Angel JORBA

Departament de Matematica aplicada i analisi
Universitat de Barcelona
Gran Via 585
08007 Barcelona
Espagne
angel@maia.ub.es
 

On quasiperiodic solutions close to the equilateral points of the real Earth-Moon system

Abstract : In this talk we will consider the existence of invariant tori near the equilateral libration points of the Earth-Moon system. We will use several models for this study, ranging from the simplest one (the spatial Restricted Three Body Problem, from now on RTBP) to the one provided by the JPL ephemeris (this is the model we refer to as "real system"). The RTBP model is an autonomous three degrees of freedom Hamiltonian system where the equilateral points are two elliptic equilibrium points. Then, KAM methods can be used to derive the existence of invariant tori near the points, and using standard Nekhoroshev estimates one can produce a region of effective stability around them. However, this estimated region is much smaller than the region obtained by purely numerical methods.

On the other hand, numerical simulations for the JPL model (where the geometrically defined equilateral points are no longer equilibrium solutions) show that there are no effective stability regions around these points. This is because the perturbations to the RTBP coming from the non circular motion of Earth and Moon plus the effect of Sun and the other planets is big enough to produce a change of stability. However, numerical simulations seem to show the existence of maximal dimension invariant tori at some distance from the points. This implies the existence of regions with effective stability.

During the talk we will discuss these issues using some intermediate models, trying to elucidate the relevant dynamics fro them. An important tool in the methodology presented is the numerical computation of normal forms and invariant tori. We will also present theoretical results about the existence of quasiperiodic solutions in order to give support to some of the numerical results presented.