The stabilty of the lagrangian points L4 and L5: recent developments

In this paper, we show few results about the Trojan stabilty problem in the Sun-Jupiter case around the L4-L5 points. In particular as considered in "On the stability of the Trojan asteroids (Giorgilli&Skokos), we present some analytical metods from the perturbation theory to describe the stability domain over a time of the order of the age of universe. Moreover as done in the quoted work, we compare the theorical stability domain with the positions of the Trojan asteroids reported in the catalogue of 14/12/94, J.D.-1440700.5. Thus we find that only 4 asteroids are in the stabilty region achieved with the therical analisys. Therefore the stability radius is unsatisfactory and has to be improved to garantee the stability of the majority of the asteroids we've considered. In the last chapter of the paper referring to the thesis "New Normal Form Approaches Adapted To The Trojan Problem"(Paez), we illustrate some recent methods to identify the numerical surfaces describing the dynamics of the Trojan asteroids. More precisely throught the use of a “Poincarè’s surface”, we obtain the dynamics of TK7 and 1872 Helenos around the L4-L5 points related to the Sun-Earth and Sun-Jupiter case for 500 periods of the bigger bodies.