Symmetries of the \(N\)-body problem, regularization of central singular solutions
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.
rectilinear and non-rectilinear solutions
scientific article; zbMATH DE number 399382
scientific article