1984


G. Gallavotti

Quasi-integrable mechanical systems

In Ph\'enom\`enes critiques, syst\`emes al\'eatoires, th\'eories de jauge, Lectures at the XLIII summer school in Les Houches, 1984, Ed. K. Osterwalder, R. Stora, Elseviers, p. 541-624.



Abstract KAM theory and renormalization group

(1) Basic definitions on integrability and canonical integrability. Examples.
(2) Canonical integrability and the Arnold-Liouville theorem.
(3) Classical perturbation theory
(4) Birkhoff theorems on harmonic oscillators.
(5) Some applications of perturbation theory. The precession of Mercury. Poincar\'e's triviality theorems.
(6) Phase space diffusion: bounds on the time scales of Arnold's diffusion. Nekhorossev theorem.
(7)Resonances and chaos
(8)Existence of non resonat invariant tori and quasi periodic motions. The Kolmogorov-Arnold-Moser theorem
(9)Concluding remarks.
References

Keyword KAM, Nekhorossev theorem, Renormalization group, Classical Mechanics, Birkhoff normal form, Poincar\'e triviality, Quasi periodic motions, Homoclinic splitting, Block waves