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# Kolmogorov-Arnold-Moser theorem

The Kolmogorov-Arnold-Moser theorem is one of the theorems in non-linear dynamics that partly resolves 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 quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Nikolaevich Kolmogorov in 1954. This was rigorously proved and extended by Vladimir Arnold (1963 for analytic Hamiltonian systems) and Moser (1962 for smooth Twist maps ), and the general result is known as the KAM theorem. The KAM theorem can be applied to astronomical three-body problems, or, with more effort, to the more general N-body problem.

The KAM theorem is usually stated in terms of trajectories in phase space of an integrable Hamiltonian system that are 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 quasiperiodic, with the independent periods possibly changed. The KAM theorem specifies quantitatively what level of perturbation can be applied for this to be true, and establishes the sufficient conditions for the motion of a nonlinear system to be regular. Most importantly, it implies that the motion remains perpetually quasiperiodic.

However, the nonresonance and nondegeneracy conditions of the KAM theorem become increasingly difficult to satisfy for systems with more degrees of freedom. So-called "converse KAM theory" also implies that in certain special circumstances the invariant tori will be destroyed and orbits may become chaotic or wander off to infinity. Destruction of invariant tori generally occurs at the locations of resonances in the original unperturbed system.

Last updated: 10-21-2005 02:43:34
03-10-2013 05:06:04