Top PDF Application of the two variable expansion procedure to the commensurable planar restricted three-body problem

Application of the two variable expansion procedure to the commensurable planar restricted three-body problem

Application of the two variable expansion procedure to the commensurable planar restricted three-body problem

The angle w-n$ will attain a maximum o r minimum only if its derivative vanishes... In the derivation of eqs..[r]

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Existence and Stability of Equilibrium Points in the Robe’s Restricted Three Body Problem with Variable Masses

Existence and Stability of Equilibrium Points in the Robe’s Restricted Three Body Problem with Variable Masses

Gelf’gat [5] examined the restricted three-body prob- lem of variable masses in which the primary bodies move within the framework of the GMP and established the existence of five libration points (collinear & triangular) analogous to the classical libration points. Bekov [6] found two additional equilibrium points, called the co- planar points. A few recent characterizations of the GMP were examined by Gurfil and Belyanin [7] and Singh and Leke [8]. The majority of the authors have been inter- ested in the stellar applications of this problem than the solar system.
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Chaos synchronization between two identical restricted three body problem via active control and adaptive control method

Chaos synchronization between two identical restricted three body problem via active control and adaptive control method

master and slave system and they demonstrated that chaotic synchronization could be achieved by driving or replacing one of the variables of a chaotic system with a variable of another similar chaotic device. Many methods for chaos synchronization of various chaotic systems have been developed, such as non linear feedback control [2], OGY approach [3], sliding mode control [4], anti synchronization method [5], adaptive synchronization [6], active control [7] and so on. The active control methods for synchronizing the chaotic systems has been applied to many practical systems such as spatiotemporal dynamical systems (Codreanu [8]), the Rikitake two-disc dynamo-a geographical systems (Vincent 9]), Complex dynamos (Mahmoud [10]) and Hyper-chaotic and time delay systems (Israr Ahmad et al. [11]) etc. Shihua Chen and Jinhu [12] proposed a new adaptive control method for adaptive synchronization of two uncertain chaotic systems, using a speci_c uncertain uni_ed chaotic model.
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Non-Uniform Motion of the Three-Body Problem When the Primaries Are Oblate Spheroids

Non-Uniform Motion of the Three-Body Problem When the Primaries Are Oblate Spheroids

Many mathematicians have made the huge contributions to the analytical, qualitative and numerical studies of the restricted three-body problem when the primaries are moving in the circular motion around there centre of mass in the uniform motion. A detailed analysis of this problem is illustrated in the work of American mathematician Szebehely [15]. In (1975) Sharma, R. K. and Subbarao[16] have discussed the collinear equilibria and their characteristic exponents in the restricted three body problem when the primaries are oblate spheroids. The Lagrangian triangular equilibria in the planar restricted three body problem where the primaries are oblate homogeneous spheroids discussed by Arredondo, J.A. et al [17]. Khan and Shahzad [18] investigated the synchronization behavior of the two identical circular restricted three body problem influenced by radiation evolving from different initial conditions via the active control. In an another paper the Complete synchronization, anti-synchronization and hybrid synchronization of two identical parabolic restricted three body problem have been studied by Khan and Rimpi pal [19]. Arif [20] studied the complete synchronization, anti-synchronization and hybrid synchronization in the planar restricted three problem by taking into consideration the small primary is ellipsoid and bigger primary an oblate spheroid via active control technique.
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Lissajous Orbits at the Collinear Libration Points in the Restricted Three Body Problem with Oblateness

Lissajous Orbits at the Collinear Libration Points in the Restricted Three Body Problem with Oblateness

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissaj- ous orbits and the phase spaces are obtained under the effect of oblateness.

