In 1905, Henri Poincaré published in the Transactions of the AMS a paper about geodesics of a convex surface. Poincaré motivated the study of geodesics of surfaces with positive curvature by the proximity of this problem with the three body problem. In particular, Poincaré stressed the analogy of closed geodesics of this kind of surfaces with periodic solutions of the three body problem.
The aim of this talk is to understand why Poincaré chose this way ; we will stress the links of this paper with preceding works of Poincaré, specially his qualitative theory of differential equations and his new methods of celestial mechanics.