On the exploration of the (N + 2)‐body ring problema
type: CreativeWork
description: In our current research, we have conducted a comprehensive analysis of the planar and three-dimensional (N + 2)-body ring problem, exploring various aspects such as the evolution of zero-velocity curves depending the values of the mass ratio, critical values of the Jacobi constant, and their impact in the dynamics of escape orbits in three dimensions. Our investigation into the planar (N + 2)-body ring problem with N = 3 to 100 peripheral bodies, and the presence of a central mass (β > 0), reveals key insights into the behavior of zero-velocity curves. For this purpose, we have accomplished a very accurate analysis of the different intersections of the so-called critical values of the Jacobi constant, which characterize the sets of stationary points of the system, when β is lower than the bifurcation value of the system, β∗N . Through detailed analysis, we have determined that up to seven distinct intersections of these critical values can occur, each corresponding to a specific β value, here denoted as βEi , i = 1, 2, . . . , 7. The arrangement and evolution of these intersections are influenced by the number of peripheralbodies, with significant implications for the system’s stationary points. Notably,the absence of intersections between zones A2 and D1 when N > 16 indicates that isolated islands around the central mass persist even as the zero-velocity curves open. Due to the fact that the set {z = 0, z˙ = 0} is invariant in this system, the results of this investigation can be applied to explain the motion of an infinitesimal particle that lies on the XY plane in the 3D (N +2)−body ring problem, which we can then conclude that will be subject to the same structure of the zero-velocity curves in the plane. Additionally, we have presented a numerical integration by means of recurrent power series method, adapting it to address the three-dimensional (N +2)-body ring problem. This method has proven both accurate and efficient, enabling the computation of the intrica…
abstract: In our current research, we have conducted a comprehensive analysis of the planar and three-dimensional (N + 2)-body ring problem, exploring various aspects such as the evolution of zero-velocity curves depending the values of the mass ratio, critical values of the Jacobi constant, and their impact in the dynamics of escape orbits in three dimensions. Our investigation into the planar (N + 2)-body ring problem with N = 3 to 100 peripheral bodies, and the presence of a central mass (β > 0), reveals key insights into the behavior of zero-velocity curves. For this purpose, we have accomplished a very accurate analysis of the different intersections of the so-called critical values of the Jacobi constant, which characterize the sets of stationary points of the system, when β is lower than the bifurcation value of the system, β∗N . Through detailed analysis, we have determined that up to seven distinct intersections of these critical values can occur, each corresponding to a specific β value, here denoted as βEi , i = 1, 2, . . . , 7. The arrangement and evolution of these intersections are influenced by the number of peripheralbodies, with significant implications for the system’s stationary points. Notably,the absence of intersections between zones A2 and D1 when N > 16 indicates that isolated islands around the central mass persist even as the zero-velocity curves open. Due to the fact that the set {z = 0, z˙ = 0} is invariant in this system, the results of this investigation can be applied to explain the motion of an infinitesimal particle that lies on the XY plane in the 3D (N +2)−body ring problem, which we can then conclude that will be subject to the same structure of the zero-velocity curves in the plane. Additionally, we have presented a numerical integration by means of recurrent power series method, adapting it to address the three-dimensional (N +2)-body ring problem. This method has proven both accurate and efficient, enabling the computation of the intrica…
author: Boureghda, Zahra
keywords: N-body ring problem, Zero-velocity curves, Celestial Mechanics, Basins of escape, Surface of section, Numerical integration, Probability of escape
date published: 2025
date modified: 2026-07-24T00:52:00Z
provider: University of Alicante
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Alicante