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Formulation of the Post Newtonian Equations of Motion of the Restricted Three Body Problem

Formulation of the Post Newtonian Equations of Motion of the Restricted Three Body Problem

The PN approximation may be described as a method for obtaining the motions of the system to one higher power of the small parameters  GM  r and v 2 than given by Newtonian mechanics. It is sometimes referred to as an expansion in inverse powers of the speed of light. We prefer to say that our expansion parameter is c  2 , note that geometric units will not be used, so that G  1 , and c  1 . We now proceed to find the equations of mo- tion of the relativistic three body problem in the PN for- malism, or more precisely the equation of the RTBP.
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Stability of Triangular Points of the Generalized Photogravitational Robes Restricted Three Body Problem

Stability of Triangular Points of the Generalized Photogravitational Robes Restricted Three Body Problem

The linear stability of the triangular points was studied for the Robes restricted three-body problem when the bigger primary (rigid shell) is oblate spheroid and the second primary is radiating. The critical mass obtained depends on the oblateness of the rigid shell and radiation of the second primary as well as the density parameter . The stability of the triangular points depends largely on the values of . The destabilizing tendencies of the oblateness and radiation fac- tors were enhanced when and weakened for

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Existence of Libration Points in the Photogravitational Elliptic Restricted Three Body Problem

Existence of Libration Points in the Photogravitational Elliptic Restricted Three Body Problem

In this paper we study the elliptical restricted three body problem which has not been studied much in comparison to the circular RTBP further we have both major bodies radiating and oblate. We obtain the equations of motion of the infinitesimal body moving under the influence of two major bodies which are both oblate and radiating and the major bodies move in elliptical orbits around their common centre of mass, the equations (1) & (2) represent the equations of motion of the infinitesimal body. The locations of triangular librational points L 4 & L 5 are obtained and equations (9)
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New periodic orbits in the solar sail restricted three body problem

New periodic orbits in the solar sail restricted three body problem

The linear solutions given in the previous section will only closely approx- imate the motion of the sail given in (1) for small amplitudes. For larger amplitude periodic orbits, we compute high order approximations using the method of Linstedt-Poincar´e. 5 This procedure is well known and is

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Halo Orbits in the Photo Gravitational Restricted Three Body Problem

Halo Orbits in the Photo Gravitational Restricted Three Body Problem

The circular restricted three-body problem (CRTBP) is a special case of the restricted three-body problem where bigger and smaller primaries move in cir- cular motion around their common center of mass. The five equilibrium points of CRTBP are known as the Lagrangian points where the gravitational forces due to two primaries and the centrifugal force on a spacecraft are balanced.

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Displaced non-Keplerian orbits using impulsive thrust

Displaced non-Keplerian orbits using impulsive thrust

In this paper, new families of displaced non-Keplerian orbits are generated using impulsive, rather than continuous thrust. The use of impulse thrust was considered by Nock (1984), Yashko and Hastings (1996), McInnes (1998), Hope and Trask (2003) and Spilker (2003) to generate small displacements away from a circular Keplerian orbit at linear order in the two-body problem. In this paper, the more general problem is considered with arbitrarily large displacements. The displaced non-Keplerian orbits are approximated by a sequence of individual Keplerian arcs connected by impulses. The sequence of arcs is therefore continuous in position, with discontinuities in velocity which are removed using the impulses. For frequent impulses the families of orbits approximate the continuous thrust non-Keplerian orbits found in previous studies. The conditions for displaced non-Keplerian orbits using impulse thrust can be found by linearising the dynamics of the two and three-body problems in the vicinity of an arbitrary reference point in a rotating frame of reference. While the motion of the spacecraft relative to the reference point is considered at linear order, the reference point can be selected arbitrarily so that, when viewed from an inertial frame, the resulting orbit has arbitrarily large displacements away from a Keplerian orbit. Finally, some applications are considered for displaced geostationary orbits.
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Effects of Radiating Oblate Spheroid and Triaxial Rigid Body in the Restricted Three Body Problem S. Dewangan 1, Poonam Duggad2

Effects of Radiating Oblate Spheroid and Triaxial Rigid Body in the Restricted Three Body Problem S. Dewangan 1, Poonam Duggad2

Therefore, there exist one and only one value of x in each of the above interval such that f x ( ) 0  . Further, f     2  0 , f (0) 0  and f     1  0 . Therefore, there are only three real roots of equation (12), one lying in each of the intervals

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Numerical study of the linear and nonlinear dynamics near l1 and l2 points in the earth moon system

