Bifurcation of spatial central configurations from planar ones. (English) Zbl 0830.34032

Some highly symmetrical spatial central configurations of the Newtonian \(N\)-body problem are considered. It is shown that spatial central configurations appear through bifurcation from planar ones as the masses are varied. As a matter of fact the central configurations are important for celestial mechanics but they are studied in the article for their own sake. Central configurations are the rest points of a certain gradient flow. It turns out that the problem of finding the central configurations is essentially that of finding the rest points of the gradient flow \(U/S\) (\(U\)-potential function of \(N\)-body problem, \(S\)-unit sphere) or, alternatively, of finding the critical points of \(U/S\). The gradient flow preserves some submanifolds of \(S\); the set of all collinear configurations and the set of all planar configurations are invariant, for example. In addition, some sets of symmetrical configurations are preserved. One of these is the main object of study in the article.


34C23 Bifurcation theory for ordinary differential equations
70F10 \(n\)-body problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
85A05 Galactic and stellar dynamics
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