Doubly-symmetric horseshoe orbits in the general planar three-body problem

Bengochea, Abimael; Galán, J.; Pérez‐Chavela, Ernesto

doi:10.1007/s10509-013-1590-3

Cited by 8 publications

(6 citation statements)

References 23 publications

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“…3. Reversing symmetries and periodic orbits The reversing symmetries [13,15] have been used successfully for studying the periodic orbits of problems described by differential equations [1,2,11,19,20]. In the following we give a brief review of some useful results.…”

Section: Equations Of Motionmentioning

confidence: 99%

“…In this Section we demonstrate that the transformation Φ θ , to be defined, is a reversing symmetry of the planar N -body problem with N = 2n + k, n, k, ∈ N. This reversing symmetry has the restriction that 2n bodies must have the same value in their masses at pairs, we mean m 2i = m 2i−1 for i = 1, • • • , n. There is not a restriction about the masses for the other k bodies 1) . Thereafter, we determine their fixed points.…”

Section: Reversing Symmetry φ θ and Its Fixed Pointsmentioning

confidence: 99%

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A restricted four-body problem for the eight figure choreography

Lara,

Bengochea

2019

Preprint

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In this work we introduce a planar restricted four-body problem where a massless particle moves under the gravitational influence due to three bodies following the eight figure choreography, and we explore some symmetric periodic orbits of this system which turns out to be non autonomous. We demonstrate that certain families of involutions are reversing symmetries of certain N -body problem. We consider a particular case of such families of reversing symmetries and we use it to study both theoretically and numerically certain type of symmetric periodic orbits of this system. The symmetric periodic orbits (initial conditions) were determined by means of solving some boundary value problems.

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Section: Equations Of Motionmentioning

confidence: 99%

Section: Reversing Symmetry φ θ and Its Fixed Pointsmentioning

confidence: 99%

A restricted four-body problem for the eight figure choreography

Lara,

Bengochea

2019

Preprint

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“…Barrabés and Mikkola (2005) computed families of symmetric periodic horseshoe orbits both in the planar RTBP and the spatial RTBP. Bengochea et al (2013) studied the numerical continuation of the doubly symmetric horseshoe orbits in the general planar three-body problem. Fitzgerald and Ross (2022) demonstrated the phase space geometry of the transit and non-transit orbits of the bicircular problem and the elliptic RTBP by linearing the Hamiltonian differential equations about the collinear Lagrange points.…”

Section: Introductionmentioning

confidence: 99%

Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

2023

Celest Mech Dyn Astron

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We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.

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“…1). Technics of reversibility have been successfully applied for studying periodic orbits of ordinary differential equations [18]; see [4,5,12,25] for the case of the N -body problem. For more details on reversibility technics, the interested reader is referred to [19], where comet and moon orbits have been computed numerically for three primary bodies following the eight choreography.…”

Section: Introductionmentioning

confidence: 99%

Comet and moon solutions in the time-dependent restricted $(n+1)$-body problem

Barrera¹,

Bengochea²,

García-Azpeitia³

2020

Preprint

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The time-dependent restricted (n + 1)-body problem concerns the study of a massless body (satellite) under the influence of the gravitational field generated by n primary bodies following a periodic solution of the n-body problem. We prove that the satellite has periodic solutions close to the large-amplitude circular orbits of the Kepler problem (comet solutions), and in the case that the primaries are in a relative equilibrium, close to small-amplitude circular orbits near a primary body (moon solutions). The comet and moon solutions are constructed with the application of a Lyapunov-Schmidt reduction to the action functional. In addition, using reversibility technics, we compute numerically the comet and moon solutions for the case of four primaries following the super-eight choreography.

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Doubly-symmetric horseshoe orbits in the general planar three-body problem

Cited by 8 publications

References 23 publications

A restricted four-body problem for the eight figure choreography

A restricted four-body problem for the eight figure choreography

Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

Comet and moon solutions in the time-dependent restricted $(n+1)$-body problem

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