Numerical study of the linear and nonlinear dynamics near l1 and l2 points in the earth moon system

body problem can help in understanding the dynamics of three-body interactions in the solar system. The Lagrangian points have important applications in astronautics, on and very good candidates to locate a satellite or a space station. Zero velocity curves were plotted for constant values of C. The curves were used to define areas of the Lagrange points of the Circular Restricted Three-Body Problem. The tion were linearized to find the eigenvectors and eigenvalues. We computing the eigenvalues to investigate the stability. The invariant manifold structures of the collinear libration body problem provide the framework for understanding complex In order to generate a trajectory around the dimensional nonlinear equations of motion were
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Triangular Libration Points in Elliptic Photogravitational Restricted Three Body Problem with Oblateness

Triangular Libration Points in Elliptic Photogravitational Restricted Three Body Problem with Oblateness

body problem (ER3BP), when one is oblate primary and other emit light energy. The positions of the triangularpoints are seen to shifted away from the line joining the primaries than in the classical case due to the effect of the introduction of the eccentricity, semi-major axis, radiation and oblateness factors.

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LOCATION OF LIBRATION POINTS IN THE GENERALISED PHOTOGRAVITATIONAL ELLIPTIC RESTRICTED THREE BODY PROBLEM

LOCATION OF LIBRATION POINTS IN THE GENERALISED PHOTOGRAVITATIONAL ELLIPTIC RESTRICTED THREE BODY PROBLEM

In the present paper we have studied the location of libration points triangular as well as libration points in the generalized photogravitational elliptic restricted three body problem. We have taken the bigger primary as an oblate spheroid and the smaller primary libration points contain eccentricity, oblateness and radiation factor which are different from those in classical case.

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Solar Sail Equilibrium Orbits in the Circular Restricted Three Body Problem with Oblateness

Solar Sail Equilibrium Orbits in the Circular Restricted Three Body Problem with Oblateness

last forever. In Figure 6(b), the left parts, the projections on ξ and η change like those in Case A; the projection on axis ζ changes as a sine curve and the amplitude remains about the same. The right parts are the variations of three angles, where α changes in a small scale about from 89.8 ˚ to 90˚, −90˚ ≤ γ , φ ≤ 90˚. Figure 8(a) displays a direct perturbation trajectory of the sail derived from L 1 point. The maximum acceleration of solar radiation

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Solar sail equilibria in the elliptical restricted three-body problem

Solar sail equilibria in the elliptical restricted three-body problem

This paper has investigated equilibria in the solar sail elliptical restricted three-body problem, and the following results obtained: There are no equilibrium surfaces in the three-dimensional, elliptical restricted three-body problem, but two-dimensional planar equilibrium curves do exist in the plane of the system, with the curve shapes in the non-uniformly rotating, pulsating system the same as the circular case; These equilibrium points are unstable; Although out-of-plane equilibrium points do not exist, when the sail can be controlled around approximated equilibrium points. Therefore, solar sails can still be utilized for practical applications at out-of-plane equilibria of the Sun-Earth elliptical restricted three-body problem.
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1.
													Syncronization behavior of restricted three body problem when bigger primary is an uniform circular disc

1. Syncronization behavior of restricted three body problem when bigger primary is an uniform circular disc

Being motivated by the above discussion, in this article we have discussed the complete synchronization behavior of the planar restricted three body problem when the bigger primary is an uniform circular disc evolving from deferent initial conditions using active control technique based on the Lyapunov-stability theory and Routh-Hurwitz criteria. It has been observed that the system is chaotic for some values of parameter. Hence the slave chaotic system completely traces the dynamics of the master system in the course of time. The paper is organized as follows.
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Perturbed Planar Restricted Four Body Problem with Repulsive Manev Potential

Perturbed Planar Restricted Four Body Problem with Repulsive Manev Potential

Therefore, the interest is to find a model that can maintain dynamical as- tronomy within the context of classical mechanics, as well as proffering justifica- tions for the observed phenomena as offered by the relativity theory. Such a model meets the theoretical needs of celestial mechanics (by preserving the sim- plicity and advantages of Newtonian mechanics), and can also describe accu- rately the orbits coming close to collisions. By using physical principles, the Bulgarian Physicist George Manev obtained a similar model in the twenties, and proposed an alternative substitute for the relativity theory [2] [3] [4] [5] [6]. In the corresponding central force problem with unit mass for the satellite, Manev’s potential gives A = µ and 2 2
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