Le séminaire a lieu le mercredi 9 juin 2004 de 14h00 à
15h00
dans la salle de réunion
du PHC (Observatoire de Nice).
Applications of
formal series in Nekhoroshev estimates of the practical stability of
the orbits
The first part of the talk is a review of the theory of normal
forms and of formal integrals of motion around an equilibrium
point in nonlinear hamiltonian systems and in symplectic mappings.
The constructions of formal integrals with a direct method, and
of normal forms with Lie series are presented. A generalization
of the Birkhoff normal form is given to cover the case when one
of the frequencies is zero. We also discuss the convergence
properties of formal series when the equilibrium point is elliptic
or hyperbolic. The second part of the talk concentrates on the
modern applications of formal series, especially in the problem
of estimating the time of practical stability of the orbits
(Nekhoroshev time) or the region of practical stability for
given time. Some examples, of interest in celestial mechanics
and in galactic dynamics, are given.