The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means...
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The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could besolved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability. (C) 2007 Elsevier B.V. All rights reserved.
We present a new computation of the Birkhoff normal form for the Hamiltonian of the restricted three body problem near the Lagrangian librationpoints. This leads to a new proof of the Lyapunov stability of these poin...
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We present a new computation of the Birkhoff normal form for the Hamiltonian of the restricted three body problem near the Lagrangian librationpoints. This leads to a new proof of the Lyapunov stability of these points. (C) 2010 Elsevier Inc. All rights reserved.
A comparison of dynamic stability determination for a celestial body is made between the Newtonian time domain and the phase-space domain. A new method of using the line of syzygies for determination of the stability ...
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A comparison of dynamic stability determination for a celestial body is made between the Newtonian time domain and the phase-space domain. A new method of using the line of syzygies for determination of the stability of a third body moving in the region of two primary bodies is reviewed for accuracy via more proven or investigated phase-space methods such as Poincare phase space surface of section methods., Phase-space surface of section methods appear to verify that the use of the line of syzygies as a stability criteria is valid, shedding new information of the stable motion of the third body, especially in the time domain.
The non-linear stability of the triangularlibration point L-4 of the restricted three-body problem is studied under the presence of third and fourth-order resonances, when the more massive primary is a triaxial rigid...
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The non-linear stability of the triangularlibration point L-4 of the restricted three-body problem is studied under the presence of third and fourth-order resonances, when the more massive primary is a triaxial rigid body and source of radiation. In this study, Markcev's theorems are applied with the help of Moser's theorem. It is found that the stability of the triangularlibration point is Unstable in the third-order resonance case and in the fourth-order resonance case, this is stable or unstable depending on A(1) and A(2), and a SOLH-CC of radiation parameter x, where A(1), A(2) depend upon the lengths of the semi-axes of the triaxial rigid body. (C) 2003 Elsevier Ltd. All rights reserved.
The non-linear stability of the triangular libration points of the restricted three-body problem is studied under the presence of third and fourth order resonance's, when the more massive primary is an oblate sphe...
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The non-linear stability of the triangular libration points of the restricted three-body problem is studied under the presence of third and fourth order resonance's, when the more massive primary is an oblate spheroid. In this study Markeev's theorem are utilised with the help of KAM theorem. It is found that the stability of the triangular libration points are unstable in the third order resonance case and stable in the fourth order resonance case, for all the values of oblateness factor A(1).
Nonlinear stability of the triangularlibration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The met...
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Nonlinear stability of the triangularlibration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The method for proper determination of cases when stability cannot be determined by four order terms of the hamiltonian was proposed.
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