An integrable case in the restricted problem of three bodies

MacMillan, William D.

doi:10.1086/103918

Cited by 76 publications

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“…Shahbaz Ullah and Hassan (2014) have studied the connection between three-body configuration and four-body configuration of the Sitnikov problem when one of the masses approaches zero: circular case. By applying the techniques of Guckenheimer and Holmes (1983) they averaged the Hamiltonian in Sitnikov three-body problem and four-body problem and established time period and series solutions following MacMillan (1913). Further they confirmed the periodicity and quasi-periodicity of the solutions by Poincaré sections.…”

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confidence: 82%

“…Later on the problem was discussed in detail by MacMillan (1913) in which he showed that the exact solution can be expressed in terms of Jacobi elliptic functions. Sitnikov (1960) studied the existence of oscillating motion of the three-body problem which is a particular case of the restricted three-body problem in which two primaries with equal masses (M 1 = M 2 = M = 1/2) are moving in a circular or in an elliptic orbit around their common center of mass under the Newtonian force of attraction and the infinitesimal mass M 3 (M 3 is much less than the mass of the other two primaries) is moving along the line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries.…”

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Sitnikov cyclic configuration of N+1-body problem

Ullah

Hassan

2014

Astrophys Space Sci

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This manuscript deals with the generalisation of all previous works on series solutions and linear stability of equilibrium points of the Sitnikov problem. Following Giacaglia (1967), in Sect. 2 we have derived the equation of motion of the infinitesimal mass moving along the z-axis about which the plane of motion is rotating with unit angular velocity. In Sects. 3, 4 and 5 the series solutions of the Sitnikov problem have been developed by the method of MacMillan, Lindstedt-Poincaré and iteration of Green's function respectively. In Sect. 6 the three series solutions have been compared graphically by putting N = 2, 3, 4. In Sect. 7 the coordinates of equilibrium points have been calculated. In Sect. 8 the linear stability of equilibrium points has been examined by the method of Murray and Dermott (Solar System Dynamics, Cambridge University Press, Cambridge, 1999) and it was found that the equilibrium points are stable in Sitnikov problem.

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confidence: 82%

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confidence: 99%

Sitnikov cyclic configuration of N+1-body problem

Ullah

Hassan

2014

Astrophys Space Sci

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“…Later on the problem was discussed in detail by MacMillan (1913) in which he showed that the exact solution can be expressed in terms of Jacobi elliptic functions. Sitnikov (1960) of the three-body problem which is a particular case of the restricted three-body problem in which two primaries with equal masses (M 1 = M 2 = M = 1/2) are moving in a circular orbit or in an elliptic orbits around their common center of mass under the Newtonian force of attraction and the infinitesimal mass M 3 (M 3 is much less than the mass of the other two primaries) is moving along the line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries (a detailed description of this work can be found in Stumpff 1965).…”

Section: Introductionmentioning

confidence: 99%

Series solutions of the Sitnikov restricted N+1-body problem: elliptic case

Majda

Ullah

2015

Astrophys Space Sci

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Following Giacaglia (1967), in Sect. 2 we have developed equation of motion of the Sitnikov restricted N + 1-body problem in elliptic case. We assumed that the primaries are at the vertices of a regular N-gon so the distances of the primaries from center of mass are time depending. In Sect. 3 we have linearized the equation of motion to obtain the Hill's type equation and then find the approximate solution. In Sects. 4 and 5 the series solutions of the Sitnikov restricted N + 1-body problem have been developed by the method of Lindstedt-Poincaré and iteration of Green's function respectively. In Sect. 6 the two series solutions have been compared graphically by putting N = 2, 3 and 4 for different eccentricity.

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“…The third body of mass m 3 is much less than the masses of the primaries. Thus the potential between two bodies m 1 and m 2 is given by An integrable case in restricted problem of three bodies by imposing further restrictions on the restricted three body problem, by supposing the two finite bodies of equal masses and an infinitesimal body moving in their common axis of revolution has been studied by MacMillan [1].…”

Section: Introductionmentioning

confidence: 99%

Stability of the Sitnikov's circular restricted three body problem when the primaries are oblate spheroid

Thapa

2014

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The Sitnikov's problem is a special case of restricted three body problem if the primaries are of equal masses (m 1 = m 2 = 1/2) moving in circular orbits under Newtonian force of attraction and the third body of mass m 3 moves along the line perpendicular to plane of motion of primaries. Here oblate spheroid primaries are taken. The solution of the Sitnikov's circular restricted three body problem has been checked when the primaries are oblate spheroid. We observed that solution is depended on oblate parameter A of the primaries and independent variable t. τ = η For this the stability of non-trivial solutions with the characteristic equation is studied. The general equation of motion of the infinitesimal mass under mutual gravitational field of two oblate primaries are seen at equilibrium points. Then the stability of infinitesimal third body m 3 has been calculated.

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An integrable case in the restricted problem of three bodies

Cited by 76 publications

References 0 publications

Sitnikov cyclic configuration of N+1-body problem

Sitnikov cyclic configuration of N+1-body problem

Series solutions of the Sitnikov restricted N+1-body problem: elliptic case

Stability of the Sitnikov's circular restricted three body problem when the primaries are oblate spheroid

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