Symmetries of the \(N\)-body problem, regularization of central singular solutions
type: Item
instance of: scholarly article
title: Symmetries of the \(N\)-body problem, regularization of central singular solutions
zbMATH Open document ID: 0818.70012
DOI: 10.1007/BF00692995
author: Edelman Colette
published in: Celestial Mechanics and Dynamical Astronomy
publication date: 1993-08-29T00:00:00Z
review text: This paper is devoted to the study of symmetries of the two-body problem and to the regularization of its central singular solutions. The extension to the case of \(N\geq 3\) bodies is also discussed. In the two-body case, the author studies two families of symmetries, \(S_ 1\) and \(S_ 2\), related to linear homographic transformations of the Lagrangian of the system; these correspond respectively to: (a) the arbitrary choice of the zero level of the potential; (b) a possible affinity of the configuration space. A symmetry of \(S_ 1\)-type may relate unbounded and bounded central solutions; a symmetry of \(S_ 2\)- type may relate rectilinear and non-rectilinear solutions. Therefore any central solution is equivalent -- via a symmetry \(S_ 1\), \(S_ 2\), \(S_ 1\cdot S_ 2\) -- to a central circular solution. In the \(N\)-body case, symmetries \(S_ 1\) and \(S_ 2\) exist whatever \(N\) would be. These symmetries extend directly to particular solutions of the \(N\)-body problems because these ones reduce to a number of synchronic two-body problems.
Mathematics Subject Classification ID: 70F10
zbMATH DE Number: 399382
zbMATH Keywords: affinity of configuration space
zbMATH Keywords: linear homographic transformations
zbMATH Keywords: Lagrangian
zbMATH Keywords: zero level of potential
zbMATH Keywords: rectilinear and non-rectilinear solutions
zbMATH Keywords: central circular solution
zbMATH Keywords: synchronic two-body problems
reviewed by: I. Bena
MaRDI profile type: Publication
cites work: Rectifications régularisantes du problème des deux corps. (Regularizing rectifications of the two-body problem)
cites work: Q4096486
cites work: Q3998801
full work available at URL: https://doi.org/10.1007/bf00692995
OpenAlex ID: W54251425
Recommended article: Symmetric central configurations of the spatial n-body problem
Recommended article: Q3424646
Recommended article: The Masses in a Symmetric Centered Solution of the n-Body Problem
Recommended article: Applying the symmetry groups to study the \(n\) body problem
Recommended article: The Lie–Poisson structure of the symmetry reduced regularizedn-body problem
Recommended article: Symmetries in the two-body problem associated to quasi-homogeneous potentials
Recommended article: Symmetric regularization, reduction and blow-up of the planar three-body problem
Recommended article: Symmetry of the restricted \(4+1\) body problem with equal masses
Recommended article: Some regularization in the N‐body problem
Recommended article: Symmetric periodic noncollision solutions forN-body-type problems
description: scientific article; zbMATH DE number 399382