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Proceedings of the Pakistan Academy of Sciences 48 (2): 117–126, 2011Copyright © Pakistan Academy of SciencesISSN: 0377 - 2969Pakistan Academy of SciencesSitnikov Problem: It’s Extension to Four Body ProblemMd. Sanam Suraj 1* and M. R. Hassan 21 Department of Mathematics, Tilka Manjhi Bhagalpur University, Bhagalpur,Bihar-812007, India2 Department of Mathematics, S.M College Bhagalpur, Bihar, IndiaOriginal ArticleAbstract: An analytical expression for the position of the infinitesimal body in the elliptic Sitnikovrestricted four body problem is presented. This solution is valid for small bounded oscillations in caseof moderate eccentricity of primaries. We have linearized the equation of motion to obtain the Hill’stype equation. Using the Courant and Snyder transformation, Hill’s equation is transformed intoHarmonic oscillator type equation. We have used the Lindstedt - poincare perturbation method andagain we have applied the Courant and Snyder transformation to obtain the final result.Keywords: Sitnikov problem, four body problem, Lindstedt- poincare method, perturbation theory,analytical solution.1. INTRODUCTIONThe Sitnikov problem is a special case of therestricted three body problem where the twoprimaries of equal masses (m 1 = m 2 = m=1/2) aremoving in circular or elliptic orbits around thecentre of mass under Newtonian force ofattraction and the third body of mass m 3 (themass of the third body is much less than themass of the primaries ) moves along the linewhich is passing through the centre of mass ofthe primaries and is perpendicular to the plane ofmotion of the primaries.This dynamical model was first described byPavanini [1].The circular problem was discussedin detail by MacMillan [2] where he showed theintegrability of the equation of motion with theaid of elliptic integrals which has beenrediscussed by Stumpff [3]. Sitnikov [4] studiedthe existence of oscillating motion of the threebody problem. Sitnikov problem has been studiedby many scientists. Perdios et al [5] have studiedstability and bifurcation of Sitnikov motion. Liu& Sun [6] have studied the mapping instead of theoriginal differential equation and discovered thatthere exist a hyperbolic invariant set. Hagel [7]has studied the problem by a new analyticalapproach. It is valid for bounded small amplitudesolution and small eccentricities of the primaries.Belbruno E., Llibre J. and Olle M. [8] havestudied the family of periodic orbits which−−−−−−−−−−−−−−−−−−−−−Received October 2010, Accepted June 2011*Corresponding author: Md. Sanam Suraj; Email: mdsanamsuraj@gmail.combifurcate from the circular Sitnikov motion. Jalali& Pourtakdoust [9] have studied the regular andchaotic solutions of the Sitnikov problem near the3/2 commensurability. Chasley [10] have studiedthe global analysis of the generalized Sitnikovproblem. Faraque [11] has established the newanalytical expression for the position of theinfinitesimal body in the elliptic Sitnikovproblem. His solution is valid for small boundedoscillation in case of moderate eccentricities ofthe primaries. Hagel [12] has studied “A highorder perturbation analysis of the Sitnikovproblem”. Perdios [13] has studied the manifoldof families of three dimensional periodic orbitsand bifurcation in the Sitnikov four bodyproblem. Soulis et al [14] have studied the“Stabality of motion in the Sitnikov problem”.Soulis et al [15] has studied the periodic orbitsand bifurcation in the Sitnikov four-bodyproblem. Hagel [16] has studied an analyticalapproach to small amplitude solutions of theextended nearly circular Sitnikov problem.Boutis & Papadakis [17] have studied “TheStability of vertical motion in the N-body circularSitnikov problem. Kovacs et al [18] studied therelativistic effects in the chaotic Sitnikovproblem.In the present paper we have extended thestudy of Sitnikov problem to four body problemin elliptic case. We have considered theprimaries moving in elliptic orbits of eccentricity