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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