Quantitative methods in classical perturbation
theory

by Antonio Giorgilli

**Incipit:**
At the beginning of the second volume of his * Méthodes nouvelles
de la Mécanique Céleste* Poincaré devoted the chapter VIII to the
problem of the reliability of the formal expansions of perturbation
theory. He proved that the series commonly used in Celestial mechanics
are typically non convergent, although their usefulness is generally
evident. In particular, he pointed out that these series could have the
same character of the Stirling's series. Recent work in perturbation
theory has enlighten this conjecture of Poincaré, bringing into
evidence that the series of perturbation theory, although non convergent
in general, furnish nevertheless valuable approximations to the true
orbits for a very large time, which in some practical cases could be
comparable with the age of the universe.

The aim of my lectures is to introduce the quantitative methods of
perturbation theory which allow to obtain such powerful results.

*From Newton to chaos: modern techniques for
understanding and coping with chaos in N--body dynamical system*, A.E. Roy e B.D. Steves eds., pp 21-38 Plenum Press, New York (1995)

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