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.

No abstract

We consider the stability and instability of an equilibrium point of a Hamiltonian system of two degrees of freedom in certain resonance cases. We also consider the stability or instability of a fixed point of an area-preserving mapping in certain resonance cases. The stability criteria are established by Moser's invariant curve theorem and the instability is established by Chetaev's theorem.

We give some results about the dynamics of a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. Our main results concern (1) singularities and (2) the dynamics in the plane that contains the circle. The study presented here is purely analytic.

We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude θ and of the number of vortices N. By using the integrals of motion, we view the system in a suitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the θ-intervals for which the Hessian of the Hamiltonian-evaluated at the equilibrium configuration-is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by