NEKHOROSHEV-STABILITY OF L 4 and L 5 IN THE SPATIAL RESTRICTED PROBLEM GIANCARLO BENETCTN, FRANCESCO FASS6 and MASSIMILIANO GUZZO Universitd di Padova, Dipartimento di Matematica Pura e Applicata, Via G. Belzoni 7, 35131 Padova, Italy The Lagrangian equilateral points £4 and L$ of the restricted circular three- body problem are elliptic for all values of the reduced mass fj, below Routh's critical mass HR « .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called 'Nekhoroshev-stability': denoting by d a convenient distance from the equilibrium point, one asks whether d(0)<e => d{t) < e a for |i|<expe -6 for any small e > 0, with positive a and 6. Until recently this problem, as more gen- erally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus on \i (see e.g .Giorgilli, 1989). Our aim was instead considering all values of fi up to HR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian ex- ists and satisfies a 'quasi-convexity' assumption (Fasso et al, 1998; Guzzo et al, 1998; Niedermann, 1998). However, in the case of L4 and Z5 such assumption is not satisfied for any \i < /J,R. Therefore, our study rests in a crucial way on two extensions of the above result: (i) The first extension replaces quasi-convexity by a weakened requirement, called directional quasi-convexity (DQC), which is specific for the case of an equilibrium point, and has no analogue in the general Nekhoroshev theorem. This property consists in testing quasi-convexity not in the whole plane of fast drift, but only in its intersection with the action space of the elliptic equilibrium, that is, the 'firstoctant' where all actions are nonnegative. DQC is a natural hypothesis for the case of elliptic equilibria, which appears to play the same role as quasi-convexity in Nekhoroshev theorem, and leads to good stability estimates (e.g. 6 = 1/ra). (ii) The second extension relaxes instead quasi-convexity to a simple' steepness' condition on the 3—jet of the sixth order Birkhoff normal form, much in the line of Nekhoroshev's original theorem. For a system with three degrees of freedom it is necessary that the Birkhoff normal form can be constructed up to order eight at least, and we obtain b — 1/20; this result improves whenever it is possible to construct higher order normal forms. Precise statements of these results are given in (Bennetin et al., 1998), where one also finds the application to £4 and L5. The result of such an analysis, which 445 J. Henrard and S. Ferraz-Mello (eds.), Impact of Modern Dynamics in Astronomy, 445-446. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0252921100073097 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.83, on 04 May 2019 at 16:21:38, subject to the Cambridge Core terms