Our idea is to imitate Smale's list of problems, in a restricted domain of mathematical aspects of Celestial Mechanics. All the problems are on the n-body problem, some with different homogeneity of the potential, addressing many aspects such as central configurations, stability of relative equilibrium, singularities, integral manifolds, etc. Following Steve Smale in his list, the criteria for our selection are: (1) Simple statement. Also preferably mathematically precise, and best even with a yes or no answer. (2) Personal acquaintance with the problem, having found it not easy. (3) A belief that the question, its solution, partial results or even attempts at its solution are likely to have great importance for the development of the mathematical aspects of Celestial Mechanics.

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We consider some questions on central configurations of five bodies in space. In the first one, we get a general result of symmetry for the restricted problem of n þ 1 bodies in dimension n À 1. After that, we made the calculation of all c.c. for n ¼ 4. In our second result, we extend a theorem of symmetry due to [Albouy, A. and Libre, I.: 2002, Contemporary Math. 292, 1-16] on non-convex central configurations with 4 unit masses and an infinite central mass. We obtain similar results in the case of a big, but finite central mass. Finally, we continue the study by [Schmidt, D.S.: 1988, Contemporary Math. 81] of the bifurcations of the configuration with four unit masses located at the vertices of a equilateral tetrahedron and a variable mass at the barycenter. Using Liapunov-Schmidt reduction and a result on bifurcation equations, which appear in [Golubitsley, M., Stewart, L. and Schaeffer, D.: 1988, Singularties and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York], we show that there exist indeed seven families of central configurations close to a regular tetrahedron parameterized by the value of central mass.

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