For a large class of concrete astronomical situations, the motion of celestial bodies is modelled by dynamical systems associated to a potential function ?/r + ?U (r = distance between particles, ? = real constant, ? = real small parameter, U = perturbing potential). In this paper the nonlinear stability of the relative equilibrium orbits corresponding to such a potential is being investigated using a less usual method, which combines a block diagonalization technique with the reduction procedure. The test points out certain nonlinearly stable orbits, and is inconclusive for the remaining equilibria. The latter ones are treated via linearization; all of them prove instability. The nonlinearly stable orbits remain stable under any perturbation that preserves the conserved momentum.

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Abstract. Astronomy is not only a branch of science but also an important part of the culture and civilisations of peoples. Starting with archeoastronomy to the present day, it has always contributed to a better understanding of life, of humanity. After 400 years of modern astronomy, it still addresses major problems such as: Why there is something rather than nothing? Why is nature comprehensible to humans? How is cosmos related to humanity? Do multiverses exist? Is there life on other planets? Are we alone in the universe? Does the universe have a beginning? If so, what does it mean? How did the universe originate? All these questions are a challenge for interdisciplinary and transdisciplinary investigations, for philosophers, physicists, cosmologists, mathematicians, theologians. The new insights gained by pursuing in depth these common investigations will shape the society we live in and have important consequences on the future we are creating.

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We consider the polarization of unstable type-IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large-N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S 3 geometry to that of a fuzzy S 4 .