Non-linear stability around the triangular libration points

McKenzie, R.; Szebehely, V.

doi:10.1007/bf01230727

Cited by 36 publications

(27 citation statements)

References 2 publications

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“…The non-vanishing character of is necessary for technical reasons. It is used in Section 2.1 to de ne the (canonical) transformation (16). Thus, condition C is needed to guarantee the di eomorphic character of this transformation.…”

Section: Methodsmentioning

confidence: 99%

“…In this section we prove Proposition 1 (see Section 2.2), used to bound the domain of de nition of the adapted system of coordinates introduced by (16).…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…The rst changes of variables are computed taking advantage of the particularities of this concrete model, but they can be extended to similar problems. The normal form is then computed by a standard recurrent procedure (based on Lie series) up to order 16. From the normal form we can easily obtain (approximate) periodic orbits (belonging to the previously mentioned Lyapunov family) as well as invariant tori of dimensions 2 and 3 that, as has beenmentioned before, generalize the linear oscillations around the orbit.…”

Section: Introductionmentioning

confidence: 99%

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Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

Jorba

Villanueva

1998

Physica D: Nonlinear Phenomena

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In this paper we introduce a general methodology for computing (numerically) the normal form around a periodic orbit of an autonomous analytic Hamiltonian system. The process follows two steps. First, we expand the Hamiltonian in suitable coordinates around the orbit and second, we perform a standard normal form scheme, based on the Lie series method. This scheme is carried out up to some nite order and, neglecting the remainder, we obtain an accurate description of the dynamics in a (small enough) neighbourhood of the orbit. In particular, we obtain the invariant tori that generalize the elliptic directions of the periodic orbit. On the other hand, bounding the remainder one obtains lower estimates for the di usion time around the orbit.This procedure is applied to an elliptic periodic orbit of the spatial Restricted Three Body Problem. The selected orbit belongs to the Lyapunov family associated to the vertical oscillation of the equilibrium point L 5 . The mass parameter has been chosen such that L 5 is unstable but the periodicorbit is still stable. This allows to show the existence of regions of e ective stability near L 5 for values of bigger that the Routh critical value. The computations have been done using formal expansions with numerical coe cients.2

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Section: Methodsmentioning

confidence: 99%

“…In this section we prove Proposition 1 (see Section 2.2), used to bound the domain of de nition of the adapted system of coordinates introduced by (16).…”

Section: Proof Of Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

Jorba

Villanueva

1998

Physica D: Nonlinear Phenomena

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“…In figure 2, we plot the synodical (x, y) coordinates corresponding to these initial conditions for the RTBP system. It is remarkable how these projected coordinates shadow the famous 'banana' shape of the stability region found in several numerical investigations [MS81].…”

Section: Initial Conditions Of the Trojan Asteroids In The Planar Rtbpmentioning

confidence: 99%

On the construction of the Kolmogorov normal form for the Trojan asteroids

2005

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In this paper, we focus on the stability of the Trojan asteroids for the planar restricted three-body problem, by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on numerical determination of the frequencies of the asteroid and effective computation of the Kolmogorov normal form for the corresponding torus. This procedure has been applied to the first 34 Trojan asteroids of the IAU Asteroid Catalogue, and it has worked successfully for 23 of them.The construction of this normal form allows computer-assisted proofs of stability. To show this, we have implemented a proof of existence of families of invariant tori close to a given asteroid, for a high order expansion of the Hamiltonian. This proof has been successfully applied to three Trojan asteroids.PACS numbers: 05.10.−a, 45.20.Jj, 45.50.Pk, 95.10.Ce

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“…McKenzie and Szebehely (1981) introduced a new idea to test the stability of the equilateral triangular libration point L 4 . They defined the maximum velocity and maximum displacement envelopes within which the third body remains for a long time starting from the suitable initial conditions so that the third body may not cross the x-axis.…”

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confidence: 99%

Position and velocity sensitivities at the triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate body

Hassan

Antia

Bhatnagar³

2013

Astrophys Space Sci

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In this paper we have examined the stability of triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate spheroid. Here we followed the time limit and computational process of Tuckness (Celest. Mech. Dyn. Mech. 61, 1-19, 1995) on the stability criteria given by McKenzie and Szebehely (Celest. Mech. 23, 223-229, 1981). In this study it was found that in comparison to other studies the value of the critical mass μ c has been reduced due to oblateness of the bigger primary, i.e. the range of stability of the equilateral triangular libration points reduced with the increase of the oblateness parameter I and hence the order of commensurability was increased.

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Non-linear stability around the triangular libration points

Cited by 36 publications

References 2 publications

Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

Numerical computation of normal forms around some periodic orbits of the restricted three-body problem

On the construction of the Kolmogorov normal form for the Trojan asteroids

Position and velocity sensitivities at the triangular libration points in the restricted problem of three bodies when the bigger primary is an oblate body

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