A Restricted Four-body Problem for the Figure-eight Choreography

Lara, Ricardo; Bengochea, Abimael

doi:10.1134/s1560354721030023

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…where K is the matrix defined in (19). The set Fix(Φ 1x ) is given by the points (22), where q j = (q jx , q jy ), v j = (v jx , v jy ), j = 1, • • • , 5 satisfy q 2x = q 1x , q 2y = −q 1y , q 3x = −q 1x , q 3y = −q 1y , q 4x = −q 2x , q 4y = −q 2y , q 5y = 0,…”

Section: Reversing Symmetries Of the Restricted Five-body Problemmentioning

confidence: 99%

“…In this case both q 5 = (α, 0), v 5 = (0, β) and ( 18) define an initial condition of a periodic orbit of comet type in the restricted 5-body problem. For details about the numerical computation of these orbits the reader is referred to the restricted 4-body problem in [19].…”

Section: Comet Orbitsmentioning

confidence: 99%

“…The case of the so-called Sitnikov problem considers the case of 2 primaries. While the analytic study of the restricted (n + 1)-body problem for n primaries following elliptic homographic solutions has been the focus of many research papers, few results have considered the problem where the primaries describe a general periodic solution [19].…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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Section: Reversing Symmetries Of the Restricted Five-body Problemmentioning

confidence: 99%

Section: Comet Orbitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Barrera¹,

Bengochea²,

García-Azpeitia³

2020

Preprint

Self Cite

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“…In recent years, researchers have conducted extensive research on the application of motion capture. Some scholars combined motion capture technology with Chinese puppet shows and proposed technical solutions for the digitalization of puppet shows [1]. Some scholars use the motion capture system to study golfers' hip and torso movements during the swing and calculate the relationship between the torso and hip movements and the swing.…”

Section: Introductionmentioning

confidence: 99%

3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System

Zheng

2022

Applied Mathematics and Nonlinear Sciences

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Visualization software provides powerful numerical calculations for the 3D mathematical modeling process. This paper uses optical motion capture mathematical modeling technology to track and detect dancers' dance movements in segments in real-time. The research results found that the visual dance choreography system proposed in the article improves the efficiency of posture analysis and real-time analysis improves the quality of teaching. This has played an important role in the digital teaching of dance movements and the development of science.

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