TITLE:
On the construction of the Kolmogorov normal form 
for the Trojan asteroids

AUTHORS:
Frederic Gabern^(1,3), Angel Jorba^(1) and Ugo Locatelli^(2)

(1) gabern@mat.ub.es, angel@maia.ub.es
    Departament de Matematica Aplicada i Analisi,
    Universitat de Barcelona,
    Gran Via 585, 08007 Barcelona, Spain.

(2) locatell@mat.uniroma2.it
    Dipartimento di Matematica, 
    Universit\`a degli Study di Roma ``Tor Vergata'',
    Via della Ricerca Scientifica 1, 00133 Roma, Italy.

(3) Present Address:
    Control and Dynamical Systems,
    California Institute of Technology, Mail Stop 107-81,
    1200 East California Blvd, Pasadena, CA 91125, USA.


ABSTRACT:
In this paper we focus on the stability of the Trojan asteroids for
the planar Restricted Three-Body Problem (RTBP), by extending the
usual techniques for the neighbourhood of an elliptic point to derive
results in a larger vicinity. Our approach is based on the numerical
determination of the frequencies of the asteroid and the effective
computation of the Kolmogorov normal form for the corresponding torus.
This procedure has been applied to the first 34 Trojan asteroids of
the IAU Asteroid Catalog, and it has worked successfully for 23 of
them.

The construction of this normal form allows for computer-assisted
proofs of stability. To show it, we have implemented a proof of
existence of families of invariant tori close to a given asteroid, for
a high order expansion of the Hamiltonian. This proof has been
successfully applied to three Trojan asteroids.