Existence of homographic solutions in non-Newtonian dynamics

Grebenikov, E. A.; Diarova, D. M.; Zemtsova, N. I.

doi:10.1063/1.3546080

Cited by 2 publications

(16 citation statements)

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“…Depending on their initial velocities, the peripheral bodies can move along an ellipse, a parabola, or a hyperbola. Apart from such single-layer configurations, in a number of studies the matter of exact solutions for configurations involving several layers was addressed (Silushik 2003 ; Grebenikov et al 2006 ; Diarova et al 2006 ; Gutsu et al 2007 ; Grebenikov 2010 ). Traditionally, the configurations of interest were treated as systems formed by mutually embedded polygons rotating, as an entity, at angular velocity ω .…”

Section: Introductionmentioning

confidence: 99%

“…At polygon vertices, material points interacting among themselves by the Newton law of gravitation are placed. Within the context of the problem under consideration, example solutions for embedded triangles, rhombs, squares, pentagons, and hexagons were reported in a generalizing study by Grebenikov (Grebenikov 2010 ). The vertices in neighbor polygons can lie either in one radius or in radii passing through the middle points of the sides of neighbor polygons.…”

Section: Introductionmentioning

confidence: 99%

“…The above problem was solved for several layers of such polygons: for triangles, up to four layers; for squares, up to three layers, and for pentagons and hexagons, up to two layers (Grebenikov 2010 ). The largest number of the interacting bodies located at polygon vertices amounted to 12 bodies.…”

Section: Introductionmentioning

confidence: 99%

“…The largest number of the interacting bodies located at polygon vertices amounted to 12 bodies. Within the Hamiltonian dynamics, the problem can be reduced to a system of algebraic equations that can be solved by computer-algebra methods (Grebenikov 2010 ; Grebenikov et al 2002 ). Each particular problem requires special consideration, and much effort has to be spent on its solution.…”

Section: Introductionmentioning

confidence: 99%

“…Each particular problem requires special consideration, and much effort has to be spent on its solution. Earlier, such problems for interacting material points were treated as homographic-dynamics problems (Grebenikov 2010 ) or problems of planar central configurations (Saari 1980 ; Perko and Walter 1985 ; Diacu 1990 ; Xia 1991 ; Albouy 1996 ; Bang and Elmabsout 2001 ; Moeckel 1990 ; Hampton and Moeckel 2006 ; Albouy et al 2008 ; Shi and Xie 2010 ). In turn, homographic dynamics itself has emerged as a new field in space dynamics (Grebenikov 2010 ).…”

Section: Introductionmentioning

confidence: 99%

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Exact solution to the problem of N bodies forming a multi-layer rotating structure

Smulsky

2015

SpringerPlus

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Exact solutions to the problem of the Newtonian gravitational interaction of N material points moving around N2 concentric circular orbits are considered. Each circular orbit contains N3 axisymmetrically located bodies having identical masses. The structure as a whole rotates around its symmetry axis. Such structures are identical to the homographic-dynamics configurations, or planar central configurations, known from literature. Conceptually, those structures can be considered as structures formed by mutually embedded polygons with point bodies placed at polygon vortices. For structures involving less than 20 bodies, solutions were obtained using Hamiltonian-mechanics methods. In the study, the forces acting on each body in the rotating structure from the side of all other bodies were found. The differential motion equations of the bodies were reduced to a system of linear algebraic equations for the body masses. Solutions in various forms were obtained. For specifying the initial parameters and for calculating all other characteristics of the structures, a computer program RtCrcSt2.for has been developed. Structures comprising up to one million bodies have been calculated. Graphical images of obtained structures are presented, and their properties are described. Stability problems for examined structures are considered, and possible application of obtained results to celestial- and space-mechanics problems is discussed.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exact solution to the problem of N bodies forming a multi-layer rotating structure

Smulsky

2015

SpringerPlus

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Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

Belgharbi

Navarro

2023

Eur. Phys. J. Plus

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In this work, we perform a numerical exploration of the escape in the N-body ring problem in absence of a central body, for $$4\le N\le 9$$ 4 ≤ N ≤ 9 and ten values of the Jacobi constant. We show how the probability of escape per interval of time varies as a function of the Jacobi constant, finding that, for values of the Jacobi constant smaller than a certain limit value, the probability of escape from the system tends to decrease with time. However, if we consider values of the Jacobi constant larger than this limit value, the probability of escape grows with time, for times of escape smaller than 100 units of time.

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Existence of homographic solutions in non-Newtonian dynamics

Cited by 2 publications

References 0 publications

Exact solution to the problem of N bodies forming a multi-layer rotating structure

Exact solution to the problem of N bodies forming a multi-layer rotating structure

Dependence of the probability of escape on the Jacobi constant in the N-body ring problem without central body

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