KAM theory and Celestial Mechanics Luigi Chierchia Mathematics Department Universit`a Roma Tre Largo San L. Murialdo, 1 I-00146 Roma Italy E-mail:
[email protected] January 20, 2005
Key words Hamiltonian systems. KAM theory. Celestial Mechanics. Perturbation techniques. Small divisor problems. n–body problems. Invariant tori.
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Introduction
Kolmogorov–Arnold–Moser (KAM) theory deals with the construction of quasi–periodic trajectories in nearly–integrable Hamiltonian systems and it was motivated by classical problems in Celestial Mechanics such as the n– body problem. Notwithstanding the formidable bulk of results, ideas and techniques produced by the founders of the modern theory of dynamical systems, most notably by H. Poincar´e and G.D. Birkhoff, the fundamental question about the persistence under small perturbations of invariant tori of an integrable Hamiltonian system remained completely open until 1954. In that year A.N Kolmogorov stated what is now usually referred to as the KAM Theorem (in the real–analytic setting) and gave a precise outline of its proof presenting a striking new and powerful method to overcome the so–called small divisor problem (resonances in Hamiltonian dynamics produce, in the 1
KAM theory and Celestial Mechanics
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perturbation series, divisors which may become arbitrarly small making convergence argument extremely delicate). Subsequently, KAM theory has been extended and applied to a large variety of different problems, including infinite dimensional dynamical systems and partial differential equations with Hamitonian structure. However, establishing the existence of quasi–periodic motions in the n–body problem turned out to be a longer story, which only very recently has reached a