MPEJ Volume 3, No.5, 25pp
Received: Mar 18, 1997, Revised: Jul 28, 1997, Accepted: Oct 1, 1997

Antonio Giorgilli, Ugo Locatelli
On classical series expansions for quasi-periodic motions

ABSTRACT:  We reconsider the problem of convergence of classical 
expansions in a parameter $\epsilon$ for quasiperiodic motions on
invariant tori in nearly integrable Hamiltonian systems. Using a
reformulation of the algorithm proposed by Kolmogorov, we show that if
the frequencies satisfy the nonresonance condition proposed by Bruno,
then one can construct a normal form such that the coefficient of
$\epsilon^s$ is a sum of $O(C^s)$ terms each of which is bounded by
$O(C^s)$. This allows us to produce a direct proof of the classical
$\epsilon$ expansions.  We also discuss some relations between our
expansions and the Lindstedt's ones.