Rigorous results on the power expansions for the integrals of a Hamiltonian system near an elliptic equilibrium point

The classical problem of the direct construction of integrals for a Hamiltonian system in the neighbourhood of an elliptic equilibrium point is revisited in the light of the rigorous Nekhoroshev's like theory. It is shown how the results about stability over exponentially large times can be recovered in a simple and effective way, at least in the non-resonant case, and in fact even more conveniently than with the usual indirect method involving normalizing canonical transformations. An application is also made to the problem of the freezing of the harmonic actions in classical models On etudie la construction d'integrales premieres d'un systeme hamiltonien au voisinage d'un point fixe elliptique par une methode de type Nekhoroshev. On montre comment les resultats de stabilite sur des temps exponentiellement longs peuvent etre retrouves simplement dans les cas non resonnants, et en fait plus aisement que par la methode habituelle des transformations canoniques. On donne une application a l'invariance de l'action harmonique dans des modeles classiques

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