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Kolmogorov-Arnold-Moser theorem

The Kolmogorov-Arnold-Moser theorem is a theorem in non-linear dynamics that solves the small-divisor problem in classical perturbation theory.

The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting periodic orbit. The solution to this problem was given by Andrey Nikolaevich Kolmogorov in 1954. This was extended and proved by Vladimir Arnold (1963 for Hamiltonian systems) and Moser (1962 for Twist maps), and the general answer is known as the KAM theorem. The KAM theorem can be applied to astronomical three-body problems.

The KAM theorem is usually stated in terms of a trajectory in phase space of an integrable Hamiltonian system that is confined to a doughnut shaped surface, an invariant torus. If the system is subjected to a weak nonlinear perturbation, this invariant torus is deformed but not destroyed. This implies that the motion continues to be periodic, with the independent periods changed. The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true, and establishes the sufficient condition for the motion of a nonlinear system to be regular. Most importantly, it implies that the motion remains multiply periodic for an arbitrarily long period.

However, the nonresonant condition of the KAM theorem becomes increasingly difficult to satisfy for systems of more degrees of freedom. The KAM theorem also implies that in certain special circumstances the invariant tori will be destroyed and the orbit may become chaotic or wander off to infinity. A destruction of invariant tori will generally occur when there are resonances in the perturbed system.