solutions of the three body problem. Eccentricity is reduced to zero for solution of first kind but it does not reduced to zero for solution of second kind. Poincare observed the first kind solution in details. Kurcheeva [2] considered the second kind solution. Hassan et al. [3] also studied the effect of perturbation due to coriolis and centrifugal force on the periodic solution of the collision orbits of the restricted three body problem. M.A.A. Khan et al./ BIBECHANA 13 (2016) 18-22 : RCOST p.19 (Online Publication: Dec., 2015) In this paper we have examined the stability of the perturbed solutions of the restricted three body problem. For these characteristic equations of the periodic solutions are used to study. 2. Characteristic equations of the periodic solutions The stability of a periodic solution depends on the variational system with periodic coefficients; and Poincare's characteristic equation become (Minorsky, [4]) 0 s1 S1 S1 S1 4 4 3 4 2 4 1 4 4 3 3 2 2 3 1 3 4 2 3 2 2 2 1 2 4 1 3 1 2 1 1 1                                      ……(1) where, in our case .4,3,2,1),,,,,(),,,,( 4141  icoxcsx iii oC4 K *S   Substituting corresponding values in (1), we get j i   0)1( * 44 * 41 * 14 * 11 2     SAA ASA S …….(2) Therefore, we get two values of S to be unity which implies that two characteristic exponents are zero; and 0)()( * 41 * 14 * 44 * 11 * 44 * 11 2  AAAAAASS …….(3) If ,1S  then the solutions will be stable otherwise they will be unstable. 3. Proof of stability of the solutions: Case I: ( )2/w,0o  (i) K-even, m-odd, S*= oC4 K M.A.A. Khan et al./ BIBECHANA 13 (2016) 18-22 : RCOST p.20 (Online Publication: Dec., 2015) m)1(A 2/)1mK(* 11   m)1(A 2/)1mK(* 44   o 2/)1mK(* 14 Cn2 m )1(A    0A* 41  Substituting these values in (3), we get 0m)1(ms2)1(S 221mK2/)1mK(2   m)1(S 2/)1mK(   m Let us change S by   1 1 . This maps the interior of the unit circle onto the left half of the complex plane, so that the condition /S/<1 is equivalent to Re .0 Applying this transformation, we get 1m 1m or 1m 1m      Which are positive as  m>1. Therefore, the solutions are unstable. (ii) K-odd, m-even, S* = oC4 K n m2/)1mK(* 44 * 11 )1(A,0A    o 2/)1mK(* 41C2 m2/)1mK(* 14 C2)1(A,)1(A o Substituting these values in (3), we get 0mSS 22 n m2   Whose roots are real. Applying S=   1 1 and taking +ve sign, we get 0)n/m1m()1m(2)n/m1m( 2222222    n/m1m )n/14(m1m n/m1m n/)m(m41m 22 222 22 2222222       Hence at least one value of  is positive as )m/Kn(,01)K/1(mn/m1m 222  Therefore, the solutions are unstable. And taking negative sign, we get 0) n m 1m()1m(2) n m 1m( 2222222      M.A.A. Khan et al./ BIBECHANA 13 (2016) 18-22 : RCOST p.21 (Online Publication: Dec., 2015)   n/m1m n/)mm41m 22 2222222    .01)K/1(m2  Therefore, the solutions are unstable. (iii) K- odd, m – odd, S* oC4 K 2/)mK( o * 41 2/)mK( C2 1* 14 2/)mK(* 44 * 11 )1(C2A )1(A,)1(n/2A,0A o     Substituting these values in equation (3), we get 0ns2ns2  Whose roots are real. Applying S=   1 1 , we get )}1n(n{)1n(n 01n2 22 2   Therefore, one value of  is positive and consequently the solutions are unstable. Case II: For ,........)2,1,0i(iw,2/o  i) K-even, m-odd, S*= oC4 K n pt L3A,0A,Am)1(A * 0 * 0II 41 * 41 * 14 * 44 2/)1mK(* 11   Substituting these values in equation (3), we get 0)1(m)1S2S 2/)1mK(2/)1mK(2   and hence by case I (i), the solutions are unstable. ii) K-odd, m-even, S*= oC4 K o 2/)1mK(* 41 o 2/)1mK(* 14 * 44 2/)1mK(* 11 Cm2)1(A, C2 m )1(A 0A,2/m)1(A       Substituting these values in (3), we get 0mS n m S 222    and hence by case I (ii) the solutions are unstable. iii) K-odd, m-odd, S*= oC4 K o 2/)mK(* 41 o 2/)mK(* 14 * 44 2/)mK(* 11 C2)1(A , C2 1 )1(A,0A, n 2 )1(A     Substituting these values in (3), we get nS2  2S – n = 0 M.A.A. Khan et al./ BIBECHANA 13 (2016) 18-22 : RCOST p.22 (Online Publication: Dec., 2015) And hence by case I (iii), the solutions are unstable. 4. Conclusion We have examined the stability of motion of the restricted three-body problem for the generating solutions applying direct method. We obtained that motions are unstable in all the cases. By virtue of Poincare's implicit function theorem we concluded that the stability would remain the same for small values of the parameters. References [1] Poincare, H. Los Method Nonvelles de la Machanique celeste, Gauthier-villars, Pairs, 1992. [2] I.V. Kurcheeve, Bull. Inst. of Theo. Astron. (1968)149. [3] M. A. A. Khan, M. R. Hassan, R.R. Thapa, A research Journal in Mathematics, M0 1, 2 (2015) 48-58. [4] N. Minorsky, Non-linear oscillations D. Van Nonstand Company, Inc. Toronto, New York, London, 1